zotero/storage/SGT3IJIQ/.zotero-ft-cache

ULTRA-SHORT-WAVE REFRACTION AND DIFFRACTION*
By T. L. ECKERSLEY, B.A., B.Sc.
(Paper first received \Oth December, 1935, and in final form 20th October, 1936.)
SUMMARY The work of G. N. Watson on the propagation of electric
waves over a spherical earth has been extended by the author
to take account of the finite resistivity of the earth, the effect of
which is of great importance in ultra-short-wave transmission. The work is in such a form that the field intensities above
the earth can be computed numerically. The effect of refrac
tion in the earth's atmosphere can also be taken into account.
The results for a range of wavelength between 2 and 10 m
and for heights up to 4 000 m and distances up to 400 km,
are published in the paper in a set of curves, the general
properties of which are discussed. The effect of atmospheric refraction is considered, and a
comparison between observation and theory is made, in which
good agreement is obtained, on the average, when neglecting
refraction. Major changes may, however, be produced
occasionally by refraction.
PROPAGATION CURVES FOR ULTRA-SHORT
WAVES
It is probable that in the near future ultra-short
waves, i.e. those below 10 m, will be extensively ex
ploited for television and for short-distance point-to-point
and aircraft communications. It is therefore essential to
know the ranges and physical transmission properties of
these waves. The curves reproduced in this paper are
intended to give this information in graphical form. The curves show the field intensity for all distances
up to 400 km and at all heights up to 4 000 metres,
produced by a vertical doublet—a radiator so short
as to give a cosine diagram—situated on the earth's
surface and radiating 1 kW. Also, by the reciprocal
theorem, the roles of receiver and sender can be reversed,
so that the curves give the field produced by a raised
transmitter at a receiver on the surface of the earth,
within the ranges of height and distance already speci
fied. Finally, they can be made to give the field at
any given height h^, from a transmitter at any other
height h^, measured above the sphericalearth surface. As in the case of the longer-wave broadcast trans
mission, the field depends to a great extent on the
earth's conductivity (a) and permittivity (K), but whereas,
in the long-wave case, oversea transmission can be
regarded as equivalent to transmission over a perfect
conductor, at the ultra-short wavelengths the behaviour
over sea departs very markedly from that for trans
mission over a perfect conductor. Curves are therefore
calculated for the following earth constants (in e.m.u.):
a = 10~13, K = 5, for overland transmission; a — 10-11,
K = 80, for oversea transmission. Although the curves
are self-sufficient, a few words of explanation are re
quired to indicate the assumptions made and the limitations involved. Ultra-short-wave transmission has generally been treated by the methods of geometrical optics. Thus, in Fig. 1, a transmitter T at a height h^ is supposed to have a visual range TR, where R is so chosen that the plane TR is tangential to the earth's surface at R. The points on the surface of the earth beyond R are in the shadow of the " bulge " of the earth, and, according to geometrical optics, nothing would be received at such distances. As a concession to the wave theory, it is generally admitted that some energy leaks into the shadow region, but the amount of spread is generally left to guesswork. Some such concession is necessary, because otherwise, according to the geometrical ray theory, a transmitter on the surface of the earth should have practically no
4H010
Fig. 1
range at all. It is clear that the problem is not one of
geometrical optics but one that requires the full wave
theory for its solution. This theory has been worked out by many eminent
mathematicians, including Poincare, Nicholson, Love,
and MacDonald. A certain amount of disagreement
was cleared up by the work of G. N. Watson, and it is
generally admitted that his mathematical results are
beyond reproach. The following well-known diffraction formula, giving
numerical results, has been derived by Van der Pol
from Watson's analytical results:
€ = A* sin *0
• Reprinted from Journal I.E.E., 1937, vol. 80, p. 286.
where e = field intensity, in millivolts per metre;
h = " effective height," in km; I = current (r.m.s.), in
amps.; A = wavelength, in km; and 6 = angular dis
tance between transmitter and receiver. In the deriva
tion of this formula it was assumed that, in effect, the
conductivity of the earth was so high that it could be
considered a perfect conductor; again, the field at a
height above the earth's surface was not considered.
Thus the formula is not adequate to describe the
behaviour of ultra-short-wave transmission where the [42]


ECKERSLEY: ULTRA-SHORT-WAVE REFRACTION AND DIFFRACTION
o
90
80
70
60
50
40 y
—
0 50 100 150 200 250 300 350 400 450 500 550 600
Fig. 2.—Height/gain curves (h and A in metres).
The following arr thefiguresof decibels to be subtracted:
Land. 2 m, Odb; 4 m, 2-2 db; 6 m, S-7db; 8 m, 4-9 db; 10 m, 5-9 db.
Sea. 2 m, 13-6 db; 4 m, 18-0 db; 6 m, 20-8 db; 8 m, 22-9 db; 10 m, 24-4 db.
o
200
180
160
140
120
80
60
40
20
/
/
/

500 1000 1500 2000 2500 3000 3500 hX- %
Fig. 3.—Height/gain curves (h and A in metres). The following are thefiguresof decibels to be subtracted:
Land. 2 m, Odb; 4m, 2-2 db; 6m, 3-7 db; 8 m, 4-9 db; 10m, 6-9 db.
S*». 2 m, 13-6 db; 4 m, 18-0 db; 6 m, 20-8 db; 8 m, 22-9 db; 10 m, 24-4 db.
50
45
40
35
30
I25
o
20
15
10
1
\
r. > S/ *
w
A
20 40 60 80 100
Fig. 4.—Height/gain curves (A and A in metres).
The following are thefiguresof decibels to be subtracted.-
Land. 2 m, Odb; 4 m, 2-2 db; 6 m, 3-7 db; 8 m, 4-9db; 10m, 6-9 db.
S*a. 2 m, 13*6 db; 4 m, 18-0 db; 6 m, 20-8 db; 8 m, 22-9 db; 10 m, 24-4db.


44 ECKERSLEY: ULTRA-SHORT-WAVE REFRACTION AND DIFFRACTION
effect of the height of the receiving and transmitting aerial requires to be known and where the effect of the earth's constants must be included. A short-cut method enabled the present author* to determine the modification of the exponential factor in the above equation due to the finite conductivity of the earth. A development of this method, by which certain characteristic numbers involved in Watson's solutions can be recalculated for the case of finite earth conductivity, has enabled the complete solution to be obtained at various heights and distances. The results are in such a form that the effect of atmospheric refraction can be included. The curves here presented are derived from these mathematical results, but the effect of refraction is neglected. They are based, therefore, on the assumption that the earth is of uniform conductivity a and permittivity K, and that the permittivity of the regions outside the earth
from the transmitter. Again, on ultra-short wavelengths the slope of the curves (loss, in db per km) is independent of the earth constants. Finally, a close scrutiny shows another general property possessed by the curves, a property which is also, of course, implicit in the mathematical analysis. It is this: above a certain height h0, which is only a slowly varying
function of the wavelength, the gain in field strength with height is, to a high degree of approximation, independent of the earth constants. It is consequently only a function of h and A in the form f(h\~l). Experimental measurement of the height/gain relation, as well as measurements of the slope of the field-intensity/distance curves, afford, therefore, a very useful check on the theory, since in this comparison between observation and theory no doubtful earth-constants are involved. A single master curve, in which the gain in decibels above hQ is plotted as a
10 12 14 16 18. 20
Fig. 5.—Height/gain curves {h and A in metres).
The following are the figures of decibels to be subtracted:
Land. 2 m, Odb; 4 m, 2-2 db; 6 m, 3-7 db; 8 m, 4-9 db; 10 m, 5-9 db.
Sea. 2 m, 13-6 db; 4 m, 18-0 db; 6 m, 20-8 db; 8 m, 22-9 db; 10 m, 24-4 db.
is constant and equal to unity. The general solution
is obtained as the sum of a number of terms of which
the first is predominant at sufficiently large distances and
for sufficiently small heights. The curves (Figs. 6-25)
are calculated in the regions where the first term is
predominant. In each figure, the broken line gives the
inverse distance field for propagation over a plane
perfectly-conducting earth. The estimated uncertainty
in the curves outside these regions is not greater than
2 or 3 db.
CHARACTERISTICS OF CURVES The most obvious feature of the curves is that in the region where the first term of the diffraction formula is predominant, the curves for various heights are all parallel. This implies that the gain in decibels resulting from an increase in height of receiver (or transmitter) above the earth's surface, is independent of the distance
• Proceedings of the Royal Society, A, 1032, vol. 136, p. 499.
function of h\~i, can therefore be constructed. This
auxiliary curve is shown in Figs. 2 and 3. It gives the
part of the curve for the height/gain relation for heights
above hQ. The value of h0 is such that ^QA"* is practically
equal to 50.
An example will make the use of this curve clearer.
Thus, if we wish to know the ratio of the field intensity
at a height h to that at hQ, we first calculate h\~l {h and
A are expressed in metres). Suppose h is 800 m and
A is 8 m; then h\~i = 200. The value on the decibel
scale corresponding to this is 61 db.- The value at hQ is
41 db. Therefore the ratio of the signal intensity at
h = 800 m, A = 8 m, to that at hQ is (61 — 41) db, i.e.
20 db, or 10 to 1. Below hQ, however, the height/gain relation is very
sensitive to the earth constants. Thus, in raising the
receiver or transmitter from the ground to a height h0,
there is a very much greater gain over land than over sea.
This is shown in Figs. 4 and 5, where the single curve


ECKERSLEY: ULTRA-SHORT-WAVE REFRACTION AND DIFFRACTION 45
giving the relation between hX~i and decibels gain, intensity above the ground intensity by using the figures which is appropriate for hX~i > 50, fans out into a appended to the curves. Thus, if we obtain directly
500
0 10 20 30 40 50 60 70 8 0 1 90 100
0 Kilometres 50 200 Fig. 6.—Results obtained over sea; A = 2 m, K = 80, a = 10"11, 1 kW radiated.
10mV/m 20db
1 • 0«
-100"
0 001- -120 50 100 J5fc) 12001 250 |300 j3£01 1400
50 200 | 1000 200013000 4000
Kilometres BOO 1500 2500 3500
Fig. 7.—Results obtained over sea; A = 2 m, K = 80, a = 10"", 1 kW radiated. Figures against curves indicate height of receiver or transmitter in metres.
sheaf of curves for different values of the earth constants from the curves the decibels corresponding to a given
and wavelengths in the region where h\~$ lies between height h, then we have only to subtract the figure
•0 and 50. Figs. 2-5 can be used to obtain the gain appropriate to the nature of the transmission (e.g. for


46 ECKERSLEY: ULTRA-SHORT-WAVE REFRACTION AND DIFFRACTION
8 m over land, 4-9 db) to obtain the gain (on a wave- curves (Figs. 6 to 25), and then the gain on raising the length of 8 m) due to raising the receiver to a height h receiver to a height h2 is given by the curves of Figs. 2,
above the land. 3, 4, and 5. The resultant field is thus found.
0 Kilometres 50
Fig. 8.—Results obtained over sea; A = 4m, K = 80, a = 10"", 1 kW radiated.
10mV/m 20db
1 » 0«
100/iV/in-20 •
10 • -40 «
1" -60'
01 • -80"
001- -100'
0-001' -120"
\\
O Q)
•51u81
J3 2
100 150 20.0 12501 „«„ ^» ^
ft 200 | 10001 2000L
Kilometres 500 1500 2500 Fig. 9.—Results obtained over sea; A = 4 m, K = 80, a = 10"u, 1 kW radiated.
We can use these curves to give the field at a height h2 The characteristics of the usual working theory, in
at the receiver due to a transmitter at height hv Thus which diffraction effects are only introduced beyond the
the field on the ground at the required distance from a geometric " visual" range, are illustrated diagrammati
transmitter at height Aj is first determined from the cally in Fig. 26. The field (curve A) maintains its inverse


ECKERSLEY: ULTRA-SHORT-WAVE REFRACTION AND DIFFRACTION
distance characteristics up to the visual range, after which distance, transmission behaves approximately as the
it drops suddenly in the shadow region. The true curve rough working theory suggests. On the other hand, for
for heights above 200 metres illustrates these charac- the lower heights, up to 200 metres or so (curve B), such
200
0 10 20 30 40 50 60 70 80 90 100
Kilometres 0 50 Fig. 10.—Results obtained over sea; A = 6 m, K = 80, a = 10~", 1 kW radiated.
lOmV/m 20db
1 ' 0•
10 • -40»
1 « -60'
01- -80«
001- -100"
0001" -120' 0 50 100 150 200
-4000
-3500
-3000
-2500
-2000
300 |350 200 5 0
4,00
0 50 200 500 1500
Kilometres 1000 Fig. 11.—Results obtained over sea; A = 6 m, K = 80, a = 10~u, 1 kW radiated.
teristics, though the point of inflection P at the visual as might be used in television practice, there is absorange is very much rounded off, and the actual field at the lutely no indication on the curve of any inflection such visual range is some 10 to 30 db below the unobstructed as occurs at the visual range in the other cases, (inverse distance) value. Thus at sufficient height and To express the results in terms of visual range, in the


48 ECKERSLEY: ULTRA-SHORT-WAVE REFRACTION AND DIFFRACTION
expectation of a sudden drop of field strength beyond This characteristic, namely the absence of marked this, is, if not false, at least meaningless. Physically, quasi-optical visual-range effect, is associated with the the meaning of the results is fairly clear. The earth following fact noted in the mathematical treatment. It
30db
lOmV/m 20 "
10 •
lmVm 0 "
-10"
2000
100/iVm -20 
-30
1000
0 10 20 30 40 50 60 70 | 80 90) 100
Kilometres 0 50
Fig. 12.—Results obtained over sea; A = 8 m, K = 80, a = 10"u, 1 kW radiated.
0 50 100 150 200 250 13001 350 400
4000
3500
3000
2500
2000
1500
Kilometres 0 I 200 I 10(J0
50 500 Fig. 13.—Results obtained over sea; A = 8 m, K = 80, a = 10"u, 1 kW radiated.
has taken such a toll of energy in the case of relatively is found that the results can be expressed in a quasi
low transmitters, even within visual range, that the optical manner with respect to a fictitious radius (RQ) of
extra shadow effect beyond this range is entirely the earth which is greater than the actual radius (R).
obscured. The difference between RQ and R is of the order of


ECKERSLEY: ULTRA-SHORT-WAVE REFRACTION AND DIFFRACTION 49
200 metres, for a wavelength of 10 metres, and is a appear. These results are illustrated in Fig. 28(a), from slowly varying function of the wavelength as well as of which it is obvious that, if the height of the transmitter is the earth's constants. The height, RQ — R, is the less than (RQ — R), no meaning can be attributed to visual
0 10 20 30 40 50 60 70 80 190 100
Kilometres 0 Fig. 14.—Results obtained over sea; A = 10 m, K = 80, a = 10~u, 1 kW radiated.
10mV/m 20db
1 - 0«
100/^V/m -20«
10 « - 4 0 «
1
0-1
-60
-80
001« -100»
0-001 • -120-0 50 100 150 200 250 30,0
Kilometres 0 350
2()0 400
50 500 Fig. 15.—Results obtained over sea; A = 10 m, K — 80, a = 10~n, 1 kW radiated.
height hQ referred to in the previous paragraphs, and is range in the altered sense described above. This leads
shown in Fig. 27 as a function of the wavelength. • It is to the following practical rule: If unobstructed transonly when the transmitter (or receiver) is well above this mission is required between points T and B, the line TB radius that the quasi-optical visual-range effects begin to must not approach the earth nearer than (Ro — R.)
VOL. 12. 4


50 ECKERSLEY: ULTRA-SHORT-WAVE REFRACTION AND DIFFRACTION
For heights less than about 500 metres, and distances field at R is calculated as the resultant of the direct less than 50 km but greater than about 3 km, the ray TR and the reflected ray TER [Fig. 28(6)]. The
lOmV/in
lmV/m
30 db
20 •
10 
0•
-10"
lOO^V/m-20
-30
-40"
\
0 UO 2,0 30 | 40 SO 60 70 80 90] 100
0 l'O 50 Kilometres200 500
Fig. 16.—Results obtained over land ; A = 2 m, * = 5, a = 10"", 1 kW radiated.
10m\/m 20 d b
1- • 0 "
lOO^V/m-20 •
10 • - 4 0 •
1 • -60•
01" -80'
001- -100«
0-001- -120" 0 50 10'0 l"50| 200 1 2 5 ^ 0 10 50 200 1000
3200000
3.5 ^ [ 4 0 0
300014000 Kilometres 500 1500 2500 3500
Fig. 17.—Results obtained over land; A = 2m, K = 5, a = 10~13, 1 kW radiated. Figures against curves indicate height of receiver or transmitter in metres.
intensity is proportional to the inverse square of the larger-scale curves for distances up to 100 km are calcu
distance. This follows from Sommerfeld's theory, or lated in this manner, and the transition to the diffraction
equally well from the approximate theory by which the curves is reasonably well defined (see Fig. 29). These


ECKERSLEY: ULTRA-SHORT-WAVE REFRACTION AND DIFFRACTION 51
results have been recently checked experimentally,* but as long ago as January, 1925, the same law was determined experimentally in a 10-metre transmission in a range
30db
lOmV/m 20
10
C.C.I.R. Documents, Lisbon (vol. 1, Propositions, p. 1294), in which the measured attenuation is shown to agree closely with the calculated value. The theoretical gain
lmV/m 0
-10
-20
-30
-40'
4000
2000
1000
0 W{20 | 30 40 50 60 | 70 80 90 | 100 . 0 10 50 Kilometres 200 500
0| 20
Fig. 18.—Results obtained over land; A = 4m, /c = 5, a = 1O~1S, 1 kW radiated.
lOmV/m 20db
0 50 100 150 200 250 3.00 3
4000
0 10 50 200 500 1000 2000 13000
Kilometres 1500 2500 3500 Fig. 19.—Results obtained over land; A = 4 m , K = 5, a = 10~15, 1 kW radiated.
between 5 and 25 miles and correctly explained as a consequence of Somtnerfeld's theory. Other experimental confirmations of these diffraction curves are given in
• B. TREVOR and P. S. CARTER: Proceedings of t he Institute of Radio Engineers, 1933, vol. 2i, p. 387; also C. R. BURROWS, L. E . HUNT, and A. DKCINO: Bell SysUm Technical jourmi, 1935, vol. 14, p. 253.
in decibels as a function of the height has also been
checked against the experimental autogyro results
obtained by Trevor and Carter and by Jones, and good agreement found. The above experimental checks COnstitute a considerable Dody oi eviaence tnat, although


52 ECKERSLEY: ULTRA-SHORT-WAVE REFRACTION AND DIFFRACTION
fading at extreme distances may imply a small degree extreme distances, which the author considers can only of refraction, the major part in ultra-short-wave trans- be caused by variation in the gradient of the refractive mission is played by diffraction. index of the air near the surface of the ground. The
30 db
lOmV/m 20 •'
10 '
lmV/m 0 •
-10 •
lOO^V/m-20 •
-30"
%V/m-40'
r 0 10 | 20 3.0 40 50 70 80 90 100
0 ID 50 Kilometres 200 5(J0
Fig. 20.—Results obtained over land; A = 6 m , K = 5, a = 10~13, 1 kW radiated.
4000 3500
3000
0 10 50 2'00 Kilometres 1000 2000 Fig. 21.—Results obtained over land; A = 6m, /c = 5, a = 10~13. 1 kW radiated.
REFRACTION alternative hypothesis that the variations are due to Although, as stated above, the major factor controlling reflections from the ionosphere seems very doubtful, for, ultra-short-wave propagation is diffraction, there is not although there is evidence of reflections at normal inciwanting evidence of variation of signal intensity at dence of waves well above the critical frequency, the


ECKERSLEY: ULTRA-SHORT-WAVE REFRACTION AND DIFFRACTION 53
reflections occur in sporadic bursts and are very unlike of the ultra-short-wave band. With the approach of the slow fadings observed in ultra-short-wave propaga- sun-spot maximum conditions, waves of this kind are tion. On the other hand the extreme ranges, recently likely to be propagated over long distances and so may
0 10 120 30 I 40 50 60 70 | 80 90 100
0 10 50 Kilometres 200 500
Fig. 22.—Results obtained over land; A = 8m, K = 5, a = 10"13, 1 kW radiated.
Uooo
3500
3000 . 2500
50 100 150 |2O;O "250 , TM _ , 
0 10 50 200 £00 I 1500,
Kilometres 1000 2000
Fig. 23.—Results obtained over land; A = 8 m, /c = 5, a = 10~13, 1 kW radiated.
recorded, of police-car signals on 9 metres from America, interfere with local reception on these waves, but regular and television signals from Germany, are evidence of long-distance working with such waves will be impossible reflection from the ionosphere at glancing incidence. on account of unreliability. Waves below 10 metres will These reflections are likely to occur at the upper limit usually be too close to the limit to give reliable working.


54 ECKERSLEY: ULTRA-SHORT-WAVE REFRACTION AND DIFFRACTION
This ultra-short-wave fading is of extreme importance the scope of such methods. If ray methods are used, from the practical point of view, for in general it sets a they require justification by a more complete wave limit to the range of a service long before the average analysis. Fortunately the wave method, by which
30db
10 •
0•
-10 •
lOO/Mn-20
-30 •
lO/tV/m-40" 10 20 30 1 40 501 60 b 10 50 Kilometres
70 180
200 90 100
Fig. 24.—Results obtained over land; A = 10 m, K = 5, a = 10 ", 1 kW radiated.
10mV/m 20(11)
0 50 100 150 20.0 ] 2 5 0 300 13501 '400
4000
3500
3000
2500
2000
Kilometres
0 10 5*0 200 1000
500 1500 Fig. 25.—Results obtained over land; A = 10 m, K = 5, a = 10"13, 1 kW radiated.
signals become too weak. Thus the study of ultra-short- the diffraction curves above were calculated, allows
wave propagation would be incomplete without an an easy extension to the case where the atmosphere analysis of the effect of refraction. This is in the main above the earth has a vertical gradient of refractive not an optical ray problem, for diffraction lies outside index.


ECKERSLEY: ULTRA-SHORT-WAVE REFRACTION AND DIFFRACTION 55
The field intensity beyond the visual distance, where fading is in evidence, can be expressed in the form
Xi
Ae
Approximate inverse
^ L distance
Visual range
Diffraction
Distance Fig. 26
where j/?n is a. constant determined in the 'diffraction
analysis, d is the distance, and RQ is the earth's radius.
The field intensity is mainly controlled by the expo
diffraction formula is now apparent. The circumstances are illustrated in Fig. 30(a), where TS is the earth's surface and TR is the tangent ray, curved downwards by the effect of the gradient of refractive index in the atmosphere above the earth's surface. If we now make a mathematical transformation such that any radius r becomes r[E1/(R1 — r)], the radius of curvature of TR,
where r = Rv becomes infinite, and that of the earth
RQR1I(R1 — RQ). In the transformed space the ray TR is
a straight line TJRJ [Fig. 30(6)], and the earth's radius is increased from Ro to RQR1I(R1 — RQ). The problem of propagation in the transformed space is now one in which there is no effective gradient of refractive index outside the earth (TJRJ being a straight line), and the solution is the same as that previously given, with RQ in place of RQ. Shelleng, Burrows, and
Ferrell have employed this method for determining the effect of refraction, using an argument similar to the above. It can now be seen that this argument is justified from the wave point of view. It should be noted, however, that the coefficient A should also be changed,
250
200
150
100
50
/
^*~
^^
56
A,,metres Fig. 27.—Values of Ro-R.
10
nential e~M. A is the amplitude factor, similar to the
0«5386A7
factor ^—. . - in Van der Pol's diffraction formula, and
A* sin*0 modified by a factor Y (< 1) which takes account of the earth constant. It is only a very slowly varying function of distance. Assuming a practically uniform gradient in the atmosphere, which would be a natural consequence of the upward decrease in molecular density in the atmosphere, it is found that the effect of this gradient is to modify the exponential coefficient by substituting an equivalent earth's radius RQ for Ro, where
7?' - P Bi
0 ~~ °R - i ?
but this change is small so long as R' is considerably
greater than Ro, as is usually the case. The change in
A cannot be allowed for by putting RQ in the place of Ro.
{a)
The distance i?,, which depends on the gradient of
refractive index (jn) being — cr-;, is readily interpreted
as the radius of curvature of the ray caused by the
vertical gradient of the refractive index. The physical meaning of the modification to the
\\\\\\\\Ng\\\\\\\\\\\\\\ Fig. 28
In (a), Tx has no effective visual range. X is expressed in metres.
One point is significant in the analysis: it appears that it is mainly, if not entirely, the gradient of the refractive index in the first few hundred metres which


56 ECKERSLEY: ULTRA-SHORT-WAVE REFRACTION AND DIFFRACTION
30db
20
10
0
-20
-30
0 10 20 30 40 50 60 70 80 90 100 110 Kilometres Fig. 29.—Continuity of reciprocal and diffraction values: results obtained over land, A = 10 m, h — 500 in.
\\\\
\
\
\\\ \ s \
S N
s\\i\
Reciprocal theorem. Diffraction theory.
counts, at least as far as transmission from one point on the ground to another on the ground is concerned. The magnitude of the refraction effects depends on the gradient of the refractive index of the atmosphere. The refractive index of air is well known, and is proportional to the density. Since the density decreases upwards in the atmosphere there is a vertical gradient of fx which can be computed when the pressure gradient T
R
10
1
01
Ri Ray
Fig. 30
(a) Actual space. (b) T f d
Actual space.
Transformed space.
is known. Water vapour has a high refractive index, and the effect of a small percentage is very pronounced.* The relative refractive indices of air and water vapour are given in the following table:
Temperature /u (air) ;u (water vapour) 45° C. 1-000574 1-01033 63° C. 1-000554 1-00965 83° C. 1-000534 1-00898
• The relevant data can be found in a paper by C. R. ENGLUND, A. B. CRAW
FORD, and W. W. MUMFORD: "Further Results of a Study of Ultra-Short
Wave Transmission Phenomena," Bell System Technical Journal. 1935, vol. 14,
p. 883. Further data may be obtained from Humphrey's " Physics of the
Air," and from an article on " The Thermodynamics of the Atmosphere " in
the " Dictionary of Applied Physics " (vol. 3, p. 44). The diagram on p. 61
of the latter is of particular interest, as it shows the large increase of saturated
water-vapour density in summer as compared with winter.
001
\\\]
I
V)
D
i \
\
:A
\\
(k\\\A
\\
i
20 40 60 80
Kilometres 100
Fig. 31.—Ultra-short-wave diffraction. Results obtained by
Jones at 61 megacycles per sec, in transmission from
Empire State Building, New York.
The diffraction curve is adjusted to fit at A.
x x Observed field intensity.
o o Calculated relative field intensity.
The difference of fx from 1 in water vapour is nearly 20 times as great as that for air, so that the effect of even a small percentage of water vapour is very considerable.


ECKERSLEY: ULTRA-SHORT-WAVE REFRACTION AND DIFFRACTION 57
Englund, Crawford, and Mumford have given the following formula for calculating R1:
R, M X 106
31 186
where M is the molecular weight of the gas involved, a the percentage of water vapour, r the absolute temperature, and h the height. A typical numerical example has been worked out for average conditions on the surface of the earth, with 1-37 per cent water vapour, giving R' = 23 000 km. In the absence of the 1 • 37 per cent of water vapour, R' would be 39 000 km. In the former case the increase in range at which a given field intensity occurs is
23 000
23 000 - 6 360 = 1-274
£0 100 150 200 250
Distance from Chelmsford, km 300
Fig. 32.—Aeroplane test. Results obtained on wavelength
of 53-4 m, 1st October, 1935.
i.e a 27*4 per cent increase. In the latter case the increase is [39 000/(39 000 — 6 360)]*, i.e. 14-2 per cent. These considerations, taking into account the considerable increase in water-vapour density in the summer, give a very plausible explanation of the now well-established result that the signal-intensity values well beyond the visual distance are much greater in summer than in winter. The effect of refraction is to increase the range, especially when, for instance, there are temperature inversions at the surface of the earth. The varying refractive index of the atmosphere is
probably the cause of the fading observed at extreme distances, which limits the reliable range. A given percentage variation in the gradient will produce a much greater percentage effect where the total attenuation is large than where it is small. Serious fading will therefore set in at ranges where the total attenuation is large. It must be left to observation to determine how large, but from the limited results obtained by Englund, Crawford, and Mumford it is suggested that
10
0-4 03
0-2
0-09 007
005
01 0-08 ,006
004 003
0-02
001
I \ \ \ \
\
<A
M
\ \w\\\\\\\\
1
\
••^
V
\
\
\s
\
A.
S
X
\\
0 100 200 300 400
Kilometres 500 600
Fig. 33.—Aeroplane test. Results obtained on wavelength
of 950 m.
— — — — Diffraction curve.
_ _ _ ^ ^ — Observed values (Fassbender, Eisner, and Kurlbaum).
_ . — - Sommerfeld curve.
an attenuation of about 30 db below the " free-space field " or inverse distance is reasonable.
EXPERIMENTAL CHECKS OF DIFFRACTION
FORMULA FOR VARIOUS WAVELENGTHS It is very desirable to check, experimentally, the theoretical conclusions discussed above, if only for the purpose of finding out whether there are important factors in ultra-short-wave transmission, e.g. loweratmosphere refractions, other than those taken into account in the main analysis.


58 ECKERSLEY: ULTRA-SHORT-WAVE REFRACTION AND DIFFRACTION
Ii
*^
s
X
Sx
X \
>y X
X 11
X <—*s
0-5
0-2
0-10 100 200 300 400 500 600 700
Distance, km
Fig. 34.—Daylight field strength of Daventry 5XX. Results
obtained at midday, July, 1931; A = 1 550 m; normal
power (25 kW).
20
10
o> 5
a*
13
P-i 2
0-5
0-2
01
V
—• \
\V
N\ s
\V
\
v.
\\ s
) \
X
vj 1 f\,r v1/l
,{T
\N
-13 s
Klfl*
.\A
-13
100 200 300 400 500 600 700
Distance, km
Fig. 35.—Results obtained on Warsaw; A = 1 400 m,
100 kW radiated.
A f x Observed results.
\ O Modified diffraction formula, a- — 1-15 x 10~13.
B - . i - . - Original Watson diffraction formula.
Sommerfeld formula.
In choosing the quantities to be compared with experimental observations it is desirable to leave out those which depend on the absolute amount of power radiated by the transmitter, since, in general, there is considerable difficulty in determining this quantity accurately. Relative, but not absolute, intensities can be checked fairly well. For the purposes of comparison use is made of the slope of the curve relating the logarithm of the field intensity to the distance, to compare with the calculated slope. Experimental data can be obtained from American results.* An example is shown here (Fig. 31) in which AAX is
the average curve of a large amount of data obtained by Jones in experimental transmissions from the Empire State Building, New York, on a frequency of 61 Me. The observed rate of attenuation (or slope) is some 14 per cent less than the calculated; this difference may
50
40
3 30
O 20
10
/
(
y
20 40 60 SO 100 120
Fig. 36.—Height/gain observations. Results obtained by
Jones at 44 megacycles per sec, in transmission to
distance of 102 km from Empire State Building, New
York. Height of transmitting aerial, 396 m.
x Calculated relative gain.
o Observed relative gain.
be due to atmospheric refraction. In the case of a normal atmospheric distribution of air density, when the effect of water vapour is neglected, the attenuation slope should be reduced approximately 15 per cent. Another example is shown in Fig. 32, which represents the ground-ray intensity from a 53*4-metre transmitter in an aeroplane flight from Croydon to Cologne. Beyond 150 km, the visual range, the slope of the log (field)/distance curve is constant and agrees closely with the slope calculated according to the diffraction theory. The measurement of the ground ray at these distances, where the reflected ray was generally predominant, was made possible by means of the impulse technique, in which the ground ray was separated from the reflected ray by an interval of about 2 milliseconds. To illustrate the scope of the diffraction theory the following examples on medium waves are given. Fig. 33 gives an example for 950 metres, and consists of field measurements of medium-wave transmissions
* Proceedings of the Institute of Radio Engineers, 1933, vol. 21, pp. 349- 464.


ECKERSLEY: ULTRA-SHORT-WAVE REFRACTION AND DIFFRACTION 59
from an aeroplane. The experimental curve is com- better agreement with the results than the Sommerfeld
pared with (a) the diffraction curve (CT = 7-5 X 10~14), curve. The scattered observations are probably caused
and (b) the Sommerfeld curve (a = 7-1 x 10-14). The agreement with the former is very good.
by the hilly country in which the measurements were taken, i.e. North England and Scotland. Finally, the
6or
40
3*30
o 20
10
0 20 40 60 80 1002 120 140 160 180 200
Fig. 37.—Height/gain observations. Results obtained by Trevor and Carter at 34 megacycles per sec, in transmission to
distance of 59*6 km from Rocky Point. Height of transmitting aerial, 39 m.
x Calculated relative gain,
o Observed relative gain.
70
60
50
^40
*OcS30 /
/
1:
ft*
20
10
0 50 100 150 200 250 300 350 ^00 450
Fig. 38.—Height/gain observations. Results obtained by Trevor and Carter at 44 megacycles per sec, in transmission to distance of 185 km from Empire State Building, New Yoak.
x Calculated relative gain.
o Observed relative gain.
In Figs. 34 and 35, measured values of Daventry and Warsaw are compared with the diffraction values. In the case of Daventry the points fall rather irregularly, but there is no doubt that the diffraction curve is in
Warsaw results (Fig. 35) show very good agreement with the diffraction values. The country is flat, and the observed curve is a smooth one agreeing closely with
the calculated curve for a = 1-15 X 10~13.


60 ECKERSLEY: ULTRA-SHORT-WAVE REFRACTION AND DIFFRACTION
With regard to the examples shown in Figs. 31 and 32, it should be emphasized that the calculated results do not involve any arbitrarily-assumed earth conductivity or permittivity. The calculated attenuation depends only on the wavelength (and, of course, the earth's radius) so long as the wavelength is so short that
a*A* < 10-7
Even with sea water this condition is satisfied, and in every case of overland transmission the quantity criA^ should be well below the limiting value. The reason for this is that in the diffraction attenuation factor e-kdj\i the parameter fc approaches a definite limit A^
when the frequency is high enough, and fc2 when the
frequency is low enough. Perhaps the best quantity to compare with experiment is the gain in signal strength with height above the earth in regions beyond the visual range. This gain of signal intensity with height has been measured by many observers in America, using either aeroplanes or autogyros. Examples of the results are given in Figs. 36,
37, and 38, which show that the calculated gains agree with the theoretical gains within a few decibels over quite a considerable range of height. The gains are plotted as a function of h\i, since the theory shows that they should be a function of this and this only. Thus it is again possible to check the theory without invoking the help of any arbitrarily-estimated earth constants. Although in the main agreement of observation and theory is very fair, individual cases of wide divergence have been recorded. Englund, Crawford, and Mumford have given an example where there was practically no gain with height well beyond the visual range. Again, they have found occasions where the field intensity just beyond the visual range has varied by as much as 20 db from one day to another. Such effects are no doubt due to refraction, which may on occasion produce profound variations, but if the present author's reading of the average results is correct such occasions are rather the exception than the rule. An estimate of the effect of atmospheric refraction can be obtained by comparing the observed results with the calculated ones.