zotero-db/storage/IKIHP7W6/.zotero-ft-cache

' /NO
.'RETURN
YWADC TECHNCAL REPORT 54.602
CIA-TYALC,UGED BY":"
1.e+- /

-'---

TFCA ...CCOo

DESTROY TO
4

flIL?E COPY

REVIEW OF SOUND PROPAGATION IN THE LOWER ATMOSPHERE

L . wpA0, 0.

NWESLEY

L. NYBORG DAVID MINTZER
BROWN, UNIVERSITY

MAY 1955

WRIIGHT AIR DEVELOPMENT CENTER

L '#P , 01 AUV ."O O'

Best A1 ,!bleCo'

NOTICE
When Government drowings, specifications, or other data amused for any purpose other than in connection with a definitely related Government procurement operation, the United States Government thereby incurs no responsibility nor any obligation whatsoever; and the fact that the Governme# may have formulated, furnished, or in any way supplied the said drawings, specifications, or other data, is not to be regarded by implication or otherwise as in any manner licensing the holder or any other person or corporation, or conveying any rights or permission to manufacture, use, or sell anypatented invention that may in any way be related thereto.
I"

VADC TECHNICAL REPORT 54- 602
REVIEW OF SOUND PROPAGATION IN THE LOWER ATMOSPHERE
Wesley L. Nybovg David Mistxer
Bvow. University
May 1955
Aero Medical L~kabooy Contract No. AF .33(616)-340)
Project No. 7212
Wrigh3t Air Development Center Air liesearcb and Development Command
United States Air Force WrigtPaersou Air Forc Base. Ohio

FOREWORD

The work herein reported was conducted by the Physics Department of Brown University, Providence, Rhode Islandunder Contract AF33 (616)-340. It was initiated under USAF Project RDO No. R-695-63 "Sonic and Mechanical Vibration Action of Air Force Personnel", continued under Project R-695-85 "Propagation
of Sound Waves Near the Earth's Surface" and finished under Task No. 71709 "Propagation of Sound Waves near the Earth's Surface" of Project No. 7212. Major Horace 0. Parrack, Mr. Wolf W. von Wittern and Dr. Henning E. von Gierke served
as project engineers for the Aeromedical Laboratory, Wright Air Development Center, during the course of this investigation.

The authors gratefully acknowledge helpful consultations

with Professors R. B. Lindsay, R. W. Morse and P. J. Westervelt

on matters contained herein. Messrs. D. Brickl, J. S. D. Y. Lee

and A. Washington provided considerable assistance in prepara-

tion of the technical material. Messrs. S. Cottrell, E. lannuccillo,

4

F. Jackson, J. Kline and L. Ray helped with numerical work and

drafting. The typing was done by Mrs. C. D. Hawes.

WADC TR 54-602

ABSTRACT
A critical review is given of available information on sound propagation through the lower atmosphere. The application is to the prediction of sound fields due to aircraft (in flight or on the ground), especially, at distances up to a few miles from the aircraft sound sources. Treatment of the prediction problem requires consideration of a number of topics including (1) absorption processes in the air, (2) boundary effects caused by the earth and (3) refraction of sound due to spatial variations in air temperature and wind. Although a fair amount of information is now available on these topics a considerable amount of research remains to be done before practical solutions will be available.
PUBLICATION REVIEW
This report has been reviewed and is approved. FOR THE COMMANDM:
Colml, WSAY (wy)
(def, As- !%dile Laborator Dlroterate of Pmemarh

WkDC 1Ti 54-602

11i

TABLE OF CONTENTS

SECTION I

Theory and Laboratory Measurements

1.1 1.2 1.2.1 1.2.2 1.2.3 1.2.4

Introduction Sound Absorption in Homogeneous Air Introduction Classical Absorption Molecular Absorption laboratory Results

* -Page 1 7 7 9 11
19

1.3

Loss Coefficients in Fog and Smoke

22

1.3.1 Introduction

22

1.3.2 Theory of Epstein and Carbart

23

1.3.3 Theory of Oswatitsch.and Wei

26

1.3.4 laboratory Measurements of Losses due to Fog

31

1.3.5 Losses in Smoke and Dust

33

1.4 Sound Propagation over a Plane Earth

33

1.4.1 Introduction

33

1.4.2 Solutions for the General Case

36

1.4.3 Solutions for Special Cases

41

1.5 Sound Propagation in a Stratified Medium

43

1.5.1 Introduction

43

1.5.2 Temperature Distribution near the Ground

44

1.5.3 Wind Velocity Distribution near the Ground

46

1.5.4 Sound Velocity as"a Function of Height

47

1.5.5 Ray Theory: Shadow Boundary

50

1.5.6 Wave Theory of Shadow Zone: Constant Velocity Gradient 56

1.5.7 Wave Theory of Shadow Zone: Constant Temperature

Gradient

61

1.5.8 Wave Theory of Shadow Zone: Logarithmic Sound

Velocity-Height Dependence

63

1.5.9 Intensity in the Normal Zone: Channelling

65

1.6 Effect of Random Temperature and Wind Inhomogeneities 66

1.6.1 Introduction

*

66

1.6.2 Averages and Correlation Functions

67

1.6.3 Temperature Variations and Fluctuations

70

1.6.4 Wind Fluctuations

73

1.6.5 Scattering of Sound by Temperature Inhomogeneities

75

*

1.6.6 i~i "

1FS.6lc.u7acttteuaritnigonsofdSueountdo

Temperature Inhomogeneities by Wind Inhomogeneities I

77 79

1.6.8 Scattering of Sound by Wind Inhomogeneities II

80

WADC TR 54-602

iv

1.7 Diffraction over a Wall 1.8 Propagation of High Amplitude Sound

Table of Contents Page, 82
91

SECTION II

Outdoor Measurements

2.1 Introduction

98

2.2 Propagation of Single-frequency Sound

100

2.2.1 Sieg (1940)

100

2.2.2 Schilling, et al (1946)

103

2.2.3 Eyring (1946)

104

2.2.4 Delsasso and leonard (1953)

107

2.3 Propagation of Aircraft Noise

117

2.3.1 Regier (1947)

117

2.3.2 Parkin and Scholes (1954)

118

2.3.3 Hayhurst (1953)

120

2.3.4 Ingard (1953)

131

" 2.*4 Acoustic Shielding by Structures

134

2.4.1 Stevens and Bolt (1954)

134

2.4.2 Hayhurst (1953)

138

2.5 Sound Transmission through Forested Areas

139

2.6 Propagation of Sound through the Ground

139

SECTION III

Applications to Practical Problems

3.1 Introduction

144

3.2 Computational Aids

145

3.2.1 The 1/R Law

145

3.2.2 The R-1 e-OR Law

147

3.2.3 Loss Coefficients for Homogeneous Air

151

3.3 Typical Propagation Problems

166

3.3.1 Propagation Vertically from Aircraft to Ground

166

3.3.2 Propagation Along the Ground

168

3.3.3 Miscellaneous Problems

175

WADC TR 54-602

v

SECTION IV

Table of Contents Page

Recommendations for Future Research

4.1 Introduction

177

4.2

Theory and Laboratory Experiments

178

4.2.1 Losses in Moist Air

178

4.2.2 Losses in Fog

178

-4

4.2.3 Ground Attenuation

179

4.2.4 Shadow Zone Problems

17'

4.2.5 Scattering by Inhomogeneities

180

4.2.6 Propagation of High Amplitude Sound

181

4.3 Experiments Under Actual or Simulated Outdoor

Conditions

182

4.3.1 Introduction

182

4.3.2 Large-scale Outdoor Experiments

183

4.3.3 Small-scale Outdoor Experiments

185

4.3.4 Model Experiments

187

Bibliography

190

Appendix I Values of Various Physical Constants

197

Appendix II Formula and Graph for Converting Units of

Humidity

199

Appendix III Computed Absorption Coefficients Based on

Weather Records at Various Geographical

I1ocations in the United States

202

Appendix IV Recent Contributions

215

WADC TR 54-602

vi

LIST OF FIGURES

Title

PaE

1.

Geometrical Variables for Describing Sound

Propagation

2

2.

Classical Pbsorption Coefficients

12

3.

Plot of Mmm x versus Frequency

15

4.

Plot of hm versus Frequency

16

5.

Plot of (amoljamax) versus (h/hm); comparison

with Knudsen Data

17

6.

Plot of (a mol/areax) versus (h/hm); comparison

with Delsasso and Leonard Data

18

7.

Comparison between Theory and Experiment for

Absorption Coefficients in Air

20

8.

Plots of Yr and Y from Epstein-Carhart Theory

25

9.

Dimensionless Loss Coefficient for Fog, from

Oswatitsch Theory

29

10.

Geometrical Relations for Source and Receiver

above a Fluid Earth and a Normal Imp-edance

Earth

35

11.

Graph of Reflection Function Ff p), I; from Ingard 39

12.

Graph of Reflection Function F( p), II; from Ingard 40

13.

Diurnal Variation of Temperature Gradients at

Porton, England; from Best

45

14.

Geometrical Relations for a Sound Ray from

Source to Receiver

49

15.

Schematic Diagram of Shadow-Zone Foiation

51

16.

Graph for Calculating Distance to Shadow Zone

Boundary

54

17.

Schematic Shadow Boundary Diagram

55

WA C T'R 54-602

i

Fi u-m

Title

List of FiguPraegse

18.

Geometrical Relations for Shadow Zone Wave

Theory

58

19.

Geometrical Parameters for Theory of Diffraction

over a Wall

83

20.

The Cornu Spiral

87

21.

Loss in Sound Level Caused by a Wall, from

Fresnel Theory

88

22.

Geometrical Parameters for Diffraction over

Wall in Presence of Vertical Gradient

90

23.

Loss Coefficients versus Frequency Out-of-Doors,

from Schilling and Co-workers

105

24.

Loss Coefficients in Open Air, from Delsasso and

Leonard: 1000 cps

113

25.

Loss Coefficiehits in Open Air, from elsasso and

Leonard: 500 cps

134

26.

Loss Coefficients in Open Air, from Delsasso and

Leonard: 250 ops

115

27.

Loss Coefficients in Open Air, from Delsasso and

Leonard: 125 cps

116

28.

Pressure Level at Ground, Due to Overhead Aircraft,

from Regier

119

29.

Loss Coefficients versus Frequency for Propagation

nearly Vertically from Aircraft to Ground,

from Parkin and Scholes

121

30.

Loss Coefficients versus Vector Wind* from

Hayhurst

124

31.

Rate of Change of Loss Coefficient with Vector

Wind, from Hayhurst

125

32.

Loss Coefficients versus Frequency for Various

Receiver Heights, from Hayhurst

126

WADCT -54-W2

viii

Figur

TiFtlieaue

at List of FiguPgres

33.

Loss Coefficients versus Frequency at Zero

!

Vector Wind, from lHayhurst

127

34. 35. 36.
37.
38.
39.
40. 41. 142.
43. 44.
45. 46
47. 48.
49.

Observed Losses in Open Air Experiments over

Two Kinds of Terrain, from Ingard

133

Losses Due to Wind, from Ingard

134

Scaled Drawing of xrrangements for Measuring

Shielding by Hangar, from Stevens and Bolt

135

Noise Reduction uzied by Hangar, from Stevens.

and Bolt

136

Loss Coeffl:i.ents in Jungle, from Eyring

140

Chart for Calculating (./A) Loss

146

Chart for Calculating Sum of (1/R) Loss and

Exponential Loss

149

Extended Plot of a max ersus Frequency

152

Nomogram for Calculating a at 500 cps

154

Nomogram for Calculating a tol at 1000 cps

155

Nomogram for Calculating mol at 2000 cps

156

Nomogram for Calculating aerel at 4000 cps

157

Nomogram for Calculating c mo I at Arbitrary

Frequency

159

Nomogram for Calculating a mol at Arbitrary

Frequency, from Pielemeier

161

Nomogram for Determining Laboratory a for Air

at 220 C (71.2 0 F); Data from Delseasso and

Leonard

164

Nomogram for Determining Laboratory a for Air
at 20C (35.6 0 F) and 350C(94.50 F); Data from

Delsao and Leonard

165

WADC TR 54-602

!iX

..4- 7i6

Fiae
50. 51. 52.

Title

List of Figures Page

Sound Level over Impedance Boundary

171

Chart for Converting Units of Humidity

201

Map of U. S. Showing Weather Stations

204

6

WAC

5460.

Table 1.

LIST OF TABLES Title Classical Absorption Coefficients

Page i0

2.

Temperature-dependent Quantities in Oswatitsch

4

Fog Theory

28

3.

Phase Velocity w t, from Oswatitsch Fog Theory

30

4.

Drop Size Distributions in an Artificial Fog,

from Knudsen, Wilson and Anderson

31

5.

Loss Coefficients in an Artificial Fog, from

Knudsen, Wilson and Anderson

32

6.

Measured and Calculated Loss Coefficients in

Fog, from Wei

32

7.

Typical Values of a in Temperature Profile

Equation

44

S8.

Typical Values of I and (u*/u2Oo) inWind

Profile Equation

46

9.

Values of Am in Shadow Zone Theory, from Ingard

and Pridmore-Brown.

60

10.

Diurnal Temperature Variations at Various Heights

above the Ground, from Sutton

71

11.

Coefficients and Phase Angles for First Two Terms

of a Fourier Series representing Diurnal

Variations, from Sutton

72

12.

The Freevel Integrals I and T

84

13.

Attenuation of high Amplitude Sound

97

14.

Loss Coefficients in the Out-of-Doors, from Sieg 100

15.

Io4s Coefficients over Various Kins of Terrain,

frm Eyring

106

16.

Loss Coefficients at High Altitudes, from Delssaso

and Leonard: 1000 cps

108

WADC mR 54-602

xi

Table 17.

I

18.

19. 41tDelsasso

1

20.

21.

22.

23.

24.

X

25.

26.

Title

List of Tables &

Loss Coefficients at High Altitudes, from

Delsasso and Leonard: 500 cps

109

Loss Coefficients at High Altitudes, from

Delsasso and Leonard: 250 CPS

110

Loss Coefficients at High Altitudes, from

and Leonard: 125 cps

Iii

Loss Coefficients over a Concrete Runway,

from Hayhurst

123

Computed Distance to Shadow Zone Boundary in

Hayhurst Experiment

131

Inverse First Power (1A) Loss for Values of (/R o )

from 1.0 to 9.9

148

Computed Loss Coefficients for Fogs

176

Values of Physical Constants

197

Code Numbers for Various Weather Stations

205

Absorption Coefficients Computed from Weather

Data

207

WADC TR 54-602

xii

INTRODUCTION

Airports and their immediate vicinities are becoming more and more subject to intense noise as the tendency to use ever more powerful aircraft continues. Because of increasing human reaction to this situation, the neighborhood aircraft noise problem presents itself as a most serious one. It is ev-ident that in the design and operation of an airport acoustical planning must henceforth play a highly important role.

"

to

be

abFleortosupchrepdlicant nwinhgat.tosobuendelfefveectlsivwe iiltl

is, of exist,

course, necessary under given condi-

tions, at various points on and in the neigh) orhood of an airport. To

do this one must, in the first place, have information on the aircraft

sound sources that will be used. That is, one must know what noise

levels exist in the ne

of the source, both when operated in the

open (either on the ground or in flight) and when modified by enclosures or other shielding structures.

One would then hope to use these near field results as a basis on which to calculate noise levels at large distances from the source, i.e., in the far field. It is, of course, obvious that to do this one must know how the sound field changes with distance. A study of these changes constitutes the subject of proa~ation of sound through the atmosphere and over the ground. An investigation of sound propagation problems must include a variety of topics for consideration including (a) absorption processes in the air, (b) viscous dissipation and condensation phenomena if fog is present, (c) effects caused by the earth as both an absorbing and a reflecting boundary, and (4) refraction of sound due to spatial variations in air temperature and wind.
It is these problems of sound propagation with which this report is concerned. The principal aim of the report is to give a review of the present state of knowledge of atmospheric acoustics, especlally as related to aircraft noise propagation. The attempt has been made to include all suitable material available on this subject, whether in the form of work published in the scientific journals, or in the form of technical reports, or in some cases, in the form of private memoranda.
All such material comes ultimately from either of three kinds of activity, namely, from theory, from laboratory exeriments, or from
measurements made out-of-doors. In this report each of these three sources makes its contribution, as discussed briefly below:

(1) There is a rather extensive amount of theoU available on special topics related to atmospheric acoustics.

WADC TR 54-602

Xiii

,'ferent

I

/ ,

In some cases the results are fairly directly applicable to actual out-of-doors situations. In others the thebry in its present form is for conditions too idealized to apply in the field. The latter kind of theory can be very useful, however, in suggesting which parameters are likely to be the important ones, and for use in estimating the order-of-magnitude of effects.
(2) Further information on particular aspects of atmospheric acoustics problems comes from the results of laboratory exPeriments. Some of these results appear to be rather directly applicable to certain field situations, but for most of the experiments this is not true, the conditions being quite difthan those obtaining in typical aircraft noise problems. The latter experiments are nevertheless of great importance. Theories can often be tested with comparative ease in the laboratory, where parameters are more readily varied and controlled than in out-of-doors. When theories have been exmined critically by means of laboratory tests they can usually be applied to field problems with more confidence and with better judgement.
(3) Finally, there is now available a fairr amount of acoustical data obtained from measurements made outof-doors. Some of these data were taken by using essentially single frequency sound generated by loudspeakers, etc; others were obtained by using noise from actual aircraft. It will be realized that, in general, it is difficult to separate the effect of different parameters in out-of-door measurements since, e.g., the weather is obviously not at the control of the experimenter. Nevertheless, in some cases the experiment was so designed and the conditions so specified that the effect of various parameters could be ascertained. In others only general or "typical" effects could be determined.
In this report Section I is a review or contributions from theox7 and laboratory measurements to our present knowledge on problems in atmospheric acoustics. In this section are presented what are felt to be the more important formulae for dealing with sound propagation in the lower atmosphere, together with related charts and tables. Ranges of applicability of the formulae are indicated, where possible; this is especially feasible when data from controlled experiments are available for domparison with the theory.

WADC T 54-602

xiv

In Section II a review is given of results from measurements made out-of-doors. Comparison is made between the results of different investigators, obtained under quite a variety of different circumstances. Also, where possible, comparison is made with the predictions of pertinent theory from Section T.
In Section III special tables and graphs a.'l p.iven for applying information reviewed in the previous sections; also, recommended procedures are described for dealing with various practical problems.
Section IV contains detailed discussion of needs for future research in the areas treated in this report. Important deficiencies in present-day knowledge are pointed out, and various methods of approach examined.
Appendices I and II contain tables of constants and a chart for converting units of humidity. Appendix III tabulates theoretical absorption coefficients for various parts of the United States. Appendix IV,. reviews briefly certain material pertinent to this report, but received too recently to be incorporated into Section II.
-5

SECTION I

THEORY AND LABORATORY MASUREMENTS

1.1

INTRODUCTION

1.1.1 Actual sound fields which exist in typical out-of-door situations are almost prohibitively difficult to describe in detail. The
atmosphere is never homogeneous - there are always variations in temperature and humidity - and it is never quiescent. Thus the medium for sound transmission is not the ideal one to which must current acoustical theory applies. In addition, the boundary conditions are often much less .simple than those used in most of currently available wave theory. Thus the terrain may be uneven, both in vegetative covering and in contour; trees and hills, as well as buildings and other man-made structures may complicate description of the lower boundary.

Hence, as would be expected, there exists at the present time no over-all theory which can be used to describe typical out-of-door sound fields with sufficient completeness. There are, however, a number of special theoretical developments which are of much interest. Each of these idealizes the total problem in order to treat some parti-
cular aspect of it, and thereby gives specific attention to certain particular parameters. By considering separately these theories for idealized cases, one can develop insight into the parts different parameters play in affecting a sound field. Also, there are a few instances where certain of the idealized theories do apply with fair accuracy to
actual 6ut-of-door situations.

The remainder of Saction I will be devoted chiefly to a discussion of the special theories and experimental results mentioned above. In order to clarify the organization of this section before
going into details, we list below the separate problems to be taken up, together with brief descriptions of them. The idealized conditions assumed in each case are stated, as are also the results from theory and/or experiment as to which parameters appear to be the most important ones.

In most of the topics to be discussed the problem Is to describe the sound field in a region of atmosphere above a flat earth. More specifically, the chosen aim is to state the sound pressure p at any point P due to a source, whose pertinent properties are assumed known, localized near another point Q. Unless otherwise stated, it will be assumed in Section I that the source is like a point source and has spherical symmetry. It is realized, however, that directional effects are very important for aircraft as noise sources and that these must finally be taken into account. Another important specialization made implicitly throughout most of Section I is that nonlinear effects are ignored, it being assumed that pressure amplitudes are small in corn-

WADC TR 54-602

i

parison to atmospheric pressure, except at points very near the source. (The situation when the pressure amplitude is not small is treated briefly in subsection 1.8.)

For describing the position of source and receiver points Q

and P with respect to the earth and each other we use the symbols de-

A

fined by the sketch in Fig. 1. Here the plane surface S represents the earth t s surface and is assumed to be-a horizontal plane. The

points 0 and P' are on the surface S and are directly below Q and P,

respectively. The source height M is z o and the receiver height ' is z. The actual distance from the source point Q to the receiver point P is R, while the horizontal component U' of this distance is r. The four above-defined quantities are related by the equation

R2 = r2 + (z- z) 2 .

(1)

As indicated on the figure, measures the angle between the wind direction and the directed line 6Pt.

P

Z

Fig. I Geometrical variables for describing sound propagation. S represents horizontal ground surface; Q and P are source and receiver points; 0 and P' are projections of Q and P on S; + measures the angle between 0' and the wind direction.

WAD TH 54-602

2

1.1.2 We now list and briefly describe the topics to be taken up in the remainder of Section I.

1.1.2.1 Sound propagation in homogeneous air

This is the simplest possible case. It is assumed that the humidity and temperature are everywhere the same, that the air is free
of particles of fog and smoke and 'hat the bounding surface presented by the earth does not affect the sound field (as if there were no bounding surfaces and the atmosphere were infirdite in extent). For typical conditions to be encountered in practical field situations, the main parameters besides the sound frequency are the air temperature, the
absolute humidity and the source-receiver distance R. The law giving the sound pressure amplitude p at any point P is assumed to be of the
form:

p = AR- 1 e-aR

(2)

where the constant A depends on the source strength and the constant a depends on air conditions.

1.1.2.2 Propagation in fog

Here the same conditions hold as in the previous subsection except that the air is assumed to hold in suspension a distribution of small spherical particles, either liquid or solid, The new parameters which prove to be important here are those describing the distribution of particle sizes, those characterizing the material composing the particles, and those, in addition to temperature and humidity, needed to specify the properties of the surrounding air. It is assumed that the sound field is of the form given by Eq. (2), so that R is the important geometrical variable.

2.1.2.3 Propagation over the ground

It is assumed here that the air is homogeneous and that the ground presents a plane unifoim surface with known acoustical properties. The air temperature and humidity are assumed relatively unimportant here. Besides the sound frequency and the geometrical quantities zo, r, and z, the important parameters are those describing the nature of the ground. The general expression for the sound pres-
sure p at any point P is rather complicated. For the special case in which the source and receiver are both very near the ground, it is found that if the ground is absorbing and r is sufficiently great the pr'essure at P is given simply by

p Br

(3)

WADO TR 54-602

3

_ 1
Jthe

where B is a constant depending particularly on the source strength, the frequency and the nature of the ground.
1.1.2.4 Propagation in a stratified medium
Here it is assumed (1) that the air is homogeneous except that the temperature and/or wind velocity varies with height and (2) that the ground surface is uniform and of known acoustical properties. The cases of special interest are those where sound shadows exist; such shadows occur when'the effective sound velocity decreases with height, so that rays from the source are bent upward. Besides
distances zo, r, and z, and the scu.id frequency, the main parameters are those which give the rate of change with height of temperature and wind velocity, and the angle # between the wind direction and the line OP (see Fig. 1). The general expression for the sound field at any point P is fairly involved. If source and receiver are at the same height (i.e., if zo and z are equal),theory for special cases indicates that a law of the form
Ce- r

holds for points inside the shadow region, where C and a are constants.
1.1.2.5 Propagation throuAh a randomly inhomogeneous atmosphere
Here the situation is considered where the wind and temperature vary in space and time, as indeed is always true in the atmosphere. However, it is assumed that in tis case (unlike that treated in the previous subsection) the time-averaged air conditions are the same everywhere. Specifically, it is assumed that (a) the time-averaged wind velocity is zero at all points, (b) the time-averaged temperature is the same at all points, and (c) each statistical index of wind and temperature fluctuations, obtained by time-averaging at a point, is the same at all points in the atmosphere.
The pressure amplitude p at any point P will vary with time in an apparently random manner. Theory for the fluctuations due to temperature variations indicate that the main parameters, besides the frequency and the distance R, are two statistical indices, one describing the mean magnitude of the temperature variations and the other the mean "grain size".
For the mean value of p at any distance R from the source there is, as yet, no adequate general theory. It has been suggested that, at least, in some cases the law may be of the form of Eq. (2), where a depends essentially on the frequency and on the same two statistical Ldices mentioned above.

WADC TR 54-602

4

1.1.2.6 Propagation over a wall
Here the classical methods for treating diffraction by a "straight edge", long known in optics, are applied, with suitable modifications, to the problem of the sound shadow cast by a long wall or building. The usual approximations of Fresnel diffraction are made. Though some consideration is given to reflections from the earth and to refraction by vertical gradients of temperature and wind it is assumed, in the main, that the atmosphere is homogeneous and
that ground effects are absent. The most important variables are the sound frequency, the distances zo, r, z, the height of the wall and its distance from source and receiver.
1.1.2.7 Propagation of high-amplitude sound waves
The problem treated here is that of sound propagation when the small-amplitude approximations of ordinary acoustics are not valid. The more exact form of the basic equations must then be considered, including nonlinear terms; solutions of these cannot be superposed as can those of the linear wave equations. A propagating sound wave, originally sinusoidal with given single frequency, will suffer distortion as it travels; harmonics are generated in such a wave at a rate which depends particularly on the source amplitude, the frequency, and the nature of the wave (e.g., whether it is plane or spherical).
1.1.3 Before proceeding with detailed discussion of the separate topics listed above in subsection 1.1.2, we pause briefly to explain certain conventions which will be followed and terminology which will be used.
In describing the'sound field for a given situation one might specify the space distribution of any of a number of quantities. such as pressure amplitude, particle velocity amplitude, etc. As in the preceding discussion we shall,throughout the report,be usually speaking of the pressure amplitude (or of some quantity proportional thereto). The reason for this choice is partly that the pressure, unlike the velocity or displacement, is a scalar quantity ard hence is comparatively easy to describe. It is also partly because both laboratory and field data are likely to be in terms of the pressure amplitude, since microphones in use tend to be essentially pressure-indicators.
One may describe any given pressure distribution by (1) stating the amplitude at some reference point P0 and (2) stating the ratio of the amplitude at any other point P to that at PO. Under the .assumption of linearity the latter ratio will be independent of the amplitude at P0 . In practice, the point P0 is often chosen near the source, so that the pressure amplitude there may be regarded as characteristic of the source and nearly unaffected by absorption or refraction in the air, and nearly

WADC TR 54-602

5

independent of the earth below. At the same time, since linearity is assumed in most of the situations to be considered here, the reference point Po, where the amplitude is to be characteristic of the source, should also be supposed sufficiently far from the source; the field at Po, and at points outward from the source relative to POO
must be weak enough to permit use of the usual acoustical approximations. (In tLe field of very powerful noise sources there may be no point P0 which is entirely satisfactory as a reference point. Thus, in such cases it may be that all points which satisfy the weak-field condition are so far from the source that the field at these' points is strongly affected by refraction in the air or by "ground effects".)

In acoustics, it is often customary to state the pressure amplitude p at any given point P by specifying a quantity, called the sound pressure level (or, simply, the level) at P, proportional to the loga-
rithm of p. Specifically, in terms of both neper and decibel (db) units we have:

Sound level in nepers ln (p/p*)

(5a)

Sound level in db = 20 logl0 (p/p*),

(5b)

where p* is an arbitrary reference amplitude.

Similarly, in stating the ratio between the amplitudes at any two points, such as P0 and P, it is convenient to specify a quantity proportional to the logarithm of the ratio. This logarithmic ratio is referred to as the loss or attenuation in sound level at P relative to that at PO, or, when appropriate, as the "loss incurred by a sound wave in traveling from Po to F", or as the transmission loss between Po and P, etc. In neper and db units, respectively, we have

Loss in nepers= ln (p/P)

(6a)

Loss in db - 20 loglO (po/p),

(6b)

whei., p is the pressure amplitude at P and po that at Po. One may convert bttween nepers and db by the following relation

(Loss in db) - 8.68 (Loss in nepers)

(7)

In the following discussion, which deals with separate problems, we shall speak of losses due to a number of different mechanisms. As an important example, if Eq. (2) holds and a is essentialty zero the loss incurred between any two points is due only to the spreading or diverence of the spherical wave. On the other hand, if a is not zero we consider Lhe loss as due to two causes. Using Eq. (2) in Eqs. (6) we may write
for the loss between P0 and P:

WAXC TH 54-602

6

LLoosss iin nepers20 -- [RR I e a(R-R(RO) o

(8a)

Loss in db =20 loglo 0 o e

0

(8b)

where R. and R are the distances from the source to Po and P, respectively. The right hand sides of each of the latter equations may be written as

Loss in nepers = in (RRo) + a (R-R o )

(9a)

Loss in db = 20 loglo (RRoI) + 8.68 a (R-R0 )

(9b)

We see that the loss consists of two parts, given by the two terms on the right hand side of either of Eqs. (9). Choosing either of these equations for our attention, the first term on the right hand side gives the loss due to spherical spreading, i.e., the loss which would occur if a were zero. The second term gives the loss associated with a . The latter loss tends to predominate at large distances from the source where the divergence loss is relatively small and the spherical wave propagates essentially like a plane wave. To facilitate discussion throughout the report we shall give distinguishing names to these two particular kinds of losses: the first we shall refer to as the divergence loss or (!/R) loss, and the second the exponential loss. From Eqs. (9) we have that the exponential loss suffered by a spherical wave in traveling radially outward from a reference point Po to
another point P is given as follows:

Exponential loss in nepers = a (R-Ro ) Exponential loss in db a* (R-Ro )

(10a) (10b)

'WherefromEq. (7), a* = 8.68 a and where (R-Ro) is the distance from Po to P. The exponential loss experienced by arP expanding spherical wave is proportional to the distance travelled. The coefficients a and a* give the loss per unit distance in units of nepers per unit distance and decibels per unit distance, respectively.

1.2

SOUND ABSORPTION IN HOMOGENEOUS AIR

1.2.1 Introduction

We treat here the idealized case of small amplitude sound propagation from a malU source in a large body of homogeneous air, re-

WADC TR.54-602

7

" A' 4We

flections from boundaries (in particular, the earth) being assumed
negligible. Under these conditions Eq. (2) holds, where the constant a depends on the temperature, pressure and molecular composi-
tion of the air. The exponential loss is in this case said to be due to absorption and a (whether in nepers or decibels per unit distance) is called the absorption coefficient. Well-known theories exist for calculating the absorption constant a . Also, experimental data taken in the laboratory under controlled conditions are available against which the theories can be checked. These data agree with theoretical expectations for some ranges of parameters. Unfortunately, however, the theory appears to be far Irom adequate for certain conditions which are very important for aircraft noise problems. Thus, for the lower audible frequencies and fairly high absolute humidities, laboratory determinations of a tend to be much in excess of present theoretical values. Still more unfortunately, adequate laboratory data do not exist at frequencies below 1000 cycles/sec. We are thus at a loss to know what values to expect for a at very low audible and,especially, at sub-audible frequencies.

give in subsection 1.2 an account of present-day knowledge about the absorption coefficient a in air under different conditions. Formulae resulting from accepted theories are presented in analytical, tabular, and graphical form; the results of laboratory experiments are
also displayed, and compared with theoretical predictions.

In Section II the results discussed here will be compared with loss coefficients measured out-of-doors.

In Section III additional tables and charts are presented for convenience in determining a for given field conditions. These computational aids are based on the findings to be discussed in the remainder of subsection 1.2.

In Appendix III average values of a , computed from the charts just mentioned, are tabulated for 80 different stations in the United States, based on average temperature and humidity data from records of
the U. S. Weather Bureau.

It is customary to regard the absorption coefficient a as boing composed of a .number of separate parts, each having a different physical origin. Thus we write

a G a-s + 0olf

(Ia)

aclass

av + ac + ad + ar ,

(lb)

where Ov, ac, Od and Or are, respectively, the absorption coefficient due to viscosity, conduction (of neat), diffusion (of w'gen and

WADC TR 54-602

8

nitrogen molecules Pmong each other), and radiation (of heat). The sum of these is here designated as aclass and is called the classical, absorption coefficient. The term a mol is due to intra-molecular causes, and is usually much larger than aclass at audible frequencies. It is sometimes referred to as the humidity loss factor because of its strong dependence on the moisture content of the air at any given frequency. We shall discuss aclass first, in subsection 1.2.2; a mol will then be taken up, in subsection 1.2.3.
1.2.2 Classical Absorption1

The classical absorption is often negligible for typical conditions in aircraft aoise propagation problems. We nevertheless shall present the main theoretical results for *class, partly because of their general interest, and partly in order that the reader may apply them t6 special problems. (For example, if the absorption at either high frequencies or low static pressures is to be considered aclass must be taken into account). The four separate terms that combine to make up
aclass, according to Eq. (llb) are given by the following expressions:

C

(12)

3 Po

dc 2 f 2 (_X-1) K

(13)

P0p

Od 71-i-1 H2(D~ 12

(4

r- H

(15)

2 cev

The symbols used in Eqs. (12 - 15) have the following meanings: iWj viscosity of air (poise) f : frequency of the sound (cycles per second) u: 2vr Po : density of air (g cm 3 ) c : velocity of sound in air (cm sea -I )
Y ; (cp/cv); ratio of specific heats for air
K: heat conductivity of air (cal ca-2 sec- rc cm-1J -1)

WADO Th 54-602

9

cp : specific heat of air at constant pressure (cal gm-l

a : molecular'constant for air (0.51)

D12 : mutual diffusion coefficient of N2 and 02 (=2 sec -1 ) H : coefficient of radiation of air 10-3 (cal sec -1 gi-1 c-1)

cv : specific heat of air at constant volume (cal gm-1(C-1)

Tabular values and empirical formulae for those constants which vary

with temperature are given in Appendix I. Using these in Eqs. (12 - 15.)

the separate components of aclass can be calculated over a range of

temperature. This has been done in preparing Table 1; here ( av + ac) ,

17

O d and ar are tabulated for temperaizures ranging from -1500 to 1000C. In converting from the units (nepers/cm) of Eqs. (12 - 15) to the units

(db/l000 ft ) of Table 1 use was made of the following conversion ratio:

(a in db/1000 ft ) = (264,500)( a in nepers/cm)

(16)

TABLE 1 Calculated Values of Classical Absorption Coefficients

(Frequency f in kc; static pressure p* in atmospheres; absorption coef-
ficients av, etc, in db/1000 ft )

Temperature

( av + ac)

Od

Or

-l500oC -100

.026 f2/p* .030 f2/p*,

.0033f2 .0035f2

.0070 .0058

-50

.033 f2/p*

.0037f2

.0051

0

.036 f2/p*

.0038f2

.0046

50

.038 f2/p*

.0040f2

.0042

100

.039 f2/p*

.004 1f2

.0039

Suppose that for definiteness we take the figure of 0.1 db/iO00 ft to be the lower limit of absorption losses which are.important in ordinary field problems. On this basis we see from Table 1 that ar is always negligible, that Ud is negligible for frequencies less than about 5 kc,

WADC TR 54-602

10

and that at ordinary static pressures the sum a v + Qc) is negligible for frequencies lesb than about 1.5 kc.

Since ar is negligible, the ciassical absorption coefficient
a class, Eq. (llb), at any given temperature can be obtained by adding av + cc) and £d at that temperature, using values given by Table 1.
Fig. 2 shows a class determined in this way, plotted against frequency for three different temperatures, the static pressure being assumed
atmospheric.

For frequencies between the abscissal limits of 1 and 10 kc

a class may be read directly from the graph. Also for frequencies out-

side the latter range the same graph may be readily used, by virtue of

Pr.!

the fact that Gclass is (omitting 4 r) proportional to the square of

the frequency. Thus suppose a given frequency f is written in the form

f = 0 n f*

(17)

where f* is a number between 1 and 10, and n is an integer. The actual

frequency f is thus lOn times the reference frequency f*, and the classi-

cal absorption coefficient for f is just 102n times that which would ob-

tain for a frequency f*. For a frequency of f* in kc the value of aclass

may be read directly from the graph in Fig. 2; the latter value is thein

only to be multiplied by 102n to yield a class for the given frequency f.

."

For example, the frequency 730 kc may be written as (7.3 x 102) kc. For

7.3 kc is 2.1

acl x 104

sdsb/i1s0020.1ftd.b/l000

ft

at

0oc.

Hence for 730 kc at OC, aclass

If the classical absorption coefficient is to be calculated at

pressures other than atmospheric, Table 1 may be used. (This may, of course, be done in any case.) As shown there av and ac vary inverse-
ly with the pressure, while a d is pressure-independent, except insofar
as the constants (other than po) appearing in Eqs. (12 - 14) vary slightly with pressure.

1.2.3 Molecular absorption2

Referring back to Eqs. (11) we now consider the second contribu-

tion to a , namely, the molecular absorption coefficient amoI . In air the absorption given by amol is due to the finite rate at which energy

is imparted to and from internal vibrations of oxygen molecules when a

disturbance, such as a sound wave, passes through air. This time for

interchange is strongly influenced by the presence of water molecules -

hence the importance of humidity in connection with this last mechanism.

Kneser's expression for amol may-be written

~mol Io

__R i

.

Cv(C + R)

f fm
2 +mf2

()

WkD TR 54-602

i. i4500.0 -

--

'

°"I
t

zI-. -. 0 0
ta

:8 "

_

_

.

.....

i ,4-
•__

C ---
0__

IA/

0
t

.0f

0I .:04e_... . .I.~.z~

03L

L2515 2 2.53 4 5 6 FREQUENCY IN KILOCYCLES

89 10

Fig. 2. Classical absorption coefficient 0.a., in air at -500' 00 and 1000 C for frequencies between 1 and 10 kc. This
graph may also be used conveniently for determination of Uclass at any frequency f outside the indicated range by carrying out the following steps (Bee accompaxiying Text): (1) Write f.- l0' f*, wher f is in ke, n is an integer
and 1lC f*-C 10; (2) Determine aclass for a frequency f* (in kce) from the
graph; (3) multiply the result of step (2) by 102n.

WADC T,f< 54-W02

12

where the symbols have the following meanings:

f : sound frequency (cps) c : phase velocity of sound (cm sec 1 ) R : molar gas constant (cal)(mole)-l(Oc)-1

Cv : total heat capacity per mole of air at constant volume (cal)(mole)-l(oC)-

fm : equal to (k/2T), where k is Kneserts rate constant

Ci : vibrational heat capacity per mole (cal)(mole)-l(Oc) -1

Kneser obtained Ci from spectroscopic data on energy levels in oxygen molecules by using the expression

Ci E2

eE/RT

(19)

RT2 (_.eEART)

r

in which T is absolute temperature and E the vibrational energy (calories

per mole) for the internal oxygen mode involved.

The quantity fm varies with humidity. According to an empirical

•

Iormula by Kneser we have

fm .01. x 103h2 ,

(20)

where f is in cycles per qecond and h is in grams per cubic meter. (Several
different kinds of units for specifying h are in common use; conversion tables and formulae are given in Appendix II.) However, as will be shown later, data by various workers are not in complete agreement and the correct relation between fm and h is not accurately known. For ease in interpretation and in application to field problemswe recast Eq. (18) in a reduced form and convert units of amol from nepers per centimeter to decibels per thousand feet, obtaining a quantity w given by

w ± M°---1 2x ..

(21a)

amax

where

Umax 264,500 f 2c Cv(Cv + R)

(21b)

WADC T 54-602

13

From Eq. (21b) we see that a ma is proportional to f. The proportionality constant depends mainly on temperature, being nearly independent of ordinary variations in pressure or humidity. Values of a max computed from Eq. (21b) and from tables in Appendix I are plotted versus
frequency for various temperatures in Section III, Fig. 41. A plot of selected values, for 200C only, is given by the straight line in Fig. 3.

The ratio w = (amol/atmax) depends only on x = (fm/f), and fm, in

turn, depends mainly on absolute humidity. When x = o, w = o; as x in-

;

creases, w rises to a maximum value of unity when x = l, then falls to

zero as x approaches infinity. The value of w for any given x is the

same as that for its reciprocal (l/x). We wish to express the ratio w

in terms of the humidity h. Let us define a frequency-dependent quan-

tity hm; the latter gives, for any frequency f, the humidity for which

amol is maximum (i.e., equal to a max) at that frequency. From Eq. (20)

J4

we have

f = 1.01 x 103 h

(22)

taking the ratio of fm to f we thus obtain

x =--/. 2.

(23)

Substituting the above into Eq. (21a) we obtain

22

2(h/hm)2

(4

1 + (h/b )4

A plot of Eq. (22) is given by the curve in Fig. 4; the function given by Eq. (24) is given by the curve in both of Figs. 5 and 6.
One may calculate aio I at 200C for a given frequency and absolute humidity h by proceeding as follows:
(i) Obtain max from Fig. 3;

(2) Obtain h from Eq. (22); form the ratio hm/h;
(3) Using the value of (hm/h) from Step 2, obtain w (amol/ amax) from Eq. (24) or from the curve of Fig. 5 or 6;
(4)Multiply the results of Steps 1 and 3 to obtain amol = w 4rax
In Section III additional graphs and other aids are given to expedite calculations of amol"

WADO TR 54-602

14

499; oootiqp uj DW

0 0 0 0 00

000

00

A

AU

0

0to0

00 o

c C; '

U.

0

r04
0 4-

0 O

00

0

0

0

sJ949wOj/qp Ul XDW a

WAC TI54-WO2

15

I. 44

+,,

2

!

Fig. 4.Soli

a0 KKnn~utdzseeln (18939°5C)(20oC)

++-+Fequenc -a

--?0

_____

o Delsosso aInnd L.ecJo..n..o.rd (22°C) - Kneser e ptrlcol formulq

' il

0

2

4

6

8

10

Frequency in k c.

';"

Fig. 4. Solid curve gives hm versus frequency from Eq. (22).

Experimental points give humidities at which peak absorption occurs in laboratory measurements, for various values of the frequency.

WADC TR 54-602

16

S0 i

hi: 0. 0

0 040 hi

0. 00
(0D0Q 0

4 1
to.4
4

OD WADC~~t

00 00

(190W ID

I

OW

HO 0R5-621

(I

O 4)

0

-

-

0)..

.

W

x00 a 00
01D1

00 000__

_

00000

-Q

in-o

*
_____

ow / lW )

10.

0*

,
In) 0
4)

00
4t
WADK

o

lo

lo

18 _____

-46o

S.-

>

I )The
4

1.2.4 Laboratory Results 3-IO
total absorption coefficient a for any specified conditions, as given by Eqs. (11), etc., would be obtained by adding the values of a class and a mol' the latter being calculated by methods described in Sections 1.2.2 and 1.2.3, respectively. These theoretical values may be compared with the results of experimental determinations of a . The laboratory value of a may be defined as the total exponential, loss (see Section 1.1.3) per unit distance. In the case of experiments where spherical wave propagation is studied, the experimentally-determined a may equally well be defined as the constant to be used in Eq. (2) in order to. fit the latter equation to the observed sound field. When plane wave propagation is used the experimental a may be defined analogously as the constant such that the sound field is fitted by the equation: p Aexp(- a x).
Measurements of a have been made in laboratory air over a wide range of conditions and with a variety of techniques. Investigators have used frequencies ranging from 1 to 2000 kc and have made determinations in the air for absolute humidities up to about 20 gm/m3 , temperatures ranging from 00 to 550C and pressures down to 0.002 atmospheres. Under some conditions the observed attenuation agrees rather closely with the a predicted by Eqd. (11). Under most conditions the former is in excess of the theoretical a ; this excess varies from a few percent up to a factor of five or more.
The situation is summarized in Fig. 7 for air at atmospheric pressure and at temperatures around 200C. In the graph, absolute humidity is plotted along the horizontal and frequency along the vertical axis. The entry at any given humidity and frequency gives the ratio of the observed to the theoretical absorption coefficient for these conditions, as found by the experimenters indicated. For example, at a humidity of 12 gm/m3 and a frequency of 21 kc Rothenberg and Pielemeier 7 meas'ued the absorption coefficient a in air and found it to be about 1.8 times the theoreti-
cal value given by ( aclass + a mol ) "
At the highest frequencies represented in Fig. 7 the absorption is due mainly to a l , and at the lowest frequencies mainiy to a tool' By extrapolating froma~gs. 2 and 3 one finds that at 200C the cross-over frequency, where a class just equals a max, is 210 kc. For all frequencies above 210 kc a class is therefore necessarily greater than amool" For lower frequencies, either term might predominate depending on the humidity; for humidity conditions usually encountered a mol is greater than a class for frequencies less than 10 kc.
In Fig. 7 we see that the observed absorption is rarely, if ever, less than the theoretical value. (Due to experimental errors the fact that a few ratios less than unity do appear may not be significant.) We

WADC TR 54-602

19

-4.

>

w

£4I

0

a

0

0.p

0

0%

zW W
\I2

0

Z W

j30r0

00

0 -

-

Iq

v

-

-

in ad m to

1

wo3 4-)
4-
x

~~~ 0

a0

10

-S313A3O1l)I

NI ADN4~fOUA

v

J

WADC TR 54-4502

20

9measurements P.,

also see that the agreement is comparatively good at the higher frequencies; for frequencies above 50 kc most of the indicated ratios are less than 1.5. Agreement is also good in the vicinity of the dashed curve; the latter gives the frequency for maximum molecular absorption as a function of humidity, from Eq. (20). Agreement is generally poor at frequencies of 10 kc or less when the humidity is greater than about 6 gn/m3 .
Because our main interest here is in frequencies below 10 kc, we shall now turn our attention to more detailed consideration of
in this range. According to theory the absorption under
these conditions should be given mainly by a mol. The most recent laboratory data for the audible range of frequencies are those of Delsasso and Leonard6. The latter present experimental plots of a versus h obtained by measuring sound decay in air at atmospheric pressure, at aix frequencies and at three different temperatures. (Data are also given for air at pressures less than atmospheric, but these will not be discussed here.) The a vs h plots exhibit peaks or tendencies toward peaks, as the theory for am9l shows they should.
The values hm of the humidity at which the peaks occur at different frequencies are plotted as open circles in Fig. 4. These b.-values tend to be appreciably higher than those given by the solid curve, the latter being plotted from Eq. (22). For comparison, the filled circles represent hm-values obtained from similar data by Knudsen3'4, and the triangles hm-values found by Kntbtzel5 to be consistent with his data. Pertinent here also, though beyond the scope of Fig. 4, are the results of Rothenberg and Pielemeier 7 . Using frequencies from 22 to 110 kc and pressures down to several cm of Hg, they found their results consistent with the assumption that hm is given by Kneser ts empirical formula, Eq. (22).

It is evident that uncertainty exists as to the humidity value for which maximum absorption occurs at any given frequency. In future work thought should be given to means of reducing this uncertainty. Present theory is of no help on this point. A basic theory for accurately predicting hmax from basic molecular considerations would require much more precise knowledge of the mechanism of molecular collisions than is now available.
The situation is otherwise with respect to the actual heights of absorption maxima. Theory for predicting amax is well developed; Eq. (21b) gives a max in terms of rather well-known thermodynamic and spectroscopic constants. Also experimental values of a max (corrected for the small contribution of a class) agree well with each other and with the theory. In Fig. 3 the solid line is plotted from Eq. (21b) for 20°C; the filled circles are from Knudsen 3,4 , while the open circles

WADC TR 54-602

21

are from Delsasso and Leonard6 ; the agreement is excellent.
Having examined the present state of knowledge regarding hm and *MAX we now consider the overall dependence of a on h. In corparing experiment with theory it will be convenient to speak of a quantity a' (a- *class) where, as before, a is the laboratory value of the absorption coefficient (i.e., a is the total exponential attenuation per unit distance). We shall also refer to a quantity wt = ( at/ Q'max) which gives the ratio of at at a given humidity and frequency to the maxim= value ( atma x ) of at at that frequency. If the laboratory value of a is just equal to the theoretical value (Eqs. (11)) the quantty a' will be just equal to emol; also'w' will then equal w and will be given by Eq. (24).
The solid curves in Figs. 5 and 6 are identical plots of
((mol/amax) versus (h/hm) , based on Eq. (24). The plotted points in Fig. 5 give w t values and are from Knudsents experimentally obtained plots of a versus h for various frequencies. Reduction of data 'forplotting at any given frequency, was accomplished by (1) dividing each a'-value by the peak value a'max for that frequency, and (2) dividing each h-value by that particular humidity value hm for which the peak occurs. The points in Fig. 6 are the data of Delsasso and Leonard, reduced in the same way. We note that the fit is fairly good near the peak, i.e., for abscissal values ranging from 0.5 to 2. However, at higher (h/hm) ratios the observed absorption exceeds that predicted by Eq. (24), frequently by factors of two or three. It is not presently known how to account for this discrepancy.
In summary, theory for amtol agrees with laboratory measurements of a' in some respects, not in others. Absorption peaks do occur whose heights are predicted rather accurately by Eq. (21b). However, it is not known with certainty at what humidity the absorption will be maximum at any given frequency. At the higher humidities absorption coefficients obtained experimeitally greatly exceed those predicted.

1.3

WSS COEFFICIENTS IN FOG AND SMOKE

1.3.1 Introduction

Information is available from both theory and experiment relative to acoustic losses due to propagation thuough fog. In our discussion of present knowledge of this subject, the idealized situation assumed is that of an infinite ocean of air, free of boundaries, in

WADC TR 54-602

22

which exists a uniform distribution of water droplets. The soiMd field is assumed to have either spherical symmetry, in which case Eq. (2) holds, or plane symmetry, in which case the pressure amplitude along the direction of propagation varies as exp(-ux). The quantity of interest in either case is the loss coefficient a
We shall suppose in the following discussion that the actually observed @, i.e., the total exponential loss per unit distance in air containing fog droplets or other suspended matter is the sum of two contributions, namely,

G = Gho m + %susp •

(25)

Here a horn represents the loss coefficient in.homogeneous air, free of liquid or solid particles; it is this contribution which was discussed in subsection 1.2. The second contribution asusp represents the additional loss per unit distance due to the suspended matter.

According to available theory for acoustic losses due to liquid
droplets we may, in turn, represent susp as due to two rather different mechanisms. One of these has to do with viscous dissipation and heat conduction which takes place near droplets (or suspended particles of any kind) in a sound field. Theory for this process has recently been made available by Epstein and Carhartll. The other loss mechanism is a relaxation process which takes place when sound-passes through air in which liquid droplets are suspended. The relaxation results from a
time lag which exists between the water vapor density in the vicinity of individual droplets and that in the surrounding air, during the cyclic pressure variations of a sound field. Theory for the attenuation due to the latter effects was given by Oswatitsch1 2 ; Wei 1 3 has recently examined the Oswatitsch theory critically and suggested modifications.

In the following subsect ion we give the Epst en-Carhart expressions for a vh, the loss coeff 1i nt due to viscosity and heat conduc-

tion. In subsection 1.3.3 the Oswatitsch-Wei results are given for the

loss coefficient ae due to relaxation effects. Certain available lab-

oratory results on transmission losses due to fog are then deucribed

*

(subsection 1.3.-4) and finally, brief consideration is given to the sub-

ject of acoustic losses in aerosols (subsection 1.3.5).

1.3.2 From the Epstein-Carhart theory we have for the loss coefficient, in db/lO00 ft.,

vh -(264500) (2*)[Y3ryj

+ 2(Y-l)nYtn

(26)

In Eq. (26) Y-9 is a function of z and YT a function of y.,where z and

WAflO Th 54-602

23

y are defined below. In Fig. (8) plots are given of Yq (z)and
YT (y) as functions of their arguments. (In the graphs x represents z for the YT plot and y for the YT plot.) All other symbols in Eq. (26) are defined below.

n = number of droplets per unit volume

radius of droplets

= density of air c,= normal velocity of sound (in homogeneous air)

9= shear viscosity coefficient for air

.= ratio of specific heats for air K = thermal conductivity coefficient for air

cp = specific heat of air at constant pressure

yz/

=

(s/w)112
(/2 V 1/2

z = 1Kc /p
O'V = Kf/p C

Eq. (26) is subject to the restrictions that the radius C of each drop-
let be small with respect to X , the wavelength of sound, and that neighboring dropleto be sufficiently far apart.
Using constants for air at 200C (see Tables in Appendix I), Eq. (26) becomes

where

Gvh 24.2nC [0.453 Y,1 (z) + 0.157 Yr (y)] Y! 4 .00 f 1 / 2

(27)

z 4.56 fl/2
In the above equation n is the nutdber of droplets per m3 of radius C
(in cm) and f is the frequency in cycles per second. In actual fogs the droplets are not unifor in size; suppose, however, the distribution can be divided into groups such that n, droplets per unit volume are approxi-

WAD. TR 54-602

24

1,

00
CM4

in

b

LIn

C

%n0

~N

(4)

4 -)

40~

t-

An

0 O

MIN

ohmo

WADC

TR 2

4-602

mately of radius C1, n2 of radius f 2 , btc. One may then determine a*vh for each of these groups separately from Eq. (26) or (27) and,

*

finally, add the group-values to obtain the resultant value of Gvh

for the distribution.

1.3.3 Theory of Oswatitsch. and modifications by Wei

*Theoretical

expressions for the loss coefficient ce' due to

evaporation processeslare rather involved. Wei ts resultsl3 are parti-

cularly so; his general expression takes into account relative motion

*

between droplets and the surrounding air and thus oontains terms in-

volwing the dimensionless quantity ( 20i/V), where , @ and U, as

before, represent the droplet radius, angular frequency and kinematic

viscosity, respectively. For typical fog droplet sizes and for frequen-

cies of the order of 100 cps or less wem find that (2/)

1

and terms involving this quantity may be dropped from Weits expression.

The remainder of Weits expression has nearly the same formal appearance as that of 0swatltsch 12 . We give below the results of Oswatitsch with

*

certain modifications by Wei.

The coefficient a e is written below in terms of a sequence of intermediate quantities which are ult-tmately expressed in terms of ordinary physical constants. Thus we have, in nepers per cm,

e AL

7__

(28)

2
'

-()

2 g'3 0 12 [1

'c

][1 2

(._k) 2
61

9t 3 0 3 +

(.4X)2

where w?, g?30, wo and wl are given in terms of new variables, as follows:

1

2

Wt

, g3

[1+Y0ls - 2f)]

c3

E

-.

(30)

930

[+,)d

r) )2 c p

~~cWD T5

2 ]k++'0X2 , -\f))

(31,

, -,WADC

T11 54-W02

26

1+ Y()k)~'

W

f

(32)

The expressions inEqs. (29-32) are given by Wei; in Oswatitsch's result the term ( f kd) in the denominators of Eqs. (30) and (32), and the term Xf in the numerator of Eq. (31), do not appear.

The quantities 4*, kf, kd and )s are defined in terms of

standard physical constants as follows:

j*

4TDnt

33

P2o_

(34)

i! :I

Ld

cp DA

(35)

xs - 2 oL (Ia)

Po p2o (dT -)

(36)

The remaining symbols have the foliowing meanings:

c : phase velocity in the limit of ifinite frequency; i.e., velocity in dry air.
w? : actual phase velocity L : latent heat of condensation for water
P20 : water vapor density c3 : specific heat of liquid water cp : specific heat of moist air at constant pressure cv : specific heat of moist air at constant volume ) : cgcv n : number of droplets per cm3
D : coefficient of dffusion of water vapor through air K : coefficient of thermal conductivity of air
To : absolute temperature

(Ld-T-:)o

treamtepeorfatuinrecrease of water vapor pressure with

WADC TR 54-602

27

Po : density of moist air in sound-free conditions

g3 0

:

rgaitvieon

of water mass in droplet form, in any volume, to the air mass in that same

volume

p2 : vapor pressure

In connection with D and K, Wei notes that for small droplets whose radius is not large compared to the mean free path of air molecules, modified values must be used for these coefficients. Thus
Oswatitsch suggests that D be replaced by a quantity equal to D divided by the factor (1 + tk /t) where t k is a constant, independent of . On the other hand, Iangmuir14 gives quite a different expression for the quantity to replace D in diffusion equations for small droplets. A theoretical result for a compensated heat conduction coefficient to be used in describing heat conduction from small droplets has been given by Howell15 . In this report we shall ignore these corrections
and assume'D and K given by their usual values (see Table 2 and Appendix I).

Calculation of ae from Eqs. (28) - (36) for a given set of conditions is facilitated by Tables 2, 3 and Fig. 9. These were prepared from charts given by Oswatitsch for his equations, which differ little from Eqs. (28) - (36). Fig. 9 shows plots of the dimensionless absorption constant (2Gew'/w t) versus the ratio (w/w ?) at two temperatures
and for two values of g30. The absorption constant has its maxdmum
value in the vicinity of (w/ w t) = 10 and falls off to zero at both large and small values of this ratio. The peak is a very broad one. For g'30 = 0.05 the quantity (2 aewl/Wt) falls to one-half its peak value about when (w/ w') 0.9 at the lower limit and when (w/ ')=250 at the upper limit. For g'30 = 0.10 the upper limit is reduced to about 110. Use of Fig. 9 requires knowledge of w' and wt . By Eqs. (29) (32) these are expressed in terms of the temperature-dependent quantities
).f, )s, )d and (1 - wo2/c2 ), values of which are given in Table 2 for temperatures fiom -l0O to 300C.
TABLE 2
Temperature-Dependent Quantities in Oswatitsch Theory

T(OC)

D

-10

.195

-5

.202

0

.209

5

.216

10

.223

15

.230

20

.238

25

.246

30

.254

kf
.0152 .0230 .0340 .0474 .0651 .0882 .118 .157 .206

.316 .467
.680 s916 1.235 1.632 2.13
2.78 3.56

1.108 1.125
1.143 1.160 1.175 1.192 1.212
1.234 1.251

l-w0 2/c2

A

.075

2.61

.096

2.75

.118

2.88

.136

2.99

.154

3.12

.168

3.22

.179

3.32

.189

3.40

.198

3.50

WADe T 54-602

28

-JEW

Ii

it

/ -
.
ILIn 0+
_ _4

"--a

0
00

0

4

0 A0lo

0

-.

Q\

WADC TR 54-602

29

L,

The phase velocity w t varies somewhat with both the temperature and the ratio w/ wt,, but is practically independent of gt30- In Table 3 values of (w?/c) at 300C and -5oC, respectively, are given for a series of values of w/wt. For temperatures less than 30°C the phase velocity w' in foggy air is always between c and 0.9c. Values of c for different Atemperatures are given in Appendix I.
TABLE 3
The Sound Velocity in Foggy Air

0

0.1

0.5

1.0

2.0

10.0

CD

(wYc3oo 0.90

0.90

0.92

0.95

0.98

(w'/c) 5o 0.95 0.95 0.96 0.98. 0.99

1.00 1.00

1.00 1.00

One may determine wt for a given fog, consisting of n particles per unit volume of radius C at a given temperature by using Eqs. (32) and (33) with values for Xs, etc. from Tables 2 and 3. More conveniently .one may use the equation

wt = A(T)nC ,

(37)

where the temperature-dependent proportionality constant A is given by

A 1 + Y(Xks8f

4 v4wD,

1 + ks kd kf Xd

(38)

and is tabulated in Table 2.

In summary, one may calculate ae from Fig. 9 and Tables 2 and 3 by the following procedure:

(1) Determine wt for given n, t and T from Eq. (37) and Table 2.
(2) Forn the ratio w', then find (2aew'/w') from Fig. 9.
(3) Estimate w' from Table 3 and Appendix I (4) Determine ae by multiplying the value of (2 aeWt/ W')
obtained from Fig. 9 in Step 2 by ( wl'/w t) x 132,200
to obtain units of db/1000 ft.

An example is given in Section III. It should be realized that the Oswatitsch-Wei theory is applicable only to a fog in which the drop-

WADC TR 54-602

30

lets are of uniform radius . Application of the result to actual distributions cannot be done in this case (unlike the Epstein-Carhart case) by dividing the droplet'sizes into groups, determining the losscoefficient separately for each group, then adding these to give the resultant loss coefficient. Further development of the theory is necessary in order to obtain a result which can be applied to a distribution of sizes.

1.3.4 Laboratory Measurements.

Measurements of loss coefficients in artificial fogs have been made by Knudsen, Wilson and Anderson 1 6 , mainly in the 500-8000 cps range, using a reverberation technique. Drop size determinations were
made by photographing droplets deposited on an oil-coated glass slide. For one set of measurements the observed droplet distribution, divided
into five groups, is given in Table 4 from Epstein and Carhart. The total volume of all drops was 2 x 10-6 cm3 per cm3 of air.

TABLE 4

Drop Size Distribution,

*1I

for Knudsen, Wilson and Anderson Measurements

Group 1 2 3 4 5

Mean Radius 3.75 x 10-4 cm 6.25 x 1010.0 x O- 4 15.0 x 10-4 21.5 x 10 - 4

Drops/cm3 55 89
121 38 21

Experimental values of a determined by Knudsen, et al for this fog at the various frequencies are given in Table 5 (from Epstein and Carhart). Also shown there are corresponding values of *vh, calculated by Epstein and Carhart from their theory, applied to the distribution given in Table 4.
The Epstein-Carhart coefficient a vh, which represents losses due to viscosity and heat conduction, is sufficient to describe the experimental results given here fairly well at about 500-1000 cps, but is too small at higher frequencies.
At frequencies less than 500 cps the viscosity and heat conduction processes treated by Epstein and Carhart plays a reduced role and

WADC TR 54-602

31

. '1I

TABLE 5
Loss Coefficients in Artificial Fog

Frequency
500 cps 1000 2000 4000 6000 8000

a
(Experiment)
4.3 db/Z000 ft.
6.1 8.2 8.8 10.7 11.6

Gvh (Theory)
4.4 db/lOO0 ft.
5.0 5.5 6.1 6.5 6.7

the evaporation mechanics considered by Oswatitsch and Wei apparently becomes the prgdominant one. Knudsen, et a116, also made preliminary measurements of the loss coefficient a in fog at lower frequencies, namely, from 27.5 to 350 cps; in this range their measured losses were considerably in excess of those given by avh.

More.recently Wei 13 has made measurements in the 30 - 100 cps
range using an impedance tube method. In one of the artificial fogs he investigated there were 5.4 x 103 droplets/cm 3 of average radius 6.6 x 10-4 cm; the ratio of water mass (in droplet form) to air mass was 6.05 x 10-3 The results are given below in Table 6.

TABLE 6 Measured and Calculated Loss Coefficients in Fog

Frequency
30 35 40 45
50 55 60 65 70 75 80 85 90 95 100

a
(Experimental) 7.3 db/l000 ft.
7.4
7.8 7.5
7.3 7.3 6.6 6.5 6.5 6.9 7.4 7.3 7.4 7.4
7.5

6 visc
(Theoretical) 0.2 db/l000 ft.
0.3
0.3 0.4
0.4 0.5 0.5 0.7 0.9 1.1 1.3 1.6 1.8 2.5 2.9

ae
(Theoretical) 5.9 db/l000 ft.
5.7
5.5 5.4 5.3 5.1 4.9
4.7

WADC TR 54-602

32

*

.

*

_ *

As seen in Table 6, measured values of a are practically independent of frequency, averaging about 7 db/lO00 ft over the frequency range given for this (rather heavy) artificial fog.
Shown for comparison is avisc , the calculated coefficient due to viscosity alone, obtained from avh by letting P, and'therefore YT equal zero. (The complete Epstein-Carhart theory was not available when Wei made his measurements.) Though negligible at 30 cps, the viscous losses increase rapidly with frequency and account for about onethird of the total exponential loss at 100 cps. Also shown in Table 6 is Ge, the coefficient due to evaporation processes, calculated from Eqs. (28) - (36). (Wei does not explain in detail how theory is applied when, as here, droplets are not uniform in size. See discussion in subsection 1.3.3.) Evidently Ge accounts for most of the observed losses, being far greater than avisc in the 30 cps region, and being about twice as great as .*visc in the 100 cps region.
1.3.5 Smoke and dust.
An equivalent analysis to that of Epstein and Carhart for fluid spheres in air has not yet been made available for small solid bodies. in air. However, upon examination one finds that avh, given by Eq. (26), does not depend significantly on the viscous or elastic properties of the inner medium of the tiny spheres, but only on the corresponding density and heat conductivity. It might therefore be argued that results for rigid spheres would not differ greatly from those, given by Eq. (26), for liquid spheres of equivalent denaity and heat conductivity. To the extent to which this is true Eq. (26) may be applied to smoke or dust composed of solid spheres. It is not obvious, however, that Eq. (26) (or any theory derived for spherical scatterers) would apply to dusts composed of rough irregularly-shaped particles, as is commonly the case. Viscous losses near rough surfaces are probably much different from those occurring at smooth boundaries.

1.4

SOUND PROPAGATION OVER A PLANE EARTH

1.4.1 Introduction
The propagation of sound through a homogeneous, isotropic atmosphere from a point source above the ground is strongly dependent upon the acoustical characteristics assumed for the ground. Irregularities of the surface (of a size of the order of the sound wavelength or larger), ground type (sand, hard-packed earth, etc.) and ground cover (bare ground,

WADC TR 54-602

33

grass, etc.) all play a role in the determination oif the intensity of sound at a distance from the source.

Expressions have been developed for the pressure from a point source above a plane earth for the case 1 7 in which the earth may be assumed a homogeneous, isotropic "fluid" medium (i.e., no shear effects are taken into account), and for the caseIB , 19 in which the earth is acoustically representable by a normal-impedance boundary condition.
The problem of a dipole source above a non-uniform surface 20 , 21 has been investigated for the case of electromagnetic waves; however, the corresponding analysis for acoustic waves has not been developed.

When there is no preferred direction at the surface of the ground, or in the ground itself, the "fluid" medium assumption should be valid; this condition has been found to be an adequate representation for sand22 o However, when the lower medium is porous, and so constituted that air in the pores moves more readily in the vertical than in any other direction, the normal impedance boundary condition should hold. In practice this situation might be approached if the earth were covered by long, vertical-stemmed vegetation, such as
meadow grass, and if this vegetation were so dense as to essentially constitute the "lower medium", the ground itself then having no effect on the sound field.

Let us consider a point source of sound, having a harmonic time dependence (sound pressure varying as exp(-iwot), at the point (0,0,zo), and a receiver at (x,y,z); see Fig. 10. We shall use the following notation for the fluid boundary condition (Case 1) and the normalimpedance boundary condition (Case 2).

w - angular frequency of source (Case 1 and 2)

k = !L propagation constant for air (Case 1 and 2)

(39)

k2 - propagation constant for earth (Case 1 and 2)

(40)

RI 0 [x2 + y2 + (z - Zo

distance from source (0,0O,z o )

to receiver (x,y,z) (Case 1 and 2)

(41)

R2 Lx2 + + (z + zo J' distance from image point

(0,O,-z O) to receiver (xy,z) (Case 1 and 2)

(42)

= specular reflection angle from horizontal (angle between horizontal and line from image point to receiver) (Case 1 and 2)

WADC TR 54-602

34

4,

I

o_>J

file

U)

0.

oo

/

c °

j

I.

/

@

00

I

WADC TR 54-602

35

-. Cos , (x2R+2 2
:R2
= pic =

sin z +R2zo
impedance of air (Case 1 and 2)

(3) (44)

Az
..
----

Z2~ p=2 cc22 = kP22 impedance of "fluid" earth (Case 1)

(45)

=
P'l

specific normal impedance of ground (normal (46) impedance of ground divided by impedance of air) (Case 2)

R(l) Z2 sin* -Zl [l-(kk 2 )2 cos 2 1 24 2 sin* +Zl[l-(kl/k2)2 Cos2']

plane wave reflection coefficient in the

specular direction for reflection from a "fluid" earth (Case 1)

R(2) , sin* - 1

(48)

p

Lsin* + l

plane wave reflection coefficient in the

specular direction for reflection from a

normal-impedance boundary (Case 2)

* (U)== 22 Uevv2-2 v

(4~9)

(u) *1

error function23 (Case' 1 and 2)

* -"53-33

1.4.2 Solutions for the General Case*
The problem of a point source of sound in a homogeneous, isotropic atmosphere above a plane boundary below which lies a homogene-
*The methods used in the acoustical case summarized here are analogous to methods developed for the problem of a dipole source of electromagnetic radiation above a plane conducting earth. The method used by Rudnick 17 and by Lawhead and Rudnick 1 9 is that developed by Sommerfeld, van der Pol and Norton; the method used by Ingard is based upon the solution of the electromagnetic problem by Weyl. For a short summary of these methods, see J. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1951), p. 573; for a complete analysis of all work done on the problem, and an extensive bibliography, see A. Banos, Jr. and J. P. Wesley, "The Horizontal Electric Dipole in a Conducting Halfspace", Scripps Institute of Oceanography (Univ. of Calif.) Reference
(September, 1953).

i"WADC

TR 554-6-022

36

i Iis
P ;

ous, isotropic "fluid" earth has been treated by Rudnickl7, following
the method of Sowmerfeld. The incident pressure from the point source, and the reflected (and transmitted) pressure fields are developed in terms of Fourier-Bessel integrals; by applying the boundary conditions at the air-earth interface (continuity of pressure and of normal particle velocity across the interface) an integral expression for the reflected field is found. The integral is then approximated under the assumption that the earth is a highly absorbing medium; specifically, it is assumed that the distance from the image point to the receiver (R2 )
large compared with the distance in which the amplitude of a plane wave traveling in the earth is diminished by a factor of e = 2.72... , i.e., it is assumed that IR2 Im(k2)1>> 1. Laboratory tests made by Lawhead and Rudnick2 4 show that the theory is valid at audible frequencies over acoustical absorbing materials such as Fiberglas.
The problem of a point source of sound in a homogeneous, isotropic atmosphere above a plane earth which may be acoustically characterized by a normal-impedance boundary condition has been treated by Ingard18 in a manner similar to that of Weyl, and by lawhead and Rudnick19 , following the method used by Rudnick in the "fluicd" earth case. Since the method of Lawhead and Rudnick yields results very close to that of Ingard (the results are the same for C >sin * ), their method will not be discussed. Ingard's method is to represent the incident and reflected fields as a (integral) superposition of plane waves; the boundary condition at the surface is that the total pressure divided by the particle velocity normal to the surface is equal tb the normal impedance of the surface. In two special cases, to be discussed in subsection 1.4.3, the resulting integral may be evaluated exactly; in the general case the approximation klR2>> 1 (distance from image point to receiver is large compared with the wavelength of sound in air) is made.
In both cases (Case 1: "fluid" boundary condition; Case 2: normal-impedance boundary condition) the sound pressure at the receiver is given by

p~xyz) eikPlRRlz +(el-iRklpR)2F(

)]

(0

(Go-" I and 2)

where the specular-direction plane wave r1flectlon ce-effioient used

is that appiv iatr to the case under coridezmtion: Rw- i) ,

(47),

ior Case 1-

Eq. (48), for Case 2 . e function 4(, i-;

WADC TR 54-602

37

where 4 is the error function, Eq. (49). The tnumerical distance",
p appearing as the argument is given in the two cases as*

kIR2

1 cos2

[sin* + (Z-1)

1- ((ikk2l)2 cOB~2

2

)

(Case 1)

, yl

-

(53)

(Case 2)

Ingard has plotted the magnitude and the phase of F( p) as functions

*

of the magnitude of the variable p , with the phase of p as a para-

meter; see Figs. Ul and 12.

The Taylor's series and the asymptotic series for F(p) are

given by Ingard as:

(Fp)

13 + 5/ ... 1W

1*3 205 30'7

(55)

2p (2p) 2 (2p)3

There are three limiting cases of importance for both the fluid boundary condition and the normal-impedance boundary condition:

(1) As R2 goes to infinity, both p(l) and (2 go to infinity, and F( p) goes to zero; the expression for the pressure becomes

eiklR1

eiklR2

p(x+y zP,,+ RI

- , R2 ..-w(Case I and 2)

(56)

R2

*

*The following changes from those formulas appeari in the references

should be noted: (a) in the numerical distance pM, the factor 1/i
replaces Rudnickts factor of i so that Ingard's graphs can be used; (b) the term I/cos2 * in p(1) is a correction for a term left out

*

of the expansion leading to Rudnick's final result; (c) in the numeri-

cal distance P(2), the factor 1/i, instead of i as given by Ingard,

is a correction to his reported result. The expression of Lawhead

and Rudnick for Case 2 Ls given by Eqs. (50) and (51), with a numeri-

cal distance

1.. kR 2 (tsin* + 1)2/t 2 cos 2 21

WADC R 54-602

38

-

.-. -

-4

1-~~-

o ul j V.-

LO

0

U--
WADO ~ 3 54-W2

AROP: -900

-20
240)
-20

WACR402

450

2

S

60

-- :

This corresponds to the usual result that, .at large distances from the source, the reflected radiation is that due to an image source having an amplitude equal to the plane-wave reflection coefficient in the specular direction.

(2) As the boundary becomes coustically soft" ( P2 40 for

Case 1, or C-t-0 for Case 2) both p

d p (2) go to infinity so that

F( p) goes to zero, while R nd Rp2 go to -1; the pressure becomes,

as expected,

l R2 p(x,y,z) eikl - eiklR2

(57)

for pressure-release surface (Case 1 and 2)

(3).As the boundary becomes "a oustic

Case 1, or

co for Case 2) both R(l and R 2

sure becomes, as expected, iklRl eiklk

p(xyp) =+

hard" (P2 -- for go to + 1; the pre-
(58
(58)

for hard surface (Case l and 2)
1.4.3 Solutions for Special Cases
Two cases of particular importance occur when the 'ource and receiver are on the ground, and when the source and the receiver are in a vertical line.
(1)Source and receiver on ground
For the source at (0,0,0) and the receiver at (xy,0), we have:

The pressure is given by

(Case 1 and 2)

p(x,y,z) = 2F( p) eiklr

(60) (Case 1 and 2)

where the numerical distances, Eqs. (52) and (53), become

p(l

klr (ZL)2 [1 (kl/k2 )1

WADCTR 54-62

41

(61) (Case 1)

... n ..

. ,W b ,..

.

,

'1ii

P(2)

kl r

2i

t

(Case 2) (62)

For large distances from source to receiver, r goes to infinity1 so

that p. l) and p(2) tend to infinity. Frcm Eq. (55), F( p)-

S~() xY,z

21

Z2

-- , -

(63)

kI [i - (kl/k2 )2] (Z1

r2

p(2) (x,y,z W3t2iklr --

r..

k1 r2

( 3) (Case 1)
(64)
(Case 2)

For a source and receiver at the boundary (or near it) and for a large distance from source to receiver, the pressure amplitude follows an inverse-square law with distance (as anticipated in Eq. (3), subsection 1.1).

(2) Source and receiver in a vertical line For the source at (OO,z o ) and the receiver at (0,O,z), we have:

R, I z - zol, R2 (z + zo); ,

90

1Z 2.+.Z

(2) -

(65) (Case 1 and 2)
(66)
(Case 1) (67)

The numerical distances become:* p(i) _ ®

(Case 2)
(68)
(Case 1)

*The numerical distance p(2) in Eq. (69) has been taken from Ingardts ap-

proximate solution, Eq. (53). However, Ingard has derived18 an exact solu-

tion for the case of 4 = 900; the exact expression for the numerical dis-

tance is p = (i/i)klR2 (i + 1/t ), just twice the expression in Eq. (69),

For the case of

1, an exact solution has also been derived for arbi-

trary angle 4 ; this is p = (i/i)klR2 (1 + sint' ). Since there is given no approximate solution for angles near 4, 90° , or impedance close to

1=, based on the exact solution for these cases, only the general ap-

proximation is used heir.

WADC TR 54-602

42

P(2) _ kl(z + z') ( +
2i

(69)
(Case 2)

The pressure is therefore given by

p(1) (x,y,z) iklz- z

+ Z eikl(z + z°)

~(2) (xpy,z) eiklIz - zoj +

z zl

C+

(Case .)
(c1) + 2F( (2))leikl(z+z0 )(7.
(z+zo)

(Case 2)

1.5

SOUND PROPAGATION IN A STRATIFIED MEDIUJM

1.5.1. Introduction

One of the most important effects to be noted in atmospheric acoustics is the change in sound intensity due to acoustic refraction by changes in air temperature and wind velocity with height. The effects of sound refraction are only partially explained by theory, due to the mathematical difficulties inherent in the calculations. These effects, although often noticed, have been carefully measured in the atmosphere in only a few cases; however, the attenuation due to refraction of sound has been well studied for the case of sound propagation in the ocean25 . The discussion here will be limited to the case of sound velocity changes (due to changes in temperature and wind velocity)
in the vertical direction only, changes due to variation of temperature and wind velocity in the horizontal direction have not yet been analyzed.

Micrometeorological measurements will first be discussed in
order to give background for obtaining a general expression for the velocity of sound in the presence of the type of wind and temperature variations most usually found to occur. The analysis of the sound field by means of ray theory (geometrical acoustics) will then be discussed. The wave theory of sound attenuation for the cases of a constant sound, velocity gradient, \nd for a linear variation of air temperature with height, over a ,;ound-absorbing earth make up the next sections. Finally, as an illustration of the general theory, an approximate sound attenuation formula is derived for the case of the sound velocity varying in a manner to be expected fron the micrometeorological measurements. This

WAe TR 54-6o2

43

I I
*
K

will be later used to analyze a recent set of measurements of sound attenuation in the atmosphere (see subsection 2.3.3).

1.5.2 Temperature Distribution near the Ground

A number of measurements of the dependence upon height of air
temperature and wind velocity have been made for the first few tens of meters above the ground26 27O0nly average values will be considered here;
the discussion of fluctuations about the mean value will be left to subsection 1.6.

0. G. Sutton28 and E. L. Deacon29 have found that the variation of air temperature with height is representable by te following empirical formula for the temperature gradient:

T -r -az 4

(72)

dz

where T is the temperature, r is the adiabatic lapse rate, and a and are experimentally determined constants. The adiabatic lapse rate, 1' Centigrade per 100 meters, is the change in atmospheric temperature with height which must exist if any mass of dry air, moving adiabatically, is to have the same temperature as its surroundings (i.e., a temperature gradient of -r is the condition for static adiabatic equilibrium of dry air).

When the temperature gradient is -r, that is, when the value of
I is zero, the temperature condition is known as an "adiabatic lapse" or tneutral stability' condition. When a is positive, the temperature de-
creases with height at a rate faster than that given by r ; this is known as a "super-adiabatic lapse"t condition (commonly c lled a "lapse" condition). When p is negative, and r is less than az the temperature increases with height; this is called an "inversion" condition. The lapse condition is found during the day, when the ground, heated by the sun, heats the lower layers of the atmosphere; this gives rise to the decrease in temperature with height. At night, the ground cools off rapidly, leading to

TABLE 7
Typical values of a (0C), derived from Fig. 13, assuming 8 1.

Day Night

Summer (June)
0.5
-0.15

Winter (January)
0.1
-0.15

WADe TR 54-602

44

'
I.
[: , I ..

r

Sunset

Dawn

I|nversion

O0La

2

6 8

44~100

s

1 1 !0--- 22 21

GUT

t

o[ ISO
io

Lapse Ca).JANUARY

Down.Invers ion

{b) JUN

Fig. 13. Diurnal variation of temperature gradients found by
Best at Porton, England. Solid line for temperatur6e1WI difference between heights of 2.5 ad 30.5 cm; daehed line same for heights of 30.5 and 120 cm. (From O.G. Sutton, Reference 26).

45

20

a temperature increase with height, or an inversion. Figure 13 showing the diurnal variation of temperature gradients, illustrates these points. From this figure, one can obtain typical values for the constant &. A few values from Fig. 13 are given in Table 7.
Deacon, using data obtained by A. C. Best, has found that 8 ,
the exponent of z in Eq. (72), is positive and approximately equal to
unity. He obtains values of 8 greater than unity for lapse conditions,
and values less than unity for-inversions (typical values being 1.15 and 0.80, respectively); near neutral stability, bol.I0.

Integration of Eq. (72) leads to:

'1

~=TT, - r (z - zl) - a~z.8 1 8

(73

where T is the temperature at height z, and T, the temperature at height zI .

1.5.3 Wind Velocity Distribution near the Ground
Under conditions of neutral staility, both theory and experiment30 *31 give a variation of mean wind velocity with height of

?-l(~

(74)

where k is the von Karman constant and is approximately equal to 0.4. The "friction velocityt u. is found to be proportional to the windspeed, at eay given reference height, its value depending upon the roughness of
ground., The "roughness length'' I is characteristic of a given surface,
TABLE 8 Representative values of I and %/u 200 for natural surfaces, where u200 is the wind speed at 200 cm above the ground (neutral stability assumed). From Sutton, Micrometeorologv p.233

Type of Surface
Very smooth (mud flats, ice) Lawn, grass up to . cm high Meadow, thin grass up to
10 cm high Thick grass, up to 10 cm high Thin grass, up to 50 cm high Thick grass, up to 50 cm high

1(cm)
0.001 0.1
0.7 2.3 5 9

*U0 0.032 0.052
0.072 0.090 0.110 0.126

WkDC TR 54-S02

46

-k-----8
-

and is roughly proportional to the height above the ground of the pro-
tuberances causing the roughness. Table 8 gives representative values
of f and of the ratio u*/u200, 'where u2 00 is the velocity at a height
of 200 cm. It is to be noted from Eq. (74) that u = 0 at z = f.

Deacon has found that under more general thermal conditions, the wind velocity gradient can be given as:

du

U z-i

(75)

dz

where u., k, I are defined as before, and is a positive constant of order unity. This expression gives the neutral stability variation of wind velocity with height for = 1. Deacon has also determined a relation between A and 8 (the corresponding constant for the temperature gradient) of:

2 - 24 0.9

(76)

Since 85!1.10 for neutral stability, this gives

as expected. Integration of Eq. (75) gives:

u o

lO

z z

1i in that case,
(77)

where u is the wind velocity at height z and u = 0 at a height equal to
the roughness length, z = . It is to be noted that z I must be taken
as the effective height of the ground.
1.5.4 Sound Velocity as a Function of Height

The results of the last two subsections may be used to determine the variation of the velocity of sound with height expected tQ occur naturally. Since the effect on the sound velocity of temperature and wind
changes will be small, Eq. (73) and (77) may be approximated to obtain a simple expression for the sound velocity.

-r Neglecting the term (z - zI ) in Eq. (73), since its effect will
be small, the sound velocity may be written as a function of height using the dependence of air temperature upon height of Eq. (73):

c N-cI

- a (zI -

)i-

(78)

c=c 1 Cl-2(1-B) T)

where cI is the velocity of sound at height zl, and T1 is the absolute temperature there.

WADCTR54-W02

47

If sound travels in a medium moving with velocity u, the sound velocity at any point (x,y,z) is given by (for u-ccc):

c(xyz) = ct (xyz)+ u(x,y,z) cost

(79)

where c? is the sound velocity at (x,y,z) in the absence of motion of

the medium, and 0 is the angle between the wind direction and the path of the sound ray through (x,y,z), i.e., if wind and sound ray are in the

same direction the sound velocity is increased by the wind. Since it is

Iassumed 4

that there are no variations in sound velocity in the x and y

directions, the projection of the ray path on the x-y plane will be a

straight line (under the approximation ucc c); the angle # is therefore

A

taken as the angle between the wind direction and the horizontal projec-

tion of the line drawn from source to receiver. (See Figure 14a.) Using

Eq. (77), Eq. (79) becomes

_,,.~

*c =t +?k(z1~.-.Ui*-)[l-

P - 1fliecos+, zt/

*

Using Eq. (78) for ct , the velocity of sound in the absence of wind:

=I !W cl

ac1(z) 4 2Tl(l-8 )

-6)+

*Cos

clk(1-1 ) -

2'

(80)

a r

Since wind velocities will be small compared with the sound velocity, and

temperature changes will be small compared -kiththe absolute temperature,

the second and third terms in Eq. (80) may be approximated without much

error in the final result. If the velocity of sound at the source (used

as a reference point here) is co C(zo), the velocity of sound at an arbitrary point z is:

c(Ze0 E

a1 (zl zo

0

) )2T,(1-

+ coku*CS

(z~-l

&C [ l az01-8 I -+B1o9

2T1

Zo

ln( ]
zo

where the approximation is valid for (1-) ln(z
The sound velocity may therefore be written as:

l, (1 l-) n(z/zo) l.

--

C(z) Co[l -1B n(JzLo)J; z Z ; B -2T1 8 - cUoAk (2)

o 0s

where c0 is the velocity of sound at the reference point zo; a, 8 0 t, I and u*/k are experimentally determined from micrometeorological informa-

WAD TR 54-602

48

J
t'I

Source 0
WWindd
Direction

Receiver Roay from Source
to Receiver
(a)

ib

P(xY,Z)

1L

Q(o1o,z,)

*':

1~ ,o,o)1

'

"

?:

(b)

'Fig. 14. Geometrical relations for a sound ray from
source to receiver. (a) Projection on x-y plane show-
ing angle between wind direction and ray direction; (b) in vertical plane, showing angles with horizontal made by ray at source and at an arbitrary point along the path.

WAD T54-60.2

49

us
*i
4

tion, and is the angle between the wind direction and the ray path
through the point P(x,y,z). The approximations are valid for 0 and
f 8 near unity, and z near zo (but, if and 8 are assumed exactly equal
to unity, z may have any value). As noted in the discussion of the
wind velocity gradient, the "ground" is at height z = f; it is at this
height that the wind velocity becomes zero.

1.5.5 Ray Theory: Shadow Boundary

An important effect due to the variation of sound velocity with height is the formation of shadows in the presence of the ground. In this subsection, a derivation by means of geometrical acoustics will be given of the position of the shadow boundary for the sound velocity varying as in Eq. (81).

A sound ray, in a medium whose sound velocity depends upon height, travels along a path given by Snell's law:

c(z) cos 80 = co Cos 9

(82)

where c(z) is the velocity of sound at a height z, is the angle made with the horizontal by the tangent to the ray path through the point (x,y,z), and co and 8o are sound velocity and angle made by the ray at some reference point (in t. s case, the point at which the source of sound is located); see Fig. 14b. For the case of the sound velocity
decreasing with height (e.g., for B>0 in Eq. (81), a given ray from the source will be bent upwards; the presence of the ground causes a region to be formed into which no rays can penetrate, the "shadow zone". Although ray theory leads to an absence of sound in the shadow zone, an application of wave theory shows that sound is diffracted across the shadow boundary, but the sound pressure decreases markedly with distance
into the shadow zone; see subsection 1.5.7. Two types of shadow zones may be formed, one in which a single ray from the source forms the shadow boundary (a constant velocity gradient forms this type of shadow zone); and one in which, due to crossing of the rays, the envelope of many rays form the shadow bcandary (the inverse-z velcoity gradient, treated above, forms this type). These are schematically illustrated in Fig. 15.

The equation of a ray (r as a function of z, zo, and 80) may be found from (see Fig. l4u)

r

dz

+_tan

((83)

where tan 9 is found from Eq. (82), z is the height of the ray at r,
and zo is the height of the source. Since a ray can pass through a

WAXG Th 54-602

50

(a)

Fig. 15. Schematic diagram of thze f'ormation of a shadow-zone due to a negative sound velocity gradient: (a) Case for one sound ray forming the shadow boundary; (b) Case for shadow boundary formed by the envelope of many rays.,

WkDC TR54-602

51

minimum point (when 8 0), the integration is carried out in two parts;
an integration from zo to Zm, and from zm to z, where zm is the minimum height of the ray, i.e., the height for which 0 0 (it should be noted that tan 8= dr/dz changes sign through the minimum). Carrying out the
analysis for the sound velocity dependence of Eq. (81) (and neglecting terms in B2 as being small compared with unity), the equation of a ray
starting from r = 0, z = zo, and making an initial angle 8o with the
horizontal, may be written as

r -zm~B

1 n z,/ ir] f w,ex (v2 dv + f wo ex(v 2 )d v]

0 0

_ [EO (z/zo)w]

(84)

w [n(z/zm ;

wo [ln(zO/zm)

where the minimum height of a ray, i.e., that height for which 0= O, is:

Zm = z° exp(- --tan2 )

(85)

~d~o

As anticipated earlier the rays given by Eq. (84) cross each other, their envelope forming a shadow boundary (see Fig. 15b). The shadow boundary for any given value of z is formed by that ray which has an initial angle giving the maximum radial distance from the source for thdt value of z; the shadow-forming rays satisfy:
(dr_) 0(86)
z=const.

As may be seen from Fig. 15b, the presence of the ground affects the shadow boundary only slightly; the ray wi h "begins" the shadow bound-
ary at the ground has an initial angle G(0o such that 0= 0 at the ground:

cos9 (0) co

(87)

oc(f)

where z I is the"ground heightt used in the velocity distribution,
Eq. (81). (For the case of the shadow zone formqd by one ray, as in Fig. 15a, the shadow-fomtng ray has an initial angle gvqn by Eq. (87).) Using Eq. (86) it is found that the initial ray angle S0s) which form the shadow boundary at height z is given approximately by:

tt an2 0(os),- 2(Bz/lzno()z2/z.)

(81

WKDC TR 54-602

52

M---------------

--

? "' : 'subject

This also approximately satisfies Eq. (87) for the shadow boundary at ground, z = E, provided that the source is not too close to the ground, i.e., Z0 ' I.

Using Eq. (88) in Eq. (84), the horizontal distance to the shadow boundary, r., is therefore:

(Zr/z 0 )T exps-v d

exp (v2) dv]

0

(z/Zo) 2-l

r ,(z/zo)

(89)

A graph of these results is given in Fig. 16 for 0.1 - (z/zo) c 10; for this range .of variable, it is found that the second term (containing the extra factor of B) is negligible for values of B usually found to occur. Figure 16 is a plot of ((r5/zo),/7]vs (z/zo); for any value of B (less than about 0.1) the distance to the shadow -one may be found from the graph,
to the condition that the source height be large compared with the roughness length ("ground Iheight") f.
The relation given by the graph, Fig. 16, may be written as:

-

D

cos f(z/z)

(90)

zo -_ os

since B, from the velocity distribution Eq. (81), is:

() cosE D - Mocos~

(91)

2T1

colc

A polar plot of r., the distance to the shadow boundary, vs , the angle between the wind direction and the line from source to receiver, may be made. It may be seen from Eq. (91)'and Fig. 14a, that the distance to the shadow boundary forms a closed curve for D.M, (i.e., for the temperature effect larger than the effect of wind), so that the source is everywhere enclosed by the shadow region. However, if the effect of wind is greater than that of the temperature variation, so that D M0 , the distance to the shadow boundary, rs, forms an open curve; this curve gives a shadow zone in directions against the wind, and goes asymptotically to the lines through
the source given by:

Cos 0 _k

(92)

These effects are i.lustrated in Fig. 17.

WADO TR 54-602

53

0

-

OD

i,:i

4L4

-

-

-

-

Cl,
Uiri
0,W

0

!o:: , o

•

i

AC

,UU

o

o

-O-

,
0
Z0

,-
\

IDtoU4

50

4-41j

-~-
DIRECTION

Shad ow
D>M 0

a)Temperature effect greater than effect of wind..

Shadow Zo
DD
WIND DIRECTION

b)Effect of wind greater than temperature effect.
Fig. 17. Schematic diagram of position of shadow bounidary in the presence of temperature and wind gr-adients; aee Eq. (91).

WALDC TR54-602

55

For general use of the graph shown in Fig. 16 let 8-up-.1 (as is
often approximately true) and let T1 be 2930A (200C); then using k = 0.4 and the value of c o given in Appendix I at 200C (i.e., c o = 34,400 cm/sec) the quantity B may be written

B = (1.71 x 10-3) [ a - 0.o (u) u2oo Cos#]

(93)

where a is in 0C and u200, the wind velocity at a height of 200 cm, is in cm/sec. The constant a may be obtained from on-site measurements of the temperatures T, To at two different heights z, zo and using the expression

a =(94) ln(z/zo)

obtained frQm integrating Eq. (72) and letting S= i, r= o; typical values of a are given in Table 7. Values of (u*/u20O ) for various types of surface are given in Table 8.
To determine r., the horizontal distance to the shadow zone bound-

ary for given z,zoo and for given meteorological and surface conditions one may proceed as follows:

1. Obtain from Fig. 16 the value of FA for

the given (z/zo)

2zo

2. Obtain B fromEq. (93) for the given u2 0 0,+, for the value of a obtained from field measurements, using Eq. (94) (see Table 8 and Fig. 13 for orders of magnitude), and the value of (u/u20 ) from Table 9 for the given surface.

3. Multiply the value obtained in Step 1 by zo(2/B)i.

1.5.6 Wave Theory of Shadow Zone: Constant Velocity Grdient

Although the ray theory of the shadow zone predicts that no sound rays enter into the shadow region, it is well known that sound does exist in this region. This may be explained on the basis of wave theory, since a shadow region is penetrated by sound waves diffracted across the shadow
boundary. The theory of shadow zone penetration has been developed by Pekeris 32 for the case of a constant sound velocity gradient in the ocean. He found that in the shadow zone formed by the surface of the ocean, in
which the sound velocity was assumed to decrease limearky with depth, the intensity of sound decreased in an exponential manner along a horizontal path. His results have been recently extended, by JIngard and Pridmore-Brown3 3 ,

il

WADC TR54-602

56

I
* jof
f .4
*1

to the cases where the ground is acoustically re.presented by a normal impedance boundary condition, and either the sound velocity or the air temperature (sound velocity proportional to the square-root of the temperature) varies linearly with height.

In this subsection the case of the constant velocity gradient will be examined; this case is found to be solvable exactly. (Since the work in the succeeding two subsections depends upon this case the method
solution will be given in some detail.) In subsection 1.5.7, a method of approximately determining th high-frequency attenuation in the shad.ow zone for an arbitrary sound velocity dependence upon height will be given, and applied to the case where the sound velocity is proportional to the square root of the height (constant air temperature gradient). Since the sound velocity determined by the micrometeorological data given in subsections 1.5.2 and 1.5.3 varies approximately as the logarithm of the height (see subsection 1.5.4), the high-frequency approximation for the sound attenuation will be applied also to the sound velocity of Eq. (81).

For analytic simplicity, the case of the constant velocity gradient is best solved in an "inverted" system of coordinates, where z is positive into the ground; the ground is at z = y; the source is at r 0, z w and the receiver is at P(rz), see Fig. ia

The sound velocity dependence upon height is:

CWz - gZ

(95)

where g is the sound velocity gradient. Although the velocity goes to

zero at z .O, i.e., at a distance * above the ground, this is taken care

of analytically by requiring outgoing waves as z approaches zero. It is

to be noted that the velocity gradient is, in practice, given by the dif-

ference in sound velocities at source and ground, divided by the source-

to-ground distancet zo

-

g ___[c(ground) - c(source)]
Zo

(96)

(So that the gradient used here is a positive number). The distance from the ground to the zero-velocity point, z O, is therefore

y

_q -

(97)

g

This very large distance, of the order of 1.1 x 105 feet for a typical atmospheric gradi.ent of g = 0.01 sec 1 , is well outside the region of physical interest.
The wave equation for the pressure in the case of linear velocity

WADC IR 54-602

57

r
V

• - +?-

"1

o

P(rz

Z
'I(a)
1t

Ground

Ground

(b)

I

Fig. 18. Geometrical relations for the shadow zone

wave theory; (a) minvertedtI coordinate system for

constant velocity gradient case; (b) "nor-nalt" co-

ordinate system for constant temperature gradient

case.

WAD T1 54-602

58

is " 2 p(r,z) + g2z22p(r,z) = 0

(98)

where the harmonic time dependence of p is exp(iwt). It is assumed that the ground is acoustically representable by a normal impedance condition:

p(ro)

z

( Iz)I

(99)

iwp

z ='0_

where p.is the density of air. By expressing the pressure as a Fourier-

Bessel integral, solving the resulting z-dependent equation, and perform-

ing a contour integration, the solution may be obtained in terms of a

Fourier-Bessel series 32 :

I !"z)m=l21

.) JD (C)+ J1 (0)

(ioo

'm

where D is a constant measuring the strength of the source, r is the radi-

al distance from the source to receiver; (Y- r) is the height of the

source above the ground (y- z) is the height of the receiver above the

ground; and r is the distance from the ground to the z = 0 plane (see Eq. (97)

and Fig. 18a); H42) is the zero-order Hankel function of the second kind;

Jin is the Bessel function of order in; Jt is its derivative with respect

to its argument; and

-in

n

(101)

[ The parameter xm is the th solution (m 1, 2,.....) of the equation:

Ji(j "ym 22 :' j

0;

-in("

Jj (xm) 0; xi ikmy

(io)

Zg

where p and co are the density and sound velocity (at the ground) of air, Z is the ground impedanc,! Eq. (99), and g is the velocity gradient.

In general, the series solution, Eq. (100), cannot be summed; however, for large ranges, the dependence of pressure on range can be found from the asympt'tic expression for the Hankel function:

H(2) (kMr)

-21 e-ikmr

kmr -' 1

(103)

ALDC TR 54-602

59

Since the propagation constant km, found from Eq. (102), has a negative imaginary part which increases with increasing m, the terms of the series depend upon distance into the shadow zone as damped cylindrical
waves. The attenuation coefficient of each term is given by:

wl/3g2/3

am

Am

2

Co

(104)

"

where the factor Am depends upon the ground impedance and (lightly) on

the frequency, and co is the sound velocity at the ground. Choosing a sound velocity of 1100 ft/sec; the attenuation is

am 2.5 Am g2/3fl/3 db/1000 ft

(105)

The dimensionless factors-Am are given for two different ground impedances in Table 9.
TABLE 9
Values of Am in Eqs. (104), (105)

m
1 2 m 3

pc-boundary

Hard boundary (z =W0)

1.85
3.242/251
2.23(m- t) 21 3

0.53 2.23(m- 3/4)2/3

Formulas for obtaining Am in other c- s are given in Reference 33.

Since the modes higher than the first are more strongly damped than

the first mode, the propagation well into the shadow zone will be charac-

terized by a single damped cylindrical mave of the type in Eq. (103) with

an attenuation given by a,, i*e., Equ (104), (105), and Table 9 for m - 1. For the region far enough into the shadow zone so that only the first mode

contributes to the sound field, the intensity at a point P(r,z) may be

written as:

',

( 1 -2a 1(r-ro)

1(P) I(B) (')e

(1o6)

r

where 1(B) is the intensity on the shadow boundary at the same height as the receiver, ro is the distance from source to B, r is the source-toreceiver distance, and @ is given by Eq. (104).

WADCT I 54-602

60

-,
--:

As an example, it may be noted that, for.a typical gradient of g 0.01 sec -1 and a 500 cps source the first mode is damped-with an attenuation factor of 8.3 db/l000 ft for Z =pc, and 2.4 db/l000 ft for
Z.=O
The theory of Pekeris (who considered the linear velocity case for a pressure-release surface) has been tested experimentally in underwater cases. It has been found that the attenuations predicted by the theory are greater (by less than a factor of two) than the measured values. This can be understood, since the effect of sound scattering from the rough sea surface, and from thermal inhomogeneities in the ocean, have not been taken into account; these would have the effect of increasing the sound intensity in the shadow zone. The attenuations found theoretically are therefore upper limits to the actual attenuations to be found experimentally, but probably give better than order-of-magnitude results.'
1.5.7 Wave Theory of Shadow Zone: Constant Temperature Gradient
From the results of Pekeris, Ingard and Pridmore-Brown32 '33 have derived an approximate result for the shadow-zone attenuation of sound propagating over a normal impedance ground when the sound velocity is an arbitrary function of height; the approximation is valid for high frequencies.
A regular coordinate system is used (in contrast to the "inverted" one of Pekeris): the z-axis is positive upwards; the ground is at z = 0, the source is at z = zo; and the receiver is at P(r,z) (see Fig. 18b). Let the sound velocity be c(z), and the velocity at the ground be c(o); form the functions:

W -

(

k)

(107)

#A(zT)

Q(z,T) dz

where T is a (complex) number to be determined. The equation:

[J... LQo,)J

.1(*/(32)OD)]

= _Z i,,eP

dodz [h{L,)J

1/ sC(zz1T1J)}

(108) 10 ))(2)

(where Z,W ,p are the normal ground impedance, the aingular sound frequency, and the density of air, respectively; 11(/23) is the Hankel function of the second kind of order 1/3) will determine an infinite number of roots rt. These will be, in general, dependent upon the sound frequency and the grourdimpedance. The (complex) propagation constant for each mode (which modes are similar to those of Eq. (100)) are given by:

WADC TR 54-602

61

-=

(110)

The attenuation factor for each mode is therefore:

a

Imkma-I

Well into the shadow zone only the first mode will be of importance, and the intensity will be given by Eq. (106). The above results are valid under the condition

4Q

2Q 3 6

((Q121) )2

In reference 33, this has been applied to the case of sound pro-

oe.

pagation in a medium having a constant temperature gradient, i.e., a

sound velocity of:

c(z) co To

(113)

where c. is the sound velocity at the ground; To is the absolute temperature there; and b = dT/dz is the (constant) temperature gradient. Eq. (113) gives a sound velocity gradient at the ground of:

go (dcZ-J

2To

(l

Applying Eqs. (107) and (108):

Q2 oW2

k2

c20 (b - kb)

(115)

It is found that for sufficiently high frequencies, such that

W/go l

(117)

Q and p satisfy the condition Eq. (112). Letting Q0o jLO ;A(or), Eq. (109) becomes, using Eq. (117):
1z(2)
1/3 (~

Q(o,t) and

WADC TR 54402

62

The roots w(m) are frequency dependent in general; however, for the case of hard ground (z =CD ), they are:

0.685; (2) - 3.90 (z C)

(119)

For high frequencies, this gives attenuations of (see Eq. (lll)) (m )j 2/3 W1/3g/3

(120)

Far enough into the shadow zone so that only the first mode is of impor-

!j

tance, Eq. (120) for m = 1 may be used in Eq. (106) for the intensity in

the shadow zone.

laboratory experiments have been performed in order to check the theoretical predictions. A small tank was heated in a manner to obtain a constant temperature gradient (b = 250°C/meter was used); measurements taken over a frequency range from 2700 cps to 10,400 cps for a hard boundary and for a pc boundary (using glass wool) showed good agreement with theory. There seems to be little doubt that, under the idealized conditions assumed in subsections (1.5.6) and (1.5.7), the theoretical predictions will be fairly accurate.

1.5.8 Wave Theory of Shadow Zone: Logarithmic Velocity Dependence
In subsection (1.5.4) an expression for the velocity of sound (Eq. (81)) in the presence of wind and temperature varying with height wis derived on the basis of the micrometeorological data of subsections 1.5.2 and 1.5.3. The problem of sound propagation in the atmosphere in the presence of a wind is no longer an isotropic one, and therefore requires the solution of a wave equation differing from the usual form, Eq. (98). It may be expected, however, that for wind velocities small compared with the sound velocity, an approximate solution may be obtained by using Eq. (81) for the sound velocity in the ordinary wave equation; an attenuation coefficient for sound in the shadow zone can therefore be derived using the methods of the previous subsection.
In the sound velocity equation, Eq. (81), the effects of wind and temperature are mixed; since the equation for the wind velocity variation
with height is valid only for distances above the ground greater than I,
an "effective" wind velocity vo will be used in the sound velocity equation, which will be assumed valid for z ? 1:

BZ

(121)

WAC Th54-602

63

c(z) Co(l -mo ln);

zmf

where c o is the sound velocity at z = f.
From Eqs. (107) and (108)
, o ln(Zi ) ]2 _ - m ( TI!)] -2 dz

(122) (123)

For small mo, the value of /5. on the ground can be approximated by

3 c0
where the value of I&(z,r) to be used in the boundary condition, Eq. (109),
is its value at the effective ground surface z = f. For sufficiently high
frequencies, and mo not too small, it can be seen that TM-; therefore

3 co

This satisfies the condition for the validity of the approximation method Eq. (112), for

WIM 0
Co

(126)

The boundary condition, Eq. (107), for determining the roots

is:

iW .-0 MolI3, 2/3 H-2D (JO)

z

Ic~

H1/3()

The roots are frequency dependent in general; however, for Z -0 they are:

---

- o0A.)685 ;

- 3.90 (Z , )

(128)

The attenuation, from Eq. (11O), (111). and (124) is:

mo(m)] 2/3 W1/3 Fo2/3 co

C0 1fl

(f29)

where go is the sound velocity gradient at the surface. This expression will be compared with data in subsection (2.3.3).

It is to be noted that all of these high-frequency approximations to the attenuation coefficient, Eq. (104), (120), (129), have the same dependence upon frequency and sound velocity gradient; however, in view of some of the results of Pekeris, this should not be considered as a general result for all frequencies of interest, It may be expected, for

WADC TR 54-602

64

example, that changes of the velocity-height profile from the simple models used above, may markedly change the frequency dependence of the attenuation, especially at the lower frequencies.

A

1.5.9 Intensity in the Normal Zone: Channelling

In the three previous sections, expressions for the sound pressure due to a point source in a stratified medium were developed by means of wave theory; the expressions, given in terms of infinite series, were shown to be useful only inside the shadow zone, where only the first term of the series is of importance. There has been no such simple solution found for the Itnormal zonet, i.e., the region between the source and the shadow boundary.

The ray-theory approximation to the normal zone sound field, valid
for infinitely-high frequency, can be developed for an infinite medium. By means of appropriate images, the effect of a hard (or soft) boundary should be obtainable; however, non-infinite (and non-zero) impedance boundaries present special difficulties, and the problem of boundary effects needs investigation.

The sound intensity at a point P(r,z) in an infinite stratified medium, associated with the ray making an angle with the horizontal of 80
at the source, see Fig. l4b, (i.e., the power per unit area at P carried
between rays emerging from the source between angles So and Go + d&o and reaching P between angles 9 and G+ d8) is given by Reference 34:

I(r,z,S,) - r(dFr/cdoOsG)sin(

(130)

where F is the source power per unit solid angle; 9 and so are connected
by Snell t s law, Eq. (82), and r is the horizontal distance from the source
(see Eq. (83)):

z, ='

Co jcose d(

(131)

cosao

(dc/dz)

For the case of the constant sound velocity gradient (c az), using the " invertet coordinates of Fig. (18a), we obtain:

FCOS2 90 4xr

r

s in 8- s in8) for c =az (132)

Cos 0s

where w (= co/a) is the distance fnoi the source to the z - o plane (see subsection 1.5.6). (It is to be noted that, for the intensity at a point
P(r,z), one determines that value of 0o which satisfies the second of Eqs. (132) and Snell's law, and uses this value of 9.in the first of
Eqs. (132).)

WADC TR 54-602

65

Due to the complexity of the equation of a ray traveling in a medium where the sound velocity depends logarithmically upon height (see Eq. (84)), it is doubtful that useful results can be obtained for this case by the above methods of analysis.
In the case of a sound velocity which increases with height above the ground (e.g., a temperature inversion), the sound rays are bent downward after leaving the source, and will all reach the ground after traveling some dis: .e from the source; the distance will depend upon the sound velocity gradient and the angle of emission of the ray. Upon reflection from the ground, the proness will be repeated, causing the sound rays to be periodically reflected from the ground as they travel outward from the source. This phenomenon, known as "channelling", can bring about a greater sound intensity at large distances from the source, than would be predicted by a simple inverse-square law. An analysis of this effect for the case of constant sound velocity gradient based on Pekerist work, has been done by Haskel13 5, but the results have not yet been extended to other cases of interest.

1.6

EFFECT OF RANDOM TEERATURE AND WIND INHOMOGENEITIES

1.6.1 Introduction

The effects on the propagation of sound of the height dependence of air temperature and wind velocity were discussed in subsection 1.5. There it was implicitly assumed that this height dependence (almost logarithmic) was constant in time and space, that is, a given height dependence was assumed to be independent of time and of horizontal position. Experimentally, it is found that the average temperature and wind velocity (speed and direction) are describable by a simple height dependence, but that thore are variations about these mean values; these variations, or sound velocity inhomogeneities, are dependent upon time and position (horizontal and vertical), and lead to sound scattering, and the associated phenomena of intensity variations and fluctuations. An important characteristic of the inhomogeneLties is their randomness, or irregularity in'space and time.

If a sound beam passes through a region ihich contains inhomogeneities, i.e., one in which the sound velocity varies randomly about its mean ' alue, sound may be scattered out of the beam, causing a decrease in intensity. Moreover, sound scatterid from a sound field by inhomogeneities may cause an increase in intensity, at some points, above what would be ex-

WADC TR 54-602

66

pected in the absence of scattering; it is believed that the scattering of sound across the shadow boundary is responsible for the observation that sound intensities in the shadow zone are considerably higher than predicted by theory.

Since the inhomogeneities are time-dependent, that is, the mag-

nitude of the deviation from the mean sound velocity depends upon time,

the combination of tdirectly-receivedtt and scattered sound will be time-

dependent; the tobal received intensity will therefore change with time.

The variations of the intensity may be subdivided into two types: tslow

variationst or changes in the average intensity taking place over a

period of minutes or more; and "fluctuations", or changes in intensity

about the (slowly varying) mean value, taking place in a period of sec-

onds, or less. (Obviously these definitions overlap, since they are

two aspects of the same general phenomenon.) This division is useful

LI

from a practical point of view, since the variations in mean sound in-

tensity may be expected to depend upon large-scale changes in sound

conditions, e.g., changes in "average" wind velocity, while the fluctua-

tions will depend upon the small-scale changes, or inhomogeneities, e.g.,

short gusts of wind.

Existing theoretical analyses of the effects of inhomogeneities usually do not take account of the presence of boundaries (the ground) or of the dependence of the meal. sound velocity with height. It is often assumed (because of mathematical difficulties) that after the mean sound pressure is determined by using the mean sound velocity (varying with height), the effect of inhomogeneities may be given by assuming an infinite medium with a constant mean sound velocity (independent of height). T+ iE expected that this should give, in most cases, the order of magnitude oi the effect, and its dependence upon important parameters, such as sound frequency. Since the inhomogeneities are random functions of space and time, the effects due to them will also be random. Theoretical treatments of those effects are therefore designed to give information about average values, and root-mean-square deviations from average values.

After a short description of the methods of averaging used, the types of temperature and wind inhomogeneities to be expected in the atmosphere near the earth wil be discussed; the analysis of specific problems will then follow.
1.6.2 Averages and Correlation Functions3 6

Consider a random function of space and time, f(x,y,zt), that is, a function whose value, at a point in space and at a given time, depends upon the variables (x,y,z) =_ a&nd t in such an irregular manner that the relationship between values at two points (in space and time)

WADC TR 54-602

67

%J
4'
1

can be given only statistical--y. f is given by:
Y~(2C,T, o
T, t =

The time average, or mean, value of

f (., t) dt

(133)

to -T

In general, the average value will depend upon the time at whirch the average is taken, t., and upon the sampling length, T, as well as the position r. This dependence on to and T will obviously be of importance; for example, if f(Z,t) is a Component of the wind velocity at a point, the average value of the wind will depend upon whether to is taken during a time of Itstrong!' or 'tweak" wind, and whether T is large enough to take account of gross changes in wind speed, or just short gusts.

The time of averaging and the sampling length are closely con-

nected to the type of changes considered, i.e., "slow variations" or

ttfluctuations". It is assumed in the theoretical developments to follow, that the length of time T is sufficiently long to include many short-time changes (fluctuations) in f(g,t), but short enough so that any long-time changes (slow variations) in f(.,t) take place in a length

of time much larger than T. -In this case, the average value T( ; T, t 0 ) will not depend upon T, but only upon the time of averaging; the fluctua-

tions are then considered as deviations from the slowly varying average

value. Since, in general, the slow variations will be assumed to depend upon ldnrge scale changes, they will not be of particular interest here;

in the study of fluctuations, the averaging process may therefore be taken as though the mean value does not change. This mean value is therefore:

T

<f(t)> lim T2T

f(r,t)dt

(134)

Since the function f(4,t) is assumed to be a random function of space as well as time, the same considerations hold in the case of a space average as held for the time average; the average should be taken over a volume large enough to include many small changes in fu,t) at any instant of time, but not so large as to include large-scale changes (e.g., the average should be over a region of space containing many gusts
bf wind, but such that every gust is ttravelinglt in a wind having the
same velocity as the others):

-f(2,t)>V -lim i f(x,y,Z,t) dx dy dz

(135)

Measurements in acoustics usually involve sound pressure at a

WADC TR 54-602

68

point in space as a function of time, so that use of the time-averaging process in theories gives results more closely related to experiment than use of space averages37 . However, since the theoretical developments are usually based upon "stationary" random variables, that is, variables whose average does not depend upon the time about which the time average is taken or upon the position about which the space average is taken (statistical homogeneity) both will give the same value for the average, that is, < f >= f>V, and the average does not depend upon position or time. This may be visualized by considering a wind with many gusts; for the time average, a stationary recording instrument averages over all the gusts carried past by the mean wind; for the space average, many recording instruments are used, placed throughout space, and the average taken by using the readings of all the instruments at one instant of time.

As will be seen in the next section, however, the micrometeorological variables are not statistically homogeneous, since they depend upon height above the ground; the time average will probably be the more useful one in theories using this fact.

The deviation from the mean is defined as:

af(,t) f(,t) -(f

(136)

The deviation has a mean value of zero. (When there is no chance of confusion the subscript will be dropped: AXt) Af(rt).)

A measure of the randomness may be found by averaging the pro-

duct of the deviation at a point in space at one time and the deviation

at another point in space at another time:

< .,t) A (r +P.

T>=lm co

a

)~

T d (137)

2T -T

where P is the vector distance between the two points in space, and T is the time interval; the mean-square value of the deviation occurs for p- 0, r- 0. As noted above, it is usual in the theoretical work to make the assumption of statistical homogeneity; the average in Eq. (137) will then depend only upon Land T , the interval in space and time between the points whose deviations are being averaged.

If the function f( t) is random, it may be expected that large
values of the average appearing on the left hand side of Eq. (137) occur for small values of p and T , since, if A (Lot) has some value, A(r + g, t + T) will have taLmostft the same value for small j, T: if A(r,t) has one 3ign (positive or negative), A(r + , t +T) will "most probably" have the same sign and the average will be made up of a series of positive terms. For large por T, the values of A(r + , t +

WADC TR 54-602

69

should bear little relation to the values of A(r,t): for a given sign
of A r,t), the sign of &(r +., t + T) may be positive or negative withl'nost equal probability". The average, for large p or T will therefore be made up of a series of terms having positive or negative signs with "almost equal probability"; the average therefore tends to zero for large p or T. (If there is a periodicity in A , the average will also tend to be periodic; this is, however, excluded from truly random functions.)

The "correlation function" is defined by normalizing the average to a maximum value of unity:

Ri(e,R"~)e6=t) < &(jat) A ((lr3r L. t +Tir

(138)

-

~(:(r,t) -,t))

Since the correlation function is often used for the case T = 0, the notation R( &) will be used for R( e.,0). Under the condition of statistical homogeneity, R( &, T ) is a function of the vector distance, .= (C ,'WC) (x2-xl, Y2-Yl, z2-zl), between the positions at. which the deviations are taken; for the case of "isotropic homogeneity", R(P., w) R(p,T) is a function of the magnitude of the distance between the points (p = jp4), and does not depend upon direction.

Since it is expected that the correlation function goes to zero as either por T go to infinity, they may be integrated from zero to infinity over either variable. Thus may be defined, for the correlation function in general, characteristic lengths and a characteristic time. For the case of isotropic inhomogeneities, the characteristic length is:

] P0

R(p,O)dp

0

(139)

with corresponding definitions for characteristic lengths in the x, y or z directions for the anisotropic correlation function. The characteristic time is:

To f R(Or)d?
0

(140)

1.6.3 Temperature Variations and Fluctuati ons

Although the slow variations in temperature are to be considered
as relatively unimportant in the problema of sound scattering (since these are related to the "slowly varying mean value of the received sound pressure), it seems best to discuso variations and fluctuations of temperature together, since they are the large-scale and small-scale aspects of the same problem of temperature changes.

WADO TR 54-602

?0

As noted in subsection 1.5.2 and Fig. (13) the temperature
gradient changes in size and sign during a 24-hour period. Table .0
TABLE 10
Diurnal temperature variations at various heights above the ground (note that the heights used are different for the two locations); from Sutton: Micrometeoroloy.

Porton, England

Leafield, England

Month

Height (meters)

Diurnal Variation (°C)

Height (meters)

Diurnal Variation (00)

December

0.025

3.7

0.30

3.3

1.20

3.1

7.10

2.7

17.10

2.4

1.20

3.2

12.40

2.2

30.50

1.6

57.40

1.2

87.70

0.9

June

0.025

0.30

1.20

7.10

17.10

11.8 10.2
9.4 8.3 7.7

1.20 12.40 30.50 57.40 87.70

10.8 8.8 8.1 7.4 7.0

shows the diurnal variation of the temperature (in °C), at various heights above the ground, on clear days in Porton (open meadows) and Leafield (hilly pasture lands), England. As might be expected, the variations are greatest near the ground; the rate of decrease with height of the diurnal variation is much greater in summer than in winter. By expressing the temperature deviation from the mean by the first two terms of a Fourier series in time, the coefficients (which will be functions of the height) may be found:

AT(z,t) . cI sin(15t +#1) + c2 sin (30t +42)

(141)

In this expression the time, t, is expressed in hours past midnight, and
* the argument of the sines are in degrees (1-hour - 15 degrees); the values
of the coefficients, cl and c2 and the phase angles + 1 and 2 are given

WADC TR 54-602

71

as functions of the height for Porton, England, and Ismailia, Egypt
(near desert conditions) in Table 11. The summer conditions at
Porton indicates that the temperature variation there can be represented, to a good approximation, by a single sinusoidal term in time; this is not true of the other conditions, however,
TABLE 11
4Coefficients and phase angles for first two
terms of a Fourier series representing the diurnal temperature variations at various heights above the ground (note that the heights used are different for the two locations, as well as the type of day used); from Sutton: Micrometeorology.

Porton, England

Month

Height (meters) cl(°C)

1

c2 (OC)

December (all days).
June (all days)
December (clear days)
August (clear days)

0.025 0.30 1.20 7.10 17.10

1.37 1.23 1.16
1.03 0.93

2440 2380 2330
2250 2180

0.025 0.30 1.20 7.10
17.10

5.78 5.14 4.72
.0
3.76

2460 2380 2350
2280
2230

IBmailia, Egypt

1.10 16.20 46.40 61.00
1.10 16.20 46.40 61.00

6.40 4.06 2.49 1.94
6.56 5.28 4.82 4.69

2280 2190 2090 2040
2250 2190 2180 2190

0.84 0.72 0.64 0.54 0.46
0.48 0.29 0.25
0.28
0.31
1.97 1.32 0.98 0.92
1.41 1.30 1.41 1.41

#2
590 550 520 450 400
1080 1070 llO0
1070 1030
610 420 280 270
500 4io 310 280

There has been little experimental work concerned with the ."luctuations of temperature. Sutton38 reports, from a single set of observations at Leafield, England, that the magnitude of the temperature deviation from the mean varies approximately as z-0 .4 , where z is the height above ground; however, this should not be considered as a general result.
There is some experimental evidence, which has a strong theoretical basis, that heat may be transferred by "bubbles" of warm air rising from the ground due to their buoyancy; this would be expected to take place under conditions of warm clear weather with a low wind, so that the ground becomes quite hot. The mean temperature field at any height would be closely related to the time-averaged effect of these bubblest passage past a point. Sutton39 estimates, from the Leafield data, that about four bubbles per minute rise from the ground, having a velocity of about 25 cm/sec at a height of 2 meters; the heat capacity per bubble, if the latter are assumed as planes, is about 0.05 cal/cm2 .
It may be expected that the temperature fluctuations are strongly related to such factors as temperature gradient, solar radiation, and wind fluctuations; the amount of data, however, is small, and is not in a form which can be used in the theoretical work (i.e., correlation functions). A statistical analysis of recent data should give useful parameters for the acoustical problems.
1.6.4. Wind Fluctuations
There is somewhat more known about the fluctuations in wind velocity than about temperature fluctuations, but not much of what is known is readily applicable to the theories of acoustic scattering. In. an analysis of turbulence over meadows (by Scrase, quoted by Sutton4O), 't has been found that of the total turbulent energy a3sociated with wind fluctuations, at least two thirds of the energy is associated with fluctuations lasting, at a point, less than five seconds. Fluctuations lasting for the order of a few minutes are found, as well as variations of a much larger time scale.
In general, it is found that the fluctuations in wind speed (in the x, y, and z directions) are approximately proportional to the mean wind speed, for small temperature-gradient conditions, at heights above about 20 meters from the ground. Below this height, the average magnitude (without regard to sign) of the fluctuations in the direction of the wind and across the wind are about equal, but the vertical fluctuation is somewhat smaller than the others; at. heights of about 2 meters, the root-mean-square cross-wind fluctuation is more than two times greater than the root-mean-square vertical fluctuation.

AtDC TR 54-602

73

The "gustinesstt may be defined as the average magnitude of the speed fluctuations, in the x, y, and z directions, divided by the mean wind speed:
gx: . In- W 1~j .y<IVIg (Mw)(12

Wu

(u)2)

where u, v, w are the components of the instantaneous wind velocity in the x, y, z directi-us, respectively, (u) is the mean wind speed
(taking the x-axis along the mean wind direction, and (v) = (w) = 0.) The gustiness istherefore, a measure of the speed fluctuations. in the direction of the mean wind and the fluctuation in mean wind direction (from gy and g.).

Best 41 has evaluated gx for winds over a level field with grass from 1 to 2 cm high; the averages were computed by taking readings of the instantaneous wind velocity every ten seconds for a total time of three minutes. The experiments were conducted under the following conditions; height above ground varied from 2.5 to 200 cm; mean wind speeds (at a height of 200 cm) varied from 50 to 800 em/sec; temperature gradients (between 10 and -10 cm height) varied from -0.015 to 0.010C/cm. It was found g. varied from 0.1 to 0.2, and was approximately independent of the mean wind (so that the average magnitude of the deviation is about 10 to 20% of the mean wind). The frequency of occurrence of values of gx had no well-defined variation with height in the above range; it was independent of the temperaturn gradient for measurements of gx between 2.5 and 10 cm above ground, but above 25 cm, gx decreased as the temperature gradient went from negative to positive.

There is leas information about the lateral (cross-wind) and
vertical gustiness gy and gz; at 200 cm above grassland the nmadimum values of gy and gz are nearly independent of mean wind speed for negative temperature gradients, but increase sharply with increasing wind speed for positive gradients; for constant mean wind, both gy and gz decrease as the temperature gradient goes from negative to positive values. Under all conditions, at this height the ratios of the maximum values of gy to gz it about 1.8. As a function of height, the ratio of the maximum values of gy to gz was found to vary from about 2.9 at 25 cm to 1.4 at 506 cm; as noted above, the ratio is about unity for heights above 20 meters.

Sutton has shown, from data taken at 200 cm above grassland (grass up to 30 cm high), that the autocorrelation function (correlation between velocity fluctuation affecting a particle at time t and
t + () may be given by:
N )n;
N+ <A2>
N 1100 cm/sec, n 0.15; (A 2 ) = 6.51 x 103 cm/sec

WADC TH 54-602

74

This autocorrelation function is approximately related to the previous-

1I

ly defined correlation function, subsection 1.6.2 by R(C) R( <u) ,C),

since a particle travels a distance (u)t in time interval .

1.6.5 Scattering of Sound by Temperature Inhomogeneities

The problem of the scattering of a plane sound wave by tempera-

")

ture inhomogeneities has been solved by Pekeris42 , and the correspondfg attenuation of a sound beam due to scattering, by Jacomini.

j

Consider a medium whose sound velocity has a mean value of co, and which has small random variation in space about the mean:

c(x,y,z) = co + Ac(xyz);

(c)= co ; --<< 1

The wave equation for the pressure amplitude is, approximately,

,Co

V 2p + k2 p= 2k2(-)p ;

k = CAo

(145)

The solution for an initial plane wave of sound traveling in the z
direction is (assuming a time dependence of exp [-iwtj ):

p~~~) e31(

ikr' __2 f_4ci?'Lyp(xvytzt) 2::-

2t

co

(146)

where (x,y,z) is the observation point, exp(ikz) is the initial plane
wave, and the integral gives the scattered pressure contribution from the inhomogeneities; (x', yt,zv) is the position of a scatterer, and
(x~x) 2 + (y-y')2 + (z~zt)2 Since the magnitude of the velocity
changes are assumed small, the pressure amplitude in the integral may
be approximated by the initial plane wave. The power scattered into
an int'initesimal solid angle dA r sin0 d d+ (where (r,0 ,*)are the
spherical coordinates of the observation point (x,y,z)) at a large dis-
tance from the scattering volume is

LedQ 1

[ ,Vi4n(2KP)2dp

K k sin/2

WADC 111 54-602

75

where 10 is the ntensity of t-Le initial piane wave, V is the volume of the medium containing the scatterers, and the space-correlation function is (see Ec (132) for r= 0)

= 1

( 1 -c(x~v~z) - 1 c(X4 6 Y+'M. z+~) 48

>

o

Cc

and statistical isotropy has been assumed:

R~)p

+TI

(149)

The plane wave attenuation coefficient may be found in the following

YA

manner: by integrating Eq. (142) over :kll angles (assuming that the

angular spread of the initial ueam is zero) the total power scattered

-'

by the inhomogeneities in the volume V may be found; the i tensity at-

t--uation coefficient, 2 (using a as the amplitude attenuation coef-

ficient) is. the ratio of the power scattered out of V to the power en-

terLng V. For initial intensity Io, this is

2 a - jfEd

(150)

iUsing Eq. (147), this becomes:

Sc _I..

o

k

For a correlation tunction:

R(p)

2 0

-a2

the intensity attenuation coefficient is:

2a -IT < (&c)2 > k02 a(1 -kO2a 2 )
Co

(152) (153)

This attenuation, increasing approximately as the square of the frequency, has not been well checked as yet by ex-erimiental work. Under the as-
sumption of the Gaussian correlation function, Eq. (152), the scattered radiation has a directionality pattern proportional to ex-p a2 k2 sin2 (9/2)] so that its half-power point occurs at angles from the incident beam axis of

9-. 2

. .radians+ 95 degrees

ak

ak

where (ak) is assumed large (sound wavelength small compared with inhomogeneity size); for audible frequencies, 8 will usually be no more than a few degrees. Since the scattering angles, Eq. (154) are so small, it is to be expected that higher-order scattering would send sound back into the acoustic beam and hence decrease the effective attenuation. Eq. (153) is therefore, probably, an upper limit to the attenuation.
A somewhat more general development of plane-wave scattering theory has been given by Ellison43 who considers the correlation between the intensity received at two points in a plane perpendicular to the direction of the incoming wave. Since there does not seem to be much application of this theory to the air acoustics problem the results shall not be given; it should be noted, however, that contributions to the Fourier transform of the intensity correlation function at a particular wave number comes from the same wave number of the Fourier transform of the refractive index correlation function.
A ray theory of attenuation due to scattering, arising from the use of a directional receiver has been developed by Givens44 , et. al; in vien of the discrepancies4 ,46 between ray and wave theoriea for the general scattering problem, there is some doubt as to whether this method is applicable quantitatively i.1 atmospheric acoustics problems.
1.6.6 Fluctuations due to Temperature Inhomogeneities
Since the temperatiuNe inhomogeneities (as well as the wind inhomogeneities) change with time, the received scattered pressure will also change with time. If a series of sound pulses aro received, the pressure aplitude of each pulse varies rapidly with time, and the timeaverage amplitude varies from pulse to pulse in a random manner about a mean value, A theory has been developed by Mintzer 45 for the fluctuation of the mean pulse amplituide from a point source of sound (in an infinite medium), and has been compared with underwater measurements. It is assmed in the development that the sound velocity at a point varies slowly enough with tite, so that there is little change in sound velocity during the passage of a pulse past the point; this is found to be true if the pulse length T is much less than the characteristic period To of the inhomogeneity (see Eq. (140), and if the characteristic length a (Eq. (139)) and period To are related by a/o< col, where c. is the mean sound velocity.
The method of analysis is similar to that of the previous section, except that the pressure fluctuations considered are those of the average pressure amplitude of a pulse, F5, and an initial spherical wave is assumed. If the coefficient of variation, V, is defined as the ratio of the standard deviation from the mean to the mean pressure amplitude

WADC TR 54-602

77

I ~

j< 12 V2 =(155)

It is found that (for distances from the source large compared with the characteristic length)

V2 . 2k2 q2r IR(C)dt

(156)

where r is the distance from source to receiver, and

o. 2R(C)

( rE&c(xyzt)]

L

Co

1 Aca + . ast
Co

(157)

Here x is the coordinate along the line from source to receiver and w is the rms value of Ac/co . For a Gaussian correlation function

the coefficient of variation is,for koa P: 1 (wavelength small cornpared with inhomogeneity size)

V [ r k02a] ri

(159)

This formula has been subject to test in the underwater case45, where acoustic measurements were made in deep water using 24 kc sound;47 measurements have also been made48 of the rms value of Ac/co, and of a,
the correlation distance. The theory and experiments agree very well.

The above work has been extended4 9 to detenine the time dependonce of the fluctuation in pulse amplitude. let the correlation function of the deviation from the mean pulse amplitude at a point in space be

OM_} -(,p(X,t)-Ap(xit +T)

-- _ <.(@pQ,t)

]2 >

(160)

and the correlation function of the deviation f-oem the mean sound velocity be

R~t

(( Ac(X.Y.Z.t)[ Ac X.Y.,t +T(61

Co

CO

It is found that

A
J ... -H
SEd
i,!

(T) R(T)

(162)

so that (under the conditions noted in the first paragraph of this subsection) a measurement of the correlation of successively received signals should give the time correlation of the velocity changes.
1.6.7 Scattering of Sound by Wind Inhomogeneities. I

As was noted in subsection 1.5, the interaction of sound waves with a wind is an anisotropic problem, and is therefore of a consider-
ably more difficult nature than the problem with a variable temperature; this is reflected in the fact that a single scalar potential for the ptaorrtipcolteent*ieallocmiutsytcbaennoitntrbeoduucseedd ians twheell.caseHowoefvaerw,inOdb,ukbhuotv50thahtas,a vbeyc-

introduction of a suitably defined scalar "quasipotentiallt, developed a simple method of taking account of vortical flow for flows of small Mach number (flow velocity much less than sound velocity). Using this method, Blokhintzev 51 has developed a single scattering approximation for the attenuation of a plane sound wave due to scattering from a turbulent region. The power scattered from a plane wave directed along the z-axis by a volume V of the turbulent medium into an infinitesimal solid angle df becomes:
Ioko d V 1 ( )2> dV e1(L-e) [rp(P) + 2

0~ 0

7c 2I

P

(163) ~-V1 2 V 2Mz (P-)
4k

where the vector is given in tems of the unit vectors in the direction of the incident wave L Rnd the scatterig direction n (at qpherical coordinate angle 8 from the z-axis):

K -k (n - 2 ;

2k sin(8/2)

(164)

and the correlptlon function of the z-component of the turbulent velocity (that iso the turbulent velocity component in the dArection of the incident wave) is defined as:

((v7

'e - ( ) vz(r -p

(165)

In generalthe correlation function depends on the direction, as well as
e. the magnitude of

From dimensional considerations, and noting that, to the approximation (v/c<< 1), used in deriving the above expression, the flow is incompressible, Blockhintzev derives a correlation function

(2 i(q.p)

2

> --

MZZ

q-1 /3 (

) dqldq2dq3

(166)

q2163

J 4greater

where Y is a constant estimated at about 0.2 (by Obukhov), and the integration is over values of the turbulent field wave number q
than a value qo where

,_ v

(167)

(The "constant" V presumably depends upon wind speed.)
This restriction on the minimum value of the turbulent wave number, is equivalent to a restriction on the maximum value of the wavelength associated with the turbulence; i.e., the turbulent wavelengths of interest should be less than a representative length of the scatter-
ing volume (VI/3 );any larger scale turbulence will have an essential-
ly constant velocity over the scattering region and will not contribute.

*i

Using Eq. (166) in Eq. (163), and determining the intensity at-

tenuation coefficient as in subsection 1.6.5, Eq. (150)

2Ga

5 (2w)113 p&5/3

2-V. 11)2
co

(168)

where p is a dimensionless parameter much greater than unity and k is the sound wavelength. Using data of Sieg's (to be discussed in Section I) Blokhintzev evaluates 1A to be equal to 10 (a reasonable value); the weak frequency dependence (as fl/3) "does not contradict Sieg's experimental results"51. This result should be the subject of further experiments.

*

1.6.8 Scattering of Sound by Wind Inhomogeneities. 11

A more exact formulation than Blockhintzev, of the scattering of 9ound by turbulence (i.e., wind inhomogeneities), has been given by Llghthill52 and independently by Kraichnan 53 , based on the general theory of the interaction of sound and turbulence by Lighthill.
Lighthill has shown (from the usual equations of a compressible fluid) that the motion of a compressible fluid may be written as a wave equation in the fluid density with a "forcing' term dependent upon the velocity of the fluid. Neglecting deviations from adiabatic flow in the turbulence and the effects of viscosity, the equation for the density of the fluid is

WADn TR 54-602

80

3

2_c2 VV22pp(=PuEiuj

(169)

,t

iJ=l xbj

where p is the density, and uju, j 4re the components of fluid velo-
city in the xi, xj direction (xI = x, x2 = y, x3 = z). By'dividing the fluid velocity into turbulent and acoustic components, and considering the turbulent-acoustic interaction term (the acoustic-acoustic term is negligible from the usual. small-amplitude acoustic assumption; the turbulent-turbulent term gives rise to turbulence-induced sound,
which has a small effect for low Mach numbers), a first approximation to the scattering can be obtained as in Eq. (146). For high frequency sound (wavelength small compared to characteristic size of the turbulence, see Eq. (139)), the power scattered from the turbulent volume into the
solid angle dU is approximately

EdA 2 1 0ok2 dAV k(vz/cO)>j Mz0,QCdt; k

1

(170)

0

where I. is the incident intensity, k the sound wave number, the correlation function is

<(vz)2>

( < (X'y'z) vz (x +, y +17, z +

(171)

andt is the characteristic length of turbulence in the z-direction, given by the integral in Eq. (170) (see Eq. (139)).

For isotropic turbulence, the high-frequency approximation (k to0C l) is not needed; the scattered power is given for any ratio of turbulence "size" to wavelength by:

1 k2

Edfat-

da V cos

ctn2(8/2) E(2k sin 0)

(172)

when £(K) is the Fourier transform (i.e., the spectrum) of the turbulent energy per unit mass of the fluid:

where the correlation function is a function only of P -W

since

isotropic turbulence is assumed. It is to be noted from Eq. (172) above,

that the scattered energy goes to zero at 9 = 900 and 8= 1800; for 8= 00A

WADC TR 54-602

81

the energy spectrum goes to zero as (2k sin 8 )4, sc that there is a zero in scattered energy at B - 00 as well.

For sound wavelengths smaller than the scale of turbulence, the intensity attenuation dae to scattering becomes:

2 a 8v.)2

~ 2

(,O

d ~

(174)

It is not known, as yet, how well the aboye theory corresponds to experiment; in view of the good agreement with experiment of Lighthill's theory of noise generation by turbulence, it seems probably that the above scattering theory should give resulto close to experimental results.

Ibounded
4"

1.7

DIFFRACTION OVER A WALL

1.7.1 Previous cases treated in Section I represent idealized situations in that, in each cases the air is assumed either unbounded, or
below by an infinite homogeneous plane. In field problems one
is compelled to consider more complex cases. Thus in practice it is important to know how sound propagates over a hilly terrain or around obstacles such as trees or buildings. We shall make no attempt to treat the general problems here, but confine ourselves to the special case of propagation over a long wall of given height. In a given field situation the "wall" might be an extended hill, a long building, a row of trees, or, of course, a man-made wall of boards or stone.

1.7.2 Appropriate methods for treating this problem are available from the classical Fresnel diffraction theory, long used in optics54 . For
definition of symbols see Fig. 19.

In Fig. 19 the source of sound is at Q, and P is a point at
which the sound level is to be calculated. The point Pc is directly above or below P and on the "line of sight" from Q; i.e., the line QPo just grazes the "wall". The distances from the wall to Q and P0 are a and b, respectively, while the vertical displacement of P below P0 is do; the latter quantity is positive for P below P', negative for P above Pc. In the Freenel theory it is assumed that a>>)k, b)'>> X, do<:-(a + b) and do < < (b/)(a+b). Define a quantity v such that

WADC TR 54-602

82

b

P

!- _

WiLi
WALL

"- P

Fig. L9. Geometrical parameters for problem of diffraction over a wall.

-b~

d0

+2a k

(175)

where ) is the acoustic wavelength. Then the sound pressure p at P is given by

o A [X2 + y2 ] Cos( Wt + 6 )

(176a)

where w is the angular frequency and

x J cosii-dx ,

(176b)

f0

2

V

9 tan-1 (Y/X)

(176d)

and A depends on a, b and the soul-ce strength, but is independent of do. Table 12 gives X and Y for positive and negative values of v

WADC TR 54-W2

83