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WAVES AND RIPPLES

A CHRISTMAS LECTURE AT THE ROYAL INSTITUTION : " WAVES AND EIPPLES IN THE AIR."

Draivn by F, C. Dickinson.] FiG. 4(J (see p. 109).

[Fnnn

the

" Graphic.'

WAVES AND EIPPLES
IN'
WATER, AIR, AND JETHKli
r.KINC
A COURSE 01' CHKIST.MAS LKC'ITIIKS I >KU VKIIKI ) AT THE ROYAL INSTITUTION OF CHEAT BRITAIN

HY

J.

A.

FLEMING,

M.A.,

D.Sc ,

F.RS.

M. INST. E.E., M.R.I., ETC., ETC.

I'UOFESSOU OF ELECTUICAL ENGINKKRING IN UNIVEKSITV COLLEGK, LONDON

PUBLISHED UNDEK THE DIRECTION OF THE GENEUAL LITERATURE COMMITTEE

LONDON
SOCIETY FOR PROMOTING CHRISTIAN KNOWLEDGE
NORTHUMBERLAND AVENUE, W.C.
43, QUEEN VICTORIA STEEET, E.G.
BRIGHTON: 129, NORTH STREET
NEW YORK: E. & J. B. YOUNG AND CO.
1902

PRINTED BY WILLIAM CLOWES AND SONS, LIMITED,
LONDON AND BECCLES.

PREFACE.

THE Christmas Lectures at the Royal Institution are,

by a time-honoured custom, invariably addressed to a

"juvenile audience." This term, however, has always
been held to be an elastic one, and to include those who

are young in spirit as well as those who are young in

years. The conditions, therefore, necessarily impose on the Lecturer the duty of treating some subject in such

a manner that, whilst not beyond the reach of youthful

minds, it may yet possess some elements of interest for

A those of maturer years.

subject which admits of abun-

dant experimental illustrations is accordingly, on these

occasions, a popular one, particularly if it has a bearing
upon topics then attracting public attention. The pro-

gress of practical invention or discovery often removes

at one stroke some fact or principle out of the region of

purely scientific investigation, and places it within the

A purview of the popular mind.

demand then arises

for explanations which shall dovetail it on to the

ordinary experiences of life. The practical use of rether

viii

PREFACE.

waves in wireless telegraphy has thus made the subject of waves in general an interesting one. Hence, when
permitted the privilege, for a second time, of addressing Christmas audiences in the Eoyal Institution, the author ventured to indulge the hope that an experimental
treatment of the subject of Waves and Ripples in various media would not be wanting in interest. Although such lectures, when reproduced in print, are destitute of the attractions furnished by successful experiments, yet, in response to the wish of many correspondents, they have
been committed to writing, in the hope that the explana-
tions given may still be useful to a circle of readers. The author trusts that the attempt to make the operations
of visible waves a key to a comprehension of some of
the effects produced by waves of an invisible kind may not be altogether without success, and that those who
lincl some of the imperfect expositions in this little book

in any degree helpful may thereby be impelled to study

the

facts

more

closely

from

that

"
open

page

of Nature "

which lies ever unfolded for the instruction of those who

have the patience and power to read it aright. J. A. F.

UNIVERSITY COLLEGE, LONDON, 1902.

CONTENTS.

CHAPTER I.

WATKlt WAVES AND WATER RIPPLES.

A visit to the seaside What is a wave ? Wave-motion on water

Definition of a wave Sea waves Various forms of wax. -

motion Wave length, velocity, and frequency Atlantic \\

I lules for speed of sea waves Illustrations of wave-motion
A stone falling on water Production of a wave-train Wav-

,. i tergy Conditions for the production of wave-motion

Distinction between wave-velocity and wave-train \< 1 -iiy

Why a wave breaks Waves in canals Rule for speed of a

A eaual wave Falling bodies

" bore " Tidal waves

Ripples Distinction between waves and ripples Surface

A tension on liquids

needle floating on water Experimental

production of ripples Reflection and refraction of ripples and waves Interference of waves and ripples Photography of

waves ami ripples

...

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nua
1

CHAPTER II.
WAVES AND RIPPLES MADE BY SHIPS.
Ship-waves The viscosity of liquids How it is demonstrated
Rotational and irrotatioual motion in fluids Kddies and whirls Smoke rings Vortex motion Professor Hele-Shaw's experi-
ments Irrotatioual or stream-line motion in water The mot ion of water round a ship The motion of water along a pipe Flow in uniform pipes and non-uniform pipes Relation between

CONTENTS.

PAGE

fluid velocity and pressure Skin resistance and wave-making resistance The movement of a fish Motion through a perfect fluid The waves made by moving objects Waves made by ducks and swans Echelon waves Ship bow waves The form of ship-waves Mr. Froude's experiments Ship-models and
experimental tanks How a ship is designed Froude's laws-
Testing ship-models The design of a racing-yacht Comparison of British and American yachts The Cup race Scott

Russell's experiments on canal-boats

...

...

...

57

CHAPTER III.

WAVES AND RIPPLES IN THE AIR.

A Air necessary for the production of sound

sounding body is in

vibration Harmonic motion The difference between noise and

music The nature of an air wave The physical qualities of

air Longitudinal or compressional waves Wave-models to illustrate the nature of sound waves Quality of a sound-
Velocity of an air wave An illustration on a gigantic scale
The voice of a volcano heard round the world The effect of

temperature on air-wave velocity Comparison of theory and

experiment Circumstances affecting distance at which sounds

can be heard Funeral guns Fog-signals and sirens Effect

of wind and density Sensitive flames as sound-detectors

Inaudible sounds The reflection and refraction of sound

A waves

sound-lens and sound-prism The interference of

sounds Two sounds producing silence The phonograph A

soap-bubble film set in vibration by air waves

...

... 103

CHAPTER IV.

SOUND AND MUSIC.

The difference between sounds and musical tones The natural

period of vibration of an elastic body The effect of accumulated

impulses Free and forced vibrations Breaking down a bridge

with a pea-shooter The vibration of a stretched string-

A Stationary waves

string vibrating in segments Acoustic

resonance Nodes and anti-nodes The musical scale or gamut

Musical intervals The natural gamuts and the scale of

equal temperament Concords and discords Musical beats

CONTENTS.

xi

PAM

IMmholtz'a theory of discord* Musical instrument* Pipc
Strings and plates A pan-pipe An organ-pipe Opm und
closed organ-pipes The distribution of uir pressure and velocity

in a sounding organ-pipe Singing flames Stringed instru-
mentsThe violin The Stroh violin The structure of the

ear The ear a wonderful :tir- \\a\v d-U-etor and analyzer

...

1 17

CHAPTER V.

ELECTRIC OSCILLATIONS AND ELECTRIC WAVES.

The conception of an tether The phenomena of light require the assumption of an aether The velocity of light Interference
of light Two rays of light can produce darkness An electric

current The phenomena of electricity require the assumption

of an electro-magnetic medium Properties and powers of an

electric current Alternating and continuous electric currents

A Electromotive force and electric strain

Leyden jar The

oscillatory discharge of a condenser Oscillatory sparks

Transformation of electric oscillations Hertz oscillator

Production of a wave of electric displacement Detection of

electric waves Metallic filings detectors The coherer-

Inductance and capacity of circuits Electro-static and electro-

magnetic energy An induction coil Electric oscillations give

rise to electric waves The electron theory of electricity

... 185

CHAPTER VI.

WAVES AND RIPPLES IN THE JETHER.

The experiments of Heinrich Hertz Electric radiation Lecture

apparatus for producing and detecting electric radiation

Why Electric transparency and opacity

this difference The

reflection of electric radiation The refraction of electric rays

An electric prism and an electric lens The electric refractive

index Interference of electric rays The velocity of electric

radiations identical with that of light Dark heat rays-

Actinic or photographic rays The cause of colour The

frequency of light waves The classification of electric or

sother waves The gamut of aether waves The eye an aether-

wave detector of limited power The electro-magnetic theory

xii

CONTENTS.

of light Artificial production of lightUse of Hertz waves in

wireless telegraphy Marconi's methods Marconi's aerial and
wave-detector The Morse alphabet How a wireless message

is sent The tuning of wireless stations Communication

between ships and shore The velocity of wireless waves

Conclusion

...

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...

...

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PAGE 223

APPENDIX ...

...

...

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...

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... 287

INDEX

293

USEFUL MEMOEANDA.

One statute mile is 5280 feet.
= One nautical mile is 608G feet 1 statute mile.
A knot is a speed of 1 nautical mile per hour.

To convert

Hence the following rules :

Knots to miles per hour multiply by 1. Miles per hour to knots multiply by f. Feet per second to miles per hour multiply by I Feet per second to knots multiply by ^5. Knots to feet per minute multiply by 100.

WAVES AND RIPPLES IN WATER, AIR, AND AETHER.

CHAPTER I.
WATER WAV1.S AND WATER RIPPLES.

WK have all stood many times by the seashore, watching the wares, crested with white foam, roll in and

break upon the rocks or beach. Every one \i-.\<

more than unce cast a stone upon still water in a lake or

] -nd, and noticed the expanding rings of ripples; and

some have voyaged over stormy seas, whereon great ships

arc tossed by mighty billows with no more seeming effort

than the rocking of a cradle. In all these things we h

been spectators of a wave-motion, as it is called, taking

place
I

upon

a

water

surface.

Perhaps it did not occur to

us at the time that the sound of the splash or thunder of

these breaking waves was conveyed to our ears as a wave-

motion of another sort in the air we breathe, nay, even

that the light by which we see these beautiful objects is

also a wave-motion of a more recondite description, pro-
duced in a medium called the cether, which fills all space.
A progressive study of Nature has shown us that we

are surrounded on all sides by wave-motions of various

descriptions waves in water, waves in air, and \\avc< in

B

V.

2

WAVES AND RIPPLES.

aether and that our most precious senses, our eyes and

ears, are really wave-detectors of a very special form.

The examination of these waves and their properties and

powers has led us to see that waves in water, air, and

aether, though differing greatly in detail, have much in

common ;

and many things

about

them

that

are

difficult

to understand become more intelligible when we compare

these various . wave-motions together. In these lectures,

therefore, I shall make use of your familiar experiences

concerning sea and water waves to assist you to understand some of the properties of air waves to which we owe our sensations of sound and music; and, as far as

possible, attempt an explanation of the nature of aether waves, created in the all-pervading aether, to which are

due not only light and sight, but also many electrical
effects, including such modern wonders as wireless telegraphy. In all departments of natural science we find ourselves confronted by the phenomena of wave-motion.

In the study of earthquakes and tides, telegraphs and telephones, as well as terrestrial temperature, no less than in the examination of water waves and ripples, sound, music, or light and heat, we are bound to consider waves

of some particular kind. Fastening our attention for the moment on surface
water waves, the first question we shall ask ourselves is What is a wave ? If we take our station on a high cliff looking down on the sea, on some clear day, when the wind is fresh, we see the waves on its surface like green
rounded ridges racing forward, and it appears at first sight as if these elevations were themselves moving masses of
water. If, however, we look instead at some patch of

seaweed, or floating cork, or seagull, as each wave passes

over it, we shall notice that this object is merely lifted up

WATER WAVES AND WATER RIPPLES.

3

and let down a.L'ain, or, at most, has a small movement to

We and fro.

are led, therefore, to infer that, even when

agitated l.y \\a\es, each particle of water never moves far

1 1 uiu its position when at rest, and that tin; real move-

ment of the water is something very different from its

apparent motion. If we place on the surface of water a

number <>f coiks or pieces of paper, and then watch them

as a wave passes over them, we shall notice that the corks

or bits of paper rise and fall successively, that is, one

A after the other, and not all together.

little more

careful scrutiny will show us that, in the case of sea waves

in deep water, the motion of the floating object as the wave

passes over it is a circular one, that is to say, it is first

lifted u] >, then pushed forward, next let down, and, lastly,

mlled
]

hack ;

and so it repeats a round-and-round motion,

with the plane of the circle in the direction in which the

wave is progressing. This may be illustrated by the

diagram in Fig. 1, where the circular dotted lines repre-

FIG.

^

sent the paths described by (forks floating on the sea-

surface when waves are travelling over it.

Accordingly, we conclude that we have to distinguish

clearly between the actual individual motion of each wain

(article and that
]

general

motion

called

the wave-motion.

We may define the latter by saying that to produce a

wave-motion, each separate particle of a medium, be it

water, or air, or any other fluid, must execute a movement

which is repeated again and again, and the several particles

4

WAVES AND HIPPIES.

along any line must perform this same motion one after

the other, that is, lagging behind each other, and not

We simultaneously.

might illustrate this performance by

supposing a row of fifty boys to stand in a line in a play-

ground, and each boy in turn to lift up his arm and let it

down again, and to continue to perform this action. If

all the boys lifted up their arms together, that would not

produce

a

wave-motion ;

but

if

each

boy did it one after

the other in order, along the rank, it would constitute a

wave-motion travelling along the line of boys. In more
learned language, we may define a wave-motion by saying

that a wave-motion exists in any medium when the sepa-

rate portions of it along any line execute in order any kind

of cyclical or repeated motion, the particles along this line

performing the movement one after the other, and with a

certain assigned delay between each adjacent particle as

regards their stage in the movement.

It will be evident, therefore, that there can be many

different kinds of waves, depending upon the sort of

repeated motion the several parts perform.

Some of the numerous forms of wave-motion can be

illustrated by mechanical models as follows :
A board has fastened to it a series of wooden wheels,
and on the edge of each wheel is fixed a white knob. The
wheels are connected together by endless bands, so that on turning one wheel round they all revolve in the same direction. If the knobs are so arranged to begin with, that each one irf a little in advance of its neighbour on the way round the wheel, then when the wheels are standing still the knobs will be arranged along a wavy line (see
Fig. 2). On turning round the first wheel, each knob will move in a circle, but every knob will be lagging a little
behind its neighbour on one side, and a little in advance

WATER WAVES AND WATER RIPPLES.

5

of its neighbour on the other side. The result will be to produce a wave-motion, and, looking at the general eil'ect

KM;. 2.
of i lie moving knobs, we shall see thai it resembles u hump moving along, just as in the case of a water wave.
The motion of the particles of the water in a deep-sea wave resembles that of the white knobs in the model

FIG. 3.
described. Those who swim will recall to mind their sensations as a sea wave surges over them. The wave

6

WAVES AND EIPPLES.

lifts up the swimmer, then pushes him a little forward, then lets him down, and, lastly, drags him back. It is this dragging-back action which is so dangerous to persons who cannot swim, when they are bathing on a steep coast where strong waves are rolling in towards the shore.
Two other kinds of wave-motion may be illustrated by
the model shown in Fig. 3. In this appliance there are a number of eccentric wheels fixed to a shaft. Each wheel
is embraced by a band carrying a long rod which ends in a white ball. The wheels are so placed on the shaft that, when at rest, the balls are arranged in a wavy line. Then, on turning round the shaft, each ball rises and falls in a vertical line, and executes a periodic motion, lagging behind that of its neighbour on one side. The result is
to produce a wave-motion along the line of balls. By slightly altering the model, each ball can be made to
describe a circle in a direction at right angles to the line
of the balls, and then we have a sort of corkscrew wave-
motion propagated along the line of balls.
Again, another form of wave-motion may be illustrated by the model shown in Fig. 4. In this case a number of

FIG. -i.
golf-balls are hung up by strings, and spiral brass springs
are interposed between each ball. On giving a slight tap to the end ball, we notice that its to-and-fro motion is handed on from ball to ball, and we have a wave-motion

WATER WAVES AND WATKK HIl'PLi .

7

in which the individual movement of the balls is in the

direction of the wave-movement, and not across it.

The kind of wave illustrated by the model in Fig. 3 is called a transverse wave, and that shown in Fig. 4 is callol

a lonyitinlinul wave.

At this stage it may be well to define the meaning of

some other expressions which will be much used in these

We lectures.

have seen that in a wave-motion each part

of the medium makes some kind of movement over and

over again; and of its neighbours on either side, one is a little ahead of it in its performance, and the other a little
in arrear. If we look along the line, we shall see that we can select portions of it which are exactly in the same si ,140 of movement that is, are moving in the same way at the same time. The distance between these portions is
called one wave-length. Thus, in the case of sea waves, the distance between two adjacent crests, or humps, is one

wave-length.
When we use the expression, a long wave, we do not
mean a wave which is of great length in the direction of the ridge, but waves in which the crests, or humps, are
separated far apart, measuring from crest to crest across the ridges.
Strictly speaking, the wave-length may be denned as
the shortest distance from crest to crest, or hollow to
hollow, or from one particle to the next one which is in the same stage of its movement at the same time.
Another way of illustrating the same thing would be
to pleat or pucker a sheet of paper into parallel ridges.
If we make these pleats very narrow, they would represent what we call short ivavcs ; but if we make these pleats very far apart, they would represent long waves.
Another phrase much used is the term ware-velocity.

8

WAVES AND RIPPLES.

Suppose that a seagull were to fly along over a set of sea

waves so as to keep always above one particular hump, or

wave-crest ;

the

speed

of

the

gull, reckoned

in miles per

hour or feet per minute, would be called the speed of the

waves. This is something very different from the actual

speed of each particle of water.
A third and constantly used expression is the term

wave-frequency. If we watch a cork floating on a wavetossed sea, we observe that it bobs up and down so many

times in a minute. The number of times per second or

per minute that each particle of the medium performs its

cycle of motion is called the wave-frequency, or simply

the frequency.
Again, we employ the term amplitude to denote the
extreme distance that each individual particle of the
medium moves from its mean position, or position of rest. In speaking of sea waves, we generally call the vertical
distance between the crest and the hollow the height of the wave, and this is twice the amplitude. With regard
to the height of sea waves, there is generally much exag-
geration. Voyagers are in the habit of speaking of " waves running mountains high," yet a sea wave which exceeds
40 feet in height is a rare sight. Waves have been
measured on the Southern Indian Ocean, between the
Cape of Good Hope and the Island of St. Paul, and of thirty waves observed the average height was found to be just under 30 feet. The highest was only 37J feet in height. On the other hand, waves of 16 to 20 feet are not uncommon. Travellers who have crossed the Atlantic

Ocean in stormy weather will often recount experiences of waves said to be 100 feet high ; but these are exceedingly rare, if even ever met with, and unless wave-heights are obtained by some accurate method of measurement,

WATER WAVES AND WATER RIPPLES.

9

the eye of the inexperienced voyager is apt to be de-
ceived.
In all cases of wave-motion there is a very close connection between the wave-velocity, or speed, the wavelength, and the wave-frequency. This connection is expressed by the numerical law that the velocity is equal to the product of the length and the frequency.
Thus, supposing we consider the case of Atlantic
waves 300 feet from crest to crest, which are travelling at the rate of 27 miles an hour, it is required to calculate the frequency or number of times per minute or per second that any floating object, say a boat, will be lifted up as these waves pass over it.
We must h'rst transform a speed of 27 miles per hour
into its equivalent in feet per second. Since one mile is 5280 feet, 27 miles per hour is equal to 2376 feet per minute. Accordingly, it is easy to see that the wave- frequency must be 7'92, or nearly 8, because 7'92 times 300 is 2376. The answer to the question is, then, that the floating object will rise and fall eight times a minute.
This rule may be embodied in a compact form, which it is
desirable to hold firmly in the memory, viz.
= Wave-velocity wave-length x wave-frequency.

This relation, which we shall have frequent occasion

We to recall, may be stated in another manner.

call the

period of a wave the time taken to make one complete

movement. The periodic time is therefore inversely pro-

portional to the frequency. Hence we can say that the

wave-length, divided by the periodic time, gives us the wave-

velocity.
In the case of water waves and ripples, the wavevelocity is determined by the wave-length. This is not

10

WAVES AND RIPPLES.

the case, as we shall see, with waves in air or waves in aether. In these latter cases, as far as we know, waves of

all wave-lengths travel at the same rate. Long sea waves,

however, on deep water travel faster than short ones.
A formal and exact proof of the law connecting speed

and wave-length for deep-sea waves requires mathematical

reasoning

of

an

advanced

character ;

but

its

results

may

be

expressed in a very simple statement, by saying that, in

the case of waves on deep water, the speed with which the

waves travel, reckoned in miles per hour, is equal to the

square root of 2^ times the wave-length measured in

feet. Thus, for instance, if we notice waves on a deep sea

which are 100 feet from crest to crest, then the speed with

which those waves are travelling, reckoned in miles per

hour, is a number obtained ,by taking the square root of

2 times 100, viz. 225. Since 15 is the square root of 225

(because 15 times 15 is 225), the speed of these waves is

therefore 15 miles an hour.

In the same way it can be found that Atlantic waves
300 feet long would travel at the rate of 26 miles an hour,
or as fast as a slow railway train, and much faster than

any ordinary ship.* The above rule for the speed of deep-sea waves, viz.
= wave-velocity square root of 2^ times the wave-length, = combined with the general rule, wave-velocity wave-length
multiplied ~by frequency, provides us with a useful practical
method of finding the speed of deep-sea waves which are
passing any fixed point. Suppose that a good way out at sea there is a fixed buoy or rock, and we notice waves

* The wave-velocity in the case of waves on deep water varies as

/ A

- , where A. is the wave-length. The rule in the text is deduced

from this formula.

WATER WAVES AND WATER RJPPLE8. 11

raring past it, and desire to know their speed, we may do it as follows : Count the number of waves which pass the

fixed point per minute, and divide the number into 198;

the quotient is the speed of the waves in miles per hour.

Thus, if ten waves per minute race past a fixed buoy, their

velocity is very nearly 20 miles an hour.*
Waves have been observed by the Challenger 420 to

480 feet long, with a period of 9 seconds. These waves were 18 to 22 feet high. Their speed was therefore

"'ii feet per second, or nearly 30 knots. Atlantic storm waves are very often 500 to 600 feet long, and have a
period of 10 to 11 seconds. Waves have been observed

by officers in the French Navy half a mile in length, and

Avitli a period of 23 seconds.

It has already been explained that in the case of deep-
sea waves the individual particles of water move in cir-

cular paths. It can be shown that the diameter of these

circular paths decreases very rapidly with the depth of

the particle below the surface, so that at a distance below

the surface equal only to one wave-length, the diameter of

I ho circle which is described by each water-particle is only

-
.';

of that

at

the

surface.f

Hence storm waves on the

sea are a purely surface effect. At a few hundred feet down a distance small compared with the depth of the ocean the water is quite still, even when the surface is

V V * If

is the velocity of the wave in feet per minute, and

is tlic

V V = v ^

^ velocity in miles per hour, then -

-

V. But

= V**A, i.nd

V = n\ t

where

A

is

the

wave-length

in

feet

and

n

the

frequency

per

V minute ; from which we have

= 1 qo
, or the rule given in the text.

7i

t The amplitude of disturbance of a particle of water at a depth equal

lo one wave-length is equal to - of its amplitude at the surface. Jjr

Lamb's

" Hydrodynamics,"

p.

189.)

(See

12

WAVES AND EIPPLES.

tossed by fearful storms, except in so far as there may be a steady movement due to ocean currents.
By a more elaborate examination of the propagation of wave-motion on a fluid, Sir George Stokes showed, many
years ago, that in addition to the circular motion of the

water-particles constituting the wave, there is also a
transfer of water in the direction in which the wave is

moving, the speed of this transfer depending on the depth, and decreasing rapidly as the depth increases. This effect,
which is known to sailors as the " heave of the sea," can
clearly be seen on watching waves on not very deep water. For the crest of the wave will be seen to advance

more rapidly than the hollow until the wave falls over

and breaks ;

and

then

a

fresh

wave

is

formed

behind it,

and the process is repeated. Hence waves break if the

depth of

water

under

them

diminishes ;

and

we know by

the presence of breakers at any place that some shallow

or sandbank is located there.

It is necessary, in the next place, to point out the difference between a mere wave-motion and a true wave.

It has been explained that in a wave-motion each one of a

series of contiguous objects executes some identical move-

We ment in turn.

have all seen the wind blowing on a

breezy day across a cornfield, and producing a sort of dark

shadow which sweeps along the field. This is clearly

caused by the wind bending down, in turn, each row of

cornstalks, and as row after row bows itself and springs

up again, we are presented with the appearance of a

wave-motion in the form of a rift rushing across the

field.
A very similar effect can be produced, and another
illustration given of a wave-motion, as follows : Coil a piece of brass wire into an open spiral like a corkscrew,

WATER WAVES AND WATER > RIPl'l.1

13

and affix to it a small fragment of sealing-wax (see Fig. 5).

Hold this in the sun, and let the shadow of it fall upon

We paper. Then turn it round like a screw.

shall see

that the shadow of the spiral is a wavy line, and that, as

FIG. 5.
it is turned round, the humps appear to move along just as do the crests of sea waves, but that the shadow of the little bit of sealing-wax simply moves up and down. Another wave-motion model may be made as follows :
Procure a painter 's comb. This is a thin steel plate, cut into long narrow teeth. Provide also a slip of glass about 3 inches wide and 12 inches long. Paint one side of
this glass with black enamel varnish, and when it is quite dry scratch a wavy line upon it (see Fig. 6). Place the

FIG. 6.
glass slip close in front of the comb before the light, and, holding the comb still, move the glass slip to and fro, lengthways. The observer will see a row of dots of light lying in a wavy line, and these, as the glass moves, will

14

WAVES AND EIPPLE8.

rise and fall. If the movement is rapid enough, the appearance of a wave moving along will be seen.* In all these exhibitions of wave-motion the movement of the
particles is due to a common cause, but the moving par-
ticles do not control each other's motion. There is no
connection or tie between them. Suppose, however, that
we suspend a series of heavy balls like pendulums, and interconnect them by elastic threads (see Fig. 7), then we

FIG. 7.

have an arrangement along which we can propagate a true wave. Draw the end ball to one side, and notice what

takes place when it is released. The first ball, being Clioplaced, pulls the second one through a less distance, and that the third one, and the third the fourth, and so on.

This happens because the balls are tied together by elastic

When threads, which resist stretching.

the first ball is

released, it is pulled back by the tension of the thread connecting it to its neighbours, and it begins to return to its old position. The ball possesses, however, a quality called inertia, and accordingly, when once set in motion,

its motion persists until an opposing force brings it to rest. Hence the returning ball overshoots the mark, and

passes to the opposite side of its original position of rest.

* This can easily be shown to an audience by projecting the apparatus on a screen by the aid of an optical lantern.

WATER WAVES AND WATER RIPPLES.

1."

Then, again, this displacement stretches the elastic threads

connecting it to its fellows, and a controlling or retarding

force is thus created, which brings it to rest, and forces it

We infill to return on its steps.

see, therefore, that each

ball must oscillate, or swing to and fro, and that its move-
A ment is gradually communicated to its neighbours.

wave-motion is thus started, and a true wave is propa-

gated along the line of balls, in consequence of the

presence of elasticity and inertia. The necessary condi-

tions for the production of a true wave in a medium of any kind are therefore : (1) that the medium must elasti-

cally resist some

sort

of

deformation ;

and (2) when

it

is

deformed at any place, and returns to its original state, it must overshoot the mark or persist in movement, in con-

sequence of inertia, or something equivalent to it.

Briefly speaking, any material or medium in or 011

which a true self-propagating wave-motion can be made

must resist and persist. It must have an elastic resistance

to some change or deformation, and it must have an inertia

which causes it to persist in movement when once set

in motion. These two qualities, or others equivalent to

them, must invariably be present if we are to have a true

wave produced in a medium.

These things may be best understood by considering,

for example, the production of surface waves on water.

Let us ask ourselves, in the first place, what alteration or

change it is that a water-surface resists. The answer is,

A that, for one thing, it resists being made unlevel.

still

water surface is everywhere a level surface. If we attempt to make it unlevel by pouring water on to it at one point,

or by heaping it up, the water surface would resist this

We process.

can dig a hole in sand, or heap up sand to

form a hillock, but we know full well we cannot do the

16

WAVES AND RIPPLES.

same thing with water. If, for instance, some water is

placed in a glass tube shaped like the letter U, then it

stands at the same level in both limbs. Again, if water

is set in motion, being a heavy substance, it cannot be

brought to rest instantly. Like every other body, it pos-
sesses inertia. Accordingly, if we do succeed by any

means in making a depression in a water-surface for an

instant, the water would immediately press in to fill up

the hole ;

but

more, it

would,

so

to

speak, overshoot

the

mark, and, in consequence of its inertia, it would create a

momentary hump, or elevation, in the place on the surface

where an instant ago there was a depression.

This elevation would again subside into a hollow, and

the process would be continued until the water-motion

was brought to rest by friction, or by the gradual disper-

sion of the original energy. The process by which a wave

is started on the surface of water, as a consequence of

these two qualities of resistance to being made unlevel

and persistence in motion, is beautifully shown by the study of waves made by throwing stones into a pond.

The events which give rise to the expanding wave are,

however, over so quickly that they can only be studied by

the aid of instantaneous photography. The most interest-

ing work on this subject is that of Professor A. M. Worthiugton, who has photographed, by the exceedingly brief

light of an electric spark, the various stages of the events
which happen when a drop of water or a stone falls into

water.* These photographs show us all that happens when

the falling object touches the water, and the manner in

which it gives rise to the wave or ripple which results.

* See "The Splash of a Drop," by Professor A. M. Worthington, F.B.S., Romance of Science Series, published by the Society for Promot-
ing Christian Knowledge,

WATER WAVES AND WATER RIPPLES.

17

8om !' lYufessor Worth! nuton's results for a* drop of

water falling into milk are repn><lu<T.l in tin- appended

:.i .

In tin- lirst place (Fig. 8) the drop is .seen

= Time after contact '0262 sec.
FIG. 8.

just entering the water. As it plunges down, it leaves behind it a cavity, or, as it may be called, a hole in the

waii'i1 (see Fig. 9).

This hole, at a certain stage, begins to fill up. The

water rushes in on all sides, and the impetus carries up

the inrushing water so that it builds up a tall pillar of

water in the place where an instant ago there was a hole
(see Fig. 1Q). No one could anticipate such an extra-

ordinary

effect ;

but the

instantaneous photographs, taken

by the light of an electric spark, which reveal it, cannot

but be truthful.

The next stage is that this pillar of water breaks up, and falls back again on the surface. Hence the water, at the place where the drop plunges into it, is subjected to two violent impulses a downward, succeeded by an uplifting, force. The effect of this is exactly analogous to that of giving a blow to the interconnected string of balls
c

18

WAVES AND KIPPLES.

in the model shown in Fig. 7 it propagates a wave. In Fig. 10 is illustrated the next stage, in which this outwardmoving initial wave-crest is shown.

= Time after contact -0391 sec.
FIG. 9.
So much for the events revealed by the flash-light of
an electric spark ; but succeeding these there is a long

= Time after contact *ioi sec,
FiG. 10.
train of interesting wave-making performance which can be watched with the eye, or the stages photographed with

WATER WAVES AND WATER HIPPIES. 19
a hand camera. This wave -production is best seen when a large stone is thrown into calm water in a lake or
pond.
A story is recorded of the great artist Turner, that he A mice spent a morning throwing stones into a pond.
i'ricud reproved him for his idleness. "No," said the painter, " I have not been idle; I have learnt how to paint
,i ripple." If the artist's eye has to be carefully trained
to notice all that there is to see when a stone is hurled
into a pond, it is not strange that a careless observer cannot grasp at once what really happens to the water in this ordinary occurrence.
The photograph in Fig. 11 will, however, show one
FIG. 11. Ripples on a lake (Sierre), produced by throwing in a stone.
stage in the event. As soon as the first wave-crest, the origin of which we have already explained, is formed, it

20

WAVES AND BIPPLES.

begins to move outwards in a circular form, and as it moves it gives rise to a wave- train, that is, it multiplies itself into a series of concentric ripples, or waves, which move outwards, multiplying in number, but getting smaller
as they move. Thus if a large stone is thrown far out into a deep,
still lake, after the first splash we shall see a circular wave spreading out from the place where the commotion was made in the water. As we look at this wave we
shall see it growing in size and multiplying itself. At
first there is but a single wave, then two, four, seven, ten,
or more concentric ripples are seen, each circular wave expanding and getting feebler, but seeming to give birth to others as it moves. Moreover, a very careful examination will show us that the whole group of waves, or the wave-train, has an outward motion with a less speed than any individual wave. This observation will serve to initiate the conceptions of a wave-train and of a wavegroup velocity. At first it is difficult to understand that a
group of waves may move more slowly than the individual waves which compose it. If, however, we cast a stone
into a pond, and look very carefully at what takes place, we shall see that the circular expanding band of ripples has an ill-defined but visible inner and outer edge, and that wavelets or ripples which compose it are being continually brought into existence on its outer edge, and dying away on its inner edge. Waves, so to speak, pass through the ripple band with a greater speed than that at which the whole band of waves moves forward. This
rather difficult, but important, idea of the distinction between the velocity of a group of waves and that of an
individual wave was first suggested by Sir George Stokes, who set a question in a Cambridge Examination on the

WATER WAVES AND WATER RIPPLES. 21

subject in 187G, and subsequently it was elucidated by Professor Osborne Reynolds * and Lord Ilayleigh.

It can be further explained as follows : Let us consider a wave-motion model such as that represented in Fig. 7,
in which a number of suspended heavy balls are connected

to one another by elastic threads. Let one ball in the centre be drawn on one side and then released. It will

swing to and fro, and will start a wave outwards in both directions. If the row of balls is sufficiently long, it will be seen that the ball by which the wave was started soon comes to rest, and that the wave-motion is confined to a certain group of balls on either side. As time goes on, the wave-motion in each group dies away on the side nearest the origin, and extends on the side furthest away. Hence the particular group of balls which are the seat of the visible wave-motion is continually being shifted along. The rate at which the centre of this active group of vibrat-

ing balls is displaced may be called the velocity of the wave-train. The velocity of the wave is, however, some-
thing greater, since the waves are all the time moving

through the group. This wave-velocity is numerically

estimated by taking the product of the wave-length and

frequency of the motion.

At this stage it is necessary to explain that waves are

not merely a mode

of

motion ;

they are

a

means

of

con-

veying energy. It is difficult to give in a compact form any simple definition of what is meant in modern scientific writings by the word JEneryy.
Briefly speaking, we may say that there are two fundamental agencies or things in Nature with which we are in

contact, manifesting themselves in many different forms,

* See Osborne Reynolds, Nature, vol. 1C, 1877, p. 343, a paper read before the British Apsociation at Plymouth ; see also Appendix, Note A.

22

WAVES AND RIPPLES.

but of which the total quantity is unchangeable by human operations. One of these is called Matter. This term is the collective name given to all the substance or stuff we can see or touch, and which can be weighed or has weight. All known solids, liquids, or gases, such things
as ice, water, steam, iron, oil, or air, are called material
substances, and they have in common the two qualities of occupying space or taking up room, and of having
weight. Experiment has shown that there are some eighty different kinds of simple matter which cannot be transformed into each other, and these forms are called
the Elements. Any other material substance is made up
of mixtures or combinations of these elements. The
elementary substances are therefore like the letters of the
alphabet, which, taken in groups, make up words, these last corresponding to compound chemical bodies. Exact research has shown that no chemical changes taking place in a closed space can alter the total weight or amount of gravitating matter in it. If a chemist and numerous
chemicals were enclosed in a great glass ball, and the ball balanced on a gigantic but very sensitive pair of scales, no operations which the chemist could conduct in the interior of his crystal laboratory would alter, by the ten-thousandth
part of a grain, the total weight of it all. He might
analyze or combine his chemicals, burn or mix them as he pleased, but as long as nothing entered or escaped from the ball, the total gravitating mass would remain precisely
the same. This great fact is called the Law of Conserva-
tion of Matter, and it teaches us that although a scuttle of
coal may seem to disappear when burnt, yet the weight
of the ashes and of all the gaseous products of combustion are together equal to the weight of the original coal and the air required to burn it.

WATER WAVES AND WATER RIPPLES.

In addition to various material substances we find

iliat we have to recognize different forms of something called Energy, associated with Matter. Thus an iron ball may be more or less hot, more or less electrified or magnetized, or moving with more or less speed. The
production of these states of heat, electrification, magneti-

zation, or movement, involves the transfer to the iron of Energy, and they are themselves forms of Energy. This Energy in all its various forms can be evaluated or measured in terms of Energy of movement. Thus the Energy required to heat a ball of iron weighing one imperial pound from a temperature of the melting point

of ice to that of boiling water, is nearly equal to the Energy required to impart to it a speed of 1000 feet a

second.

In the same way, every definite state of electrification

or magnetization can be expressed in its mechanical

/>nralent, as it is called. Moreover, it is found that we

can never create any amount of heat or mechanical motion

or other form of energy without putting out of existence

We an equivalent of energy in some other form.

are

therefore compelled to consider that Energy stands on the same footing as Matter in regard to our inability to create or destroy it, and its constancy in total amount, as far as we can ascertain, gives it the same character of permanence.
The difference, however, is that we cannot, so to speak,
ear-mark any given quantity of energy and follow it through all its transformations in the same manner in
which we can mark and identify a certain portion of Matter. The moment, however, that we pass beyond
these merely quantitative ideas and proceed to ask further
questions about the nature of Energy and Matter, we find
We ourselves in the presence of inscrutable mysteries.

24

WAVES AND BIPPLES.

are not able as yet to analyze into anything simpler this

"

"

something

we

call

Energy which presents itself

in

the

guise of heat or light, electricity or magnetism, movement

or chemical action. It is protean in form, intangible, yet

measurable in magnitude, and all its changes are by definite

equivalent amount and value. There is a most rigid

system of book-keeping in the transactions of the physical
universe. You may have anything you like in the way

of Energy served out to you, but the amount of it is

debited to your account immediately, and the bill has to

be discharged by paying an equivalent in some other form

of Energy before you can remove the goods from the

counter.

Matter in its various forms serves as the vehicle of

We Energy.

have no experience of Energy apart from

Matter of some kind, nor of Matter altogether devoid of

Energy. "We do not even know whether these two things

can exist separately, and we can give no definition of the

one which does not in some way presuppose the existence

of the other. Eeturning, then, to the subject of waves, we

may say that a true wave can only exist when Energy is

capable of being associated with a medium in two forms,

and the wave is a means by which that Energy is trans-

ferred from place to place.

It has already been explained that a true wave can

only be created in a medium which elastically resists

some kind of deformation, and persists in motion in virtue

of inertia. When any material possesses such a quality

of resistance to some kind of strain or deformation of such

a character that the deformation disappears when the force creating it is withdrawn, it is called an elastic material. This elasticity may arise from various causes. Thus air resists being compressed, and if the compressing force is

WATER WAVES AND WATER RIPPLES.

2r,

removed the air expands again. It possesses so-called elasticity of bulk. In the case of water having a free
surface there is, as we have seen, a resistance to any change of level in the surface. This may be called an elasticity of surface form. Whenever an elastic material
i- strained or deformed, energy has to be expended on it to create the deformation. Thus to wind up a watchspring, stretch a piece of indiarubber, compress some air, or bend a bow, requires an energy expenditure.
As long as the material is kept strained, it is said to
have potential energy associated with it. This term is not a very expressive one, and it would be better to call it
Kiwrgy of strain, or deformation. If, however, we relax the bent bow or release the compressed air, the Energy of Strain disappears, and we have it replaced by Energy of Motion. The arrow which flies from a bow carries with
it, as energy of motion, some part of the energy of strain associated with the bent bow.
A little examination of wave-motion shows us, there-
fore, that we always have at any instant associated with the material in which the wave is being propagated, both Energy of Strain and Energy of Motion. It can be shown that in a true wave of permanent type, the whole energy at any one moment is half energy of strain and half energy of motion, or, as it is called, half potential and half kinetic.
Thus if we consider a wave being propagated along a line of balls elastically connected, at any one moment some of the balls are moving with their greatest velocity, and some are at the extremity of their swing. The former
have energy of motion, and the latter energy of strain. Or, look at a train of sea waves. Some parts of the
water are at any moment lifted high above the average level of the sea, or are much below it, but are otherwise

26

WA VES A ND EIPPLES.

nearly at rest. These portions possess what is called potential energy, or energy of position. Other parts of the water are at the average level of the sea, but are moving with considerable velocity, and these portions possess energy of motion. Every other part of the wave has in some degree both energy of motion and energy of position, and it can be shown that the energy of the whole wave is half of one kind and half of the other.
As a wave progresses over the surface, wave-energy is
continually being imparted to portions of the water in
front, and it is transferred away from others in the rear.
In the very act of setting a fresh particle of water in oscillation, the portions already vibrating must diminish
their own motion. They may hand on the whole of their
energy or only a part of it to their neighbours. This distinction is a very important one, and it determines whether a single act of disturbance shall create a solitary wave or wave-train in a medium.
The difference may be illustrated as follows : Consider
a row of glass or steel balls suspended by threads so hung as to be quite close to each other (see Fig. 12). Withdraw

FIG. 12.
the first ball, and let it fall against the second one. The result is that the last ball of the row flies off with a jerk. In this case the whole energy imparted to the first ball is transmitted along the row of balls. The first ball, on falling against the second one, exerts on it a pressure

WATER WAVES AND WATER RIPPLES. 27
which slightly squeezes both out of shape. This pressure just sufficient to bring the first ball to rest. The
second ball, in turn, expands after the blow and squeezes tlit- third, and so on. Hence, in virtue of Newton's Third
Law of Motion, that "action and reaction are equal and
opposite," it follows that the pressure produced by the blow of the first ball is handed on from ball to ball, and
finally causes the last ball to fly off.
In this case, owing to the rigid connection between the elastic balls, each one hands on to its neighbour the whole
of the energy it receives. Supposing, however, that we
separate the balls slightly, and give the first ball a transverse, or side-to-side swing. Then, owing to the fact that there is no connection between the balls, the energy imparted to the first ball would not be handed on at all, and no wave would be propagated.
Between these two extremes of the whole energy transferred and a solitary wave produced, and no energy transferred and no wave produced, we have a condition in which an initial disturbance of one ball gives rise to a wave-train and part of the energy is transferred.
For if we interconnect the balls by loose elastic threads,
and then give, as before, a transverse or sideways impulse to the first ball, this will pull the second one and set it swinging, but it will be pulled back itself, and will be to some extent deprived of its motion. The same sharing or division of energy will take place between the second and third, and third and fourth balls, and so on. Hence the initial solitary vibration of the first ball draws out into a wave-train, and the originally imparted energy is spread out over a number of balls, and not concentrated in one of them. Accordingly, as time goes on, the wavenaiii is ever extending in length and the oscillatory motion

28

WAVES AND BIPPLES.

of each ball is dying away, and the original energy gets spread over a wider and wider area or number of balls, but is propagated with less speed than the wave-velocity for that medium.
There need be no difficulty in distinguishing between the notion of a wave-velocity and a wave-train velocity,
if we remember that the wave travels a distance equal to a wave-length in the time taken by one oscillation. Hence the wave-velocity is measured by taking the quotient of the wave-length by the time of one complete
vibration.
If, for example, the wave-length of a water wave is 4 inches, and we observe that twelve waves pass any given point in 3 seconds, we can at once infer that the wavevelocity is 16 inches per second. The transference of energy may, however, take place so that the whole group of waves moves forward much more slowly. They move forward because the waves are dying out in the rear of the group and being created in the front, and the rate of movement of the group is, in the case of deep-water waves, equal to half that of the single-wave velocity.
A very rough illustration of this difference between
a group velocity and an individual velocity may be given
by supposing a barge to be slowly towed along a river. Let a group of boys run along the barge, dive over the bows, and reappear at the stern and climb in again. Then the velocity of the group of boys on the barge is the same as the speed of the barge, but the speed of each individual boy in space is equal to the speed of the barge added to the speed of each boy relatively to the barge. If the barge is being towed at 3 miles an hour, and the boys run along the boat also at 3 miles an hour, then the velocity of the group of boys is only half that of

WATER WAVES AND WATER RIPPLES. 29

the individual buy, because the former is o miles ail hour ami I lit- latin- is ', miles an hour.

lie tore leaving the subject of sea waves there are two or three interesting matters which must be considered. In the first place, the breaking of a wave on the shore or
on shallow water calls for an explanation. If we watch a sea wave rolling in towards the beach, we shall notice
i hat, as it nears the shore, it gets steeper on the shore side, ami xnidually curls over until it falls and breaks into spray. The reason is because, as the wave gets into the shallow water, the top part of the wave advances more rapidly than the bottom portion. It has already been

explained that the path of the water-particle is a circle, with its plane vertical and perpendicular to the wave-front

or line.

Accordingly, if the wave is moving in shallow water,

the friction of the water against the bottom retards the

backward movement at the lowest position of the water,

but no such obstacle exists to the forward movement of

An the water at its highest position.

additional reason

for the deformation of the wave on a gently sloping shore

may be found in the fact that the front part of the wave

is then in shallower water, and hence moves more slowly than the rearward portion in deeper water. From both

causes, however, the wave continually gets steeper and

steeper on its landward side until it curls over and tumbles

down like a house which leans too much on one side.

The act of curling over in a breaking wave is a beautiful thing to watch, and one which attracts the eye of every artist who paints seascapes and storm waves, or of any lover of Nature who lingers by the shore.
Another matter of interest is the origin of sea waves. Undoubtedly they are due originally to the action of the

30

WAVES AND KIPPLES.

wind upon the water. Whenever two layers of fluid lie in contact with each other, and one moves faster than the
other, the faster-moving layer will throw the other into waves. This is seen, not only in the action of moving air or wind upon water, but even in the action of air upon
air or water upon water. From the tops of high mountains we may sometimes look down upon a flat surface of cloud beneath. On one occasion the author enjoyed a curious spectacle from the summit of an Alpine peak. The climb up had been through damp and misty air, but on reaching the summit the clouds were left behind, and a canopy of
blue sky and glorious sunshine were found overhead.
Beneath the clouds lay closely packed like a sea of white
vapour, and through this ocean of cloud the peaks of many
high mountains projected and stood up like islands. The
surface of this sea of white cloud, brilliantly illuminated
by the sunshine, was not, however, perfectly smooth. It was tossed into cloud waves and billows by the action of currents of air blowing over its upper surface, and it had
a striking resemblance to the surface of a rough sea.
When such a cloud layer is not too thick, the ruffling of its upper or under surface into cloud waves may thin it away into regular cloud rolls, and these cloud rollers may
then be cut up again by cross air-currents into patches, and we have the appearance known as a " mackerel

sky."

Another familiar phenomenon is that known as the

"

"

ripple-mark

on

wet sand.

As the tide ebbs out over

a smooth bank of sea-sand, it leaves the surface ploughed

into regular rounded ridges and furrows, which are

stationary waves on the sand. This is called the ripple-

mark. It is due to the fact that the sand, when covered

by the water, forms a surface which in a certain sense is

WATER WAVES AND WATER RIPPLES. 31
fluid, being saturated and filled with water, but the movement df this bottom sand-logged water is hindered l)y the sand, mid hence the layer of overlying water moves over ii at a different speed in ebbing out, and carves it into what are virtually sand waves.
Even a dry sand or snow surface may in this manner
T 1 moulded into a wave-form by the wind, and very curious effects of this kind have been noticed and described by I >r. Vaughan Cornish, who has made a great study of the science of waves.*
The production of waves on water by means of a current of air blowing over it is easily exhibited on a small scale by blowing through an indiarubber pipe, the end of which is held near the surface of the water in a tub or
tank. The exact manner in which the moving air gets a grip of the water is not quite plain, but it is clear that, if once an inequality of level is set up, the moving air has then an oblique surface against which it can press, and so increase the inequality by heaping up the water in some places, and hollowing it out in others.
Hence oscillations of the water-surface are set up, which go on accumulating. These waves then travel away with a speed depending upon their wave-length, and we may have great disturbances of the sea-surface at places where there is no actual storm- wind. These " echoes of a far-off storm " are known as a " ground swell." In some localities the inhabitants are able to apprise themselves of the coming of a storm by noticing movements of the sea which indicate the arrival of waves which have
travelled more quickly than the storm-centre itself.
A * very interesting article on " Kumatology, or the Science of Waves,"
appeared in a number of Pearson's Magazine for July, 1901. In this article, by Mr. Marcus Tindal, many interesting facts about, and pictures of, sea waves are given.

32

WAVES AND RIPPLES.

Every visitor to the seaside will have noticed occasions on which the sea is violently disturbed by waves, and yet the air in the locality is tolerably calm. In this case the waves have been propagated from some point of
disturbance at a distance.
A study of breaking waves shows us that the cause of
their great power to effect damage to coast structures, such as piers, harbour works, and shipping in harbours, is really due to the forward motion of the water as the wave is
breaking. Every cubic foot of water weighs 63J Ibs., so that a cubic yard of water weighs about three-quarters
of a ton. If this water is moving with a speed of many
feet per second in a forward direction, the energy of motion stored up in it is tremendous, and fully sufficient to account for the destructive power of storm waves on a coast.
The total volume of water which is comprised in the space occupied by even one sea-storm wave of moderate
dimensions may have a mass of many hundreds of tons, and its energy of motion may easily amount to that of an
express train in motion. Hence when, in the last stage of its career, this mass of water is hurled forward on the
shore, its destructive effects are not a matter for surprise.
We must now leave the subject of waves in the open
sea on a large level surface, and consider that of waves in narrow channels, such as canals or rivers. The laws
whioh govern water-wave production in a canal can best be studied by placing some water in a long tank with glass
sides. If at one end we insert a flat piece of wood and give it a push forward, we shall start what is called a long wave in the tank. The characteristic of this kind of wave is that the oscillatory motion is chiefly to- and- fro, and not up-and-down. This may be very easily seen by placing some Iran in the water, or floating in it some

WATER WAVES AND WATER RIPPLES. 33
glass balls which have been adjusted so as to just float
anywhere in the water. When this is done, and a wave
n si a ci I in the tank, it runs up and down, being reflected at each end (see Fig. 13).

Fio. 13. Water-wave produced in a tank.

From the motion of the bran we can see that the water

swingi^s backwards and forwards in a horizontal line with

a pendulum-like motion, but its up-and-down or vertical

A motion is much more restricted.

wave of this kind

navels along a canal with a speed which depends upon the depth of the canal. If waves of this kind are started in a very long trough, the wave-length being large compared with the depth of the trough,* it can be shown that the speed of the wave is equal to the velocity which would be gained by a stone or other heavy body in falling through half the depth of the canal. Hence, the deeper the water, the quicker the wave travels. This can be shown as an experimental fact as follows : Let two galvanized iron tanks be provided, each about G feet long and 1 foot wide and deep.
At one end of each tank a hollow cylinder, such as a coffee-canister or ball made water-tight, is floated, and it may be prevented from moving from its place by being
attached to a hinged rod like the ball-cock of a cistern. The two tanks are placed side by side, and one is filled to

*

Lord

Kelvin

(see

lecture

on

" Ship Waves," Popular Lectures, vol.

iii. i>. 4G8) says the wave-length must be at least fifty times the depth of

the canal.

31

WAVES AND RIPPLES.

a depth of 6 inches, and the other to a depth of 3 inches,
with water. Two pieces of wood are then provided and
joined together as in Fig. 14, so as to form a double paddle.
By pushing this through the water simultaneously in both
tanks at the end opposite to that at which the floating cylinders are placed, it is possible to start two solitary waves, one in each tank, at the same instant. These
waves rush up to the other end and cause the floats to bob up. It will easily be seen that the float on the deeper water bobs up first, thus showing that the wave

FIG. 14.
on the deeper water has travelled along the tank more quickly than the wave on the shallower water. * In order to calculate the speed of the waves, we must call to mind the law governing the speed of falling bodies. If a stone falls from a height its speed increases as it falls. It can be shown that the speed in feet per second after falling from any height is obtained by multiplying together the number 8 and a number which is the square
root of the height in feet.
Thus, for instance, if we desire to know the speed attained by falling from a height of 25 feet above the earth's surface, we multiply 8 by 5, this last number being the square root of 25. Accordingly, we find the

WATER WAVES AND WATER RIPPLES. 35

velocity to be 40 feet per second, or about 20 miles an

hour.

The force of the blow which a body administers and

sutlers on striking the ground depends on the energy of motion it has acquired during the fall, and as this varies

as the square of the speed, it varies also as the height

fallen through.

Let us apply these rules to calculate the speed of a

long wave in a canal having water 8 feet deep in it.

The half-depth of the canal is therefore 4 feet. The

square root of 4 is

2 ;

hence

the

speed

of the wave is

that of a body which has fallen from a height of 4 feet, and

is therefore 16 feet per second, or nearly 11 miles an hour.

AVhen we come to consider the question of waves made

by ships, in the next chapter, a story will be related of a scientific discovery made by a horse employed in dragging

canal-boats, which depended on the fact that the speed

of long waves in this canal was nearly the same as the

trotting speed of the horse.

FIG. 15.
It may be well, as a little digression, to point out how
the law connecting height fallen through and velocity
acquired by the falling body may be experimentally
illustrated for teaching purposes.
The apparatus is shown in Fig. 15. It consists of a

36

WAVES AND RIPPLES.

long board placed in a horizontal position and held with

the face vertical. This board is about 16 feet long.

Attached to this board is a grooved railway, part of which

A is on a slope and part is horizontal.

smooth iron ball,

A, about 2 inches in diameter, can run down this railway,

and is stopped by a movable buffer or bell, B, which can

be clamped at various positions on the horizontal rail.

At the bottom of the inclined plane is a light lever, T,

which is touched by the ball on reaching the bottom of the

hill. The trigger releases a pendulum, P, which is held

engaged on one side, and, when released, it takes one swing

and strikes a bell, G. The pendulum occupies half a
second in making its swing. An experiment is then per-

formed in the following manner : The iron ball is placed

at a distance, say, of 1 foot up the hill and released. It rolls down, detaches the pendulum at the moment it

arrives at the bottom of the hill, and then expends its

momentum in running along the flat part of the railway.

The buffer must be so placed by trial that the iron ball

hits it at the instant when the pendulum strikes the bell.

The distance which the buffer has to be placed from the

bottom of the hill is a measure of the velocity acquired

by the iron ball in falling down the set distance along the hill. The experiment is then repeated with the iron

ball placed respectively four times and nine times higher

up the hill, and it will be found that the distances which

the ball runs along the flat part in one half-second are
in the ratio of 1, 2, and 3, when the heights fallen through

down the hill are in the ratio of 1, 4, and 9.

The inference we make from this experiment is that

the velocity acquired by a body in falling through any

distance is proportional to the square root of the height.
The same law holds good, no matter how steep the hill,

WATER WAVES AND WATER RIPPLES. 37

and therefore it holds good when the body, such as a

stone or ball, falls freely through the air.
The experiment with the ball rolling down a slope is

an instructive one to make, because it brings clearly

1 -I -lore the mind what is meant by saying, in scientific

language, that one thing "varies as the square root" of

We another.

meet with so many instances of this mode

of variation in the study of physics, that the reader,

especially the young reader, should not be content until

tin- idea conveyed by these words has become quite clear to him or her.

Thus, for instance, the time of vibration of a simple clock

pendulum " varies as the square

root of the

" length ;

the

velocity of a canal wave " varies as the square root of the

depth

of

the

canal ;

"

and

the

velocity or

speed

acquired

by

a falling ball "varies as the square root of the distance

fallen through." These phrases mean that if we have

pendulums whose lengths are in the ratio of 1 to 4 to 9,

then the respective times of their vibration are in the ratio

of 1 to 2 to 3. Also a similar relation connects the canal-

depth and wave-velocity, or the ball-velocity and height
of fall.

Returning again to canal waves, it should be pointed

out that the real path of a particle of water in the canal,
when long waves are passing along it, is a very flat oval

curve called an ellipse. In the extreme cases, when the

canal is very wide and deep, this ellipse will become

nearly

a

circle ;

and,

on

the

other

hand,

when

narrow

and

shallow, it will be nearly a straight line. Hence, if long

waves are created in a canal which is shallow compared

with the length of the wave, the water-particles simply

oscillate to and fro in a horizontal line. There is, how-

ever, one important fact connected with wave-propagation

38

WAVES AND RIPPLES.

in a canal, which has a great bearing on the mode of

formation

of

what

is

called

a

" bore."

As a wave travels along a canal, it can be shown, both experimentally and theoretically, that the crest of the wave travels faster than the hollow, and as a consequence the wave tends to become steeper on its front side, and its
shape then resembles a saw-tooth.
A very well known and striking natural phenomenon
is the so-called " bore " in certain tidal rivers or estuaries.

It is well seen on the Severn in certain states of the tide

and wind. The tidal wave returning along the Severn
channel, which narrows rapidly as it leaves the coast, becomes converted into a " canal wave," and travels with

great rapidity up the channel. The front side of this great wave takes an almost vertical position, resembling an advancing wall of water, and works great havoc with boats and shipping which have had the misfortune to be left in its path. To understand more completely how a " bore " is formed, the reader must be reminded of the
cause of all tidal phenomena. Any one who lives by the
sea or an estuary knows well that the sea-level rises and falls twice every 24 hours, and that the average interval of time between high water and high water is nearly 12 J hours. The cause of this change of level in the watersurface is the attraction exerted by the sun and moon upon the ocean. The earth is, so to speak, clothed with a flexible garment of water, and this garment is pulled
out of shape by the attractive force of our luminaries;
very roughly speaking, we may say that the ocean-surface
is distorted into a shape called an ellipsoid, and that there are therefore two elevations of water which march across

the sea-covered regions of the earth as it revolves on its axis. These elevations are called the tidal waves. The

\VATER WAVES AND WATER RIPPLES. 39
u Herts, however, are much complicated by the i'act that
the ocean does not cover all parts of the earth. There is no difficulty in showing that, as the tidal wave progresses round the earth across each great ocean, it produces an elevation of the sea-surface which is not simultaneous at
all places. The time when the crest of the tidal wave reaches any place is called the " time of high tide." Thus if we consider an estuary, such as that of the Thames, tl it-re is a marked difference between the time of high tide as we ascend the estuary.
Taking three places, Margate, Gravesend, and London Bridge, we find that if the time of high tide at Margate is at noon on any day, then it is high tide at Gravesend at 2.15 p.m., and at London Bridge a little before three o'clock. This difference is due to the time required for the tidal wave to travel up the estuary of the Thames.
When an estuary contracts considerably as it proceeds,
as is the case with the Bristol Channel, then the range of
the tide or the height of the tidal wave becomes greatly increased as it travels up the gradually narrowing channel, because the wave is squeezed into a smaller space. For
example, the range of spring tides at the entrance of the Bristol Channel is about 18 feet, but at Chepstow it is
about 50 feet.* At oceanic ports in open sea the range
of the tide is generally only 2 or 3 feet.
If we look at the map of England, we shall see how
rapidly the Bristol Channel contracts, and hence, as the tidal wave advances from the Atlantic Ocean, it gets jam bed up in this rapidly contracting channel, and as the depth of the channel in which it moves rapidly shallows, the rear portion of this tidal wave, being in deeper water,
* See article " Tides," by G. H. Darwin, " Encyclopaedia Britannica," 9th edit., vol. 23, p. 353.

40

WAVES AND RIPPLES.

travels faster than the front part and overtakes it, pro-

ducing thus a flat or straight-fronted wave which goes

forward with tremendous speed.*

We must, in the next place, turn our attention to the

study of water ripples.

The

term

"

"

ripple

is

generally

used to signify a very small and short wave, and in

ordinary language it is not distinguished from what might

be called a wavelet, or little wave. There is, however, a

scientific distinction between a wave and a ripple, of a very

fundamental character.

It has already been stated that a wave can only exist, or be created, in or on a medium which resists in an

elastic manner some displacement. The ordinary water-

surface wave is termed a gravitation wave, and it exists because the water-surface resists being made unlevel.

There is, however, another thing which a water-surface

resists. It offers an opposition to small stretching, in

virtue of what is called its surface tension. In a popular

manner the matter may thus be stated: The surface of

every liquid is covered with a sort of skin which, like a

sheet of indiarubber, resists stretching, and in fact con-

tracts under existing conditions so as to become as small

We as possible.

can see an illustration of this in the

case of a soap-bubble. If a bubble is blown on a rather

wide glass tube, on removing the mouth the bubble rapidly

shrinks up, and the contained air is squeezed out of the

tube with sufficient force to blow out a candle held near

the end of the tube.

Again, if a dry steel sewing-needle is laid gently in a horizontal position on clean water, it will float, although

* The progress of the Severn "bore" has been photographed and reproduced by a kinematograph by Dr. Vaughan Cornish. For a series of papers bearing on this sort of wave, by Lord Kelvin, see the Philosophical Magazine for 1886 and 1887.

WATER WAVES AND WATER RIPPLES. 41

the metal itself is heavier than water. It floats because

the weight of the needle is not sufficient to break through the surface film. It is for this reason that very small and light insects can run freely over the surface of water in a pond.
This surface tension is, however, destroyed or diminished by placing various substances on the water. Thus if a small disc of writing-paper the size of a wafer is placed on the surface of clean water in a saucer, it will rest in the middle. The surface film of the water on

which it rests is, however, strained or pulled equally in

different directions. If a wire is dipped in strong spirits

of wine or whisky, and one side of the wafer touched with

the drop of spirit, the paper shoots away with great speed

in the opposite direction. The surface tension on one side

has been diminished by the spirit, and the equality of

tension destroyed.

These experiments and many others show us that we

must regard the surface of a liquid as covered with an

invisible film, which is in a state of stretch, or which

resists stretching. If we imagine a jam-pot closed with

a cover of thin sheet indiarubber pulled tightly over it,
it is clear that any attempt to make puckers, pleats, or

wrinkles "in it would involve stretching the indiarubber.

It is exactly the same with water. If very small wrinkles

or pleats, as waves, are made on its surface, the resistance

which is brought into play is that due to the surface

tension, and not merely the resistance of the surface to

being made unlevel. Wavelets so made, or due to the

above cause, are called ripples.

It

can

be

shown

by

mathematical

*
reasoning

that

on

*

See

Lord

Kelvin,

" Hydrokinetic

Solutions and Observations,"

Philo-

sophical Magazine, November, 1871.

42

WAVES AND 1UPPLES.

the free surface of a liquid, like water, what arc called capillary ripples can be made by agitations or movements of a certain kind, and the characteristic of these surface-
tension waves or capillary ripples, as compared with gravitation waves, is that the velocity of propagation of the capillary ripple is less the greater the wave-length, whereas the velocity of gravitation on ordinary surface waves is greater the greater the wave-length.
It follows from this that for any liquid, such as water, there is a certain length of wave which travels most slowly. This slowest wave is the dividing line between what are properly called ripples, and those that are properly called waves. In the case of water this slowest wave has a wave-length of about two-thirds of an inch (0*68 inch), and a speed of travel approximately of 9 inches (078 foot) per second.
More strictly speaking, the matter should bs explained as follows : Sir George Stokes showed, as far back as 1848,
that the surface tension of a liquid should be taken into
account in finding the pressure at the free surface of a liquid. It was not, however, until 1871 that Lord Kelvin discussed the bearing of this fact on the formation of waves, and gave a mathematical expression for the velocity
of a wave of oscillatory type on a liquid surface, in which the wave-length, surface tension, density, and the acceleration of gravity were taken into account. The result was to show that when waves are very short, viz. a small fraction of an inch, they are principally due to surface tension, and when long are entirely due to gravity.
It can easily be seen that ripples run faster the
smaller their wave-length. If we take a thin wire and hold it perpendicularly in water, and then move it quickly parallel to itself, we shall see a stationary pattern of

WATER WAVES AND WATER RIPPLES. 43

ripples round the wire which moves with it. Tli-
ripples are smaller aiid closer together the faster the wire
is moved.

Hippies on water are formed in circular expanding
liii^s when rain-drops fall upon the still surface of a lake or pond, or when drops of water formed in any other way i'all in the same manner. On the other hand, a stone

Hung into quiet and deep water will, in general, create \\avesofwave-lengthgreater than two-thirds of an inch, so that they are no longer within the limits entitling them to be called ripples. Hence we have a perfectly scientific distinction between a ripple and a wave, and a simple measurement of the wave-length will decide whether disturbances of oscillatory type on a liquid surface should be called ripples or waves in the proper

sense of the words.

The production of water ripples and their properties,

and a beautiful illustration of wave properties in general,

can be made by allowing a steady stream of water from a

very small jet to fall on the surface of still water in a

tank. In order to see the ripples so formed, it is necessary

to illuminate them in a particular manner.

The following is a description of an apparatus,

designed by the author for exhibiting all these effects to

a large audience :

The instrument consists essentially of an electric lan-

A tern.

hand-regulated or self-regulating arc lamp is

employed to produce a powerful beam of light. This is

collected by a suitable condensing-lens, and it then falls

upon a mirror placed at an angle of 45, which throws it

vertically upwards. The light is then concentrated by a

plain convex lens placed horizontally, and passes through

a trough of metal having a plane glass bottom. This

44

WAVES AND SIPPLES.

trough is filled to a depth of half an inch with water, and it

has an overflow pipe to remove waste water. Above the

tank, at the proper distance, is placed a focussing-lens,

and another mirror at an angle of 45 to throw an image

of the water-surface upon a screen. The last lens is so

arranged that ripples on the surface of the water appear

like dark lines flitting across the bright disc of light which
appears upon the screen. Two small brass jets are also

arranged to drop water into the tank, and these jets must

be supplied with water from a cistern elevated about

4 feet above the trough. The jets must be controlled by

screw-taps which permit of very accurate adjustment.

These jets should work on swivels, so that they may be

turned about to drop the water at any point in the tank.

The capillary ripples which are produced on the water-

surface by allowing water to drop on it from a jet, flit

across the surface so rapidly that they cannot be followed

by the eye. They may, however, be rendered visible as

A follows:

zinc disc, having holes in it, is arranged in

front of the focussing-lens, and turned by hand or by

means of a small electric motor. This disc is called a

stroboscopic disc. When turned round it eclipses the

light at intervals, so that the image on the screen is inter-

mittent. If, now, one of the water-jets is adjusted so as

to originate at the centre of the tank a set of diverging

circular ripples, they can be projected as shadows upon

the screen. These ripples move at the rate of 1 or 2

feet per second, and their shadows move so rapidly across

the field of view that we cannot well observe their

behaviour. If, however, the metal disc with holes in it is
made to revolve and to intermittently obscure the view, it
is possible to adjust its speed so that the interval of time between two eclipses is just equal to that required by the

WATER WAVES AND WATER RIPPLES. 45

ripples to move forward through one wave-length. When
this exact speed is obtained, the image of the ripples on the screen becomes stationary, and we see a series of concentric dark circles with intermediate bright spaces (see Fig. 16), which are the shadows of the ripples. In this
manner we can study many
of their effects. If, for in-
stance, the jet of water is
made to fall, not in the
centre of the trough, but
nearer one side, we shall
notice that there are two sets

of ripples which intersect one of these is the direct or

original set, and the other is

a set produced by the re-

flection of the original rip-

FIG. 16.

ples from the side of the trough. These direct and reflected ripple- shadows intersect and produce a cross-hatched pattern. If a slip of

metal or glass is inserted into the trough, it is very easy
to show that when a circular ripple meets a plane hard

surface it is reflected, and that the reflected ripple is also a

circular one which proceeds as if it came from a point, Q,

on the opposite side of the boundary, just as far behind

that boundary as the real centre of disturbance or origin

P of the ripple

is in front of it (see Fig. 17). In the

diagram the dotted curves represent the reflected ripple-

crests.
If we make two sets of ripples from origins P and Q
(see Fig. 18), at different distances from a flat reflecting boundary, it is not difficult to trace out that each set of ripples is reflected independently, and according to the

46

WAVES AND EIPPLES.

We above-mentioned rule.

here obtain a glimpse of a

principle which will come before us again in speaking of

FIG. 17.

o.
Reflection of circular ripples.

aether waves, and furnishes an explanation of the familiar
optical fact that when we view our own reflection in a

FIG. 18.
looking-glass, the image appears to be as far behind the glass as we are in front of it,

WATEIi WAVES AND WATER HIITLES.

47

A very pretty experiment can be shown by fitting into

the trough an oval band of metal bent into the form of an

ellipse. If two pins are stuck into a sheet of card, and a

loop of thread fitted loosely round them, and a pencil

cm ployed to trace out a curve by using it to strain the

loop of thread tight and moving it round the pin, we

obtain a closed curve called an ellipse (see Fig. 19).
A positions of the two pins

The

and B are called the foci.

It is a property of the

ellipse that the two lines
AP and BP, called radii

vectores, drawn from the

foci to any point P on the

curve, make equal angles

with a line TT' called a tan-

FIG

gent, drawn to touch the selected point on the ellipse. If we draw the tangent TT' to

the ellipse at P, then it needs only a small knowledge of
geometry to see that the line PB is in the same position and direction as if it were drawn through P from a false

focus A', which is as far behind the tangent TT' as the real

A focus

is in front of it. Accordingly, it follows that

A circular ripples diverging from one focus of an ellipse

must, after reflection at the elliptical boundary, be con-

verged to the other focus B. This can be shown by the

use of the above described apparatus in a pretty manner.
A strip of thin metal is bent into an elliptical band

and placed in the lantern trough. The band is so wide

that the water in the trough is about halfway up it. At a

point corresponding to oue focus of the ellipse, drops of

water are then allowed to fall on the water-surface and

start a series of divergent ripples. When the stroboscopic

48

WAVES AND PTPPLES.

disc is set in revolution and its speed properly adjusted,
we see that the divergent ripples proceeding from one focus

of the ellipse are all converged or concentrated to the other focus. In fact, the ripples seem to set out from one
focus, and to be, as it were, swallowed up at the other. When, in a later chapter, we are discussing the production and reflection of sound waves in the air, you will be able to bring this statement to mind, and it will be clear to you that if, instead of dealing with waves on water, we were to create waves in air in the interior of a similar

elliptically shaped room, the waves being created at one focus, they would all be collected at the other focus, and the tick of a watch or a whisper would be heard at the

point corresponding to the other focus, though it might not be heard elsewhere in the room.

With the appliances here described many beautiful

effects can be shown, illustrating the independence of

different wave-trains and their interference. If we hurl

two stones into a lake a little way apart, and thus create

two sets of circular ripples (see Fig. 20), we shall notice

that these two ripple-trains pass freely through each other,

A and each behave as if the other did not exist.

careful

examination will, however, show that at some places the water-surface is not elevated or disturbed at all, and at

others that the disturbance is increased.

If two sets of waves set out from different origins and arrive simultaneously at the same spot, then it is clear that if the crests or hollows of both waves reach that

point at the same instant, the agitation of the water will be increased. If, however, the crest of a wave from one

source reaches it at the same time as the hollow of another

equal wave from the other origin, then it is not difficult to see that the two waves will obliterate each other. This

WATER WAVES AND WATER RJPPLEB. 49
mutual destruction of wave by wave is called inter/' and it is a very important fact in connection with wavemotion. It is not too much to say that whenever we can

FiG. 20. Intersecting ripples produced on a luke by throwing in simultaneously two stones.

prove the existence of interference, that alone is an almost

crucial proof that we are dealing with wave-motion. The

conditions under which interference can take place must be

examined a little more closely. Let us suppose that two

wave-trains, having equal velocity, equal wave-length, and

equal amplitude or wave-height, are started from two

A points,

and B (see Fig. 21). Consider any point, P.

What is the condition that the waves from the two

sources shall destroy each other at that point ? Obviously

AP it is that the difference of the distances

and BP shall

be an odd number of half wave-lengths. For if in the
length AP there are 100 waves, and in the distance BP

50

WAVES AND RIPPLES.

there are 100| waves, or 101J or 103J, etc., waves, then
A the crest of a wave from will reach P at the same time
as the hollow of a wave from B, and there will be no

FIG. 21.

wave at all at the point P. This is true for all such
A positions of P that the difference of its distances from
and B are constant.

But again, we may choose a point, Q, such, that the

A difference of its distances from

and B is equal to an

even number of half wave-lengths, so that whilst in the

AQ BQ length

there are, say, 100 waves, in the distance

When there are 101, 102, 103, etc., waves.

this is the

case, the wave-effects will conspire or assist each other at
Q, and the wave-height will be doubled. If, then, we have
A any two points, and B, which are origins of equal waves,
we can mark out curved lines such that the difference of

the distances of all points on these lines from these origins is constant. These curves are called hyperbolas

(see Fig. 22).
All along each hyperbola the disturbance due to the combined effect of the waves is either doubled or annulled
when compared with that due to each wave-train separately. With the apparatus described, we can arrange

AMI WATER //'.I '//./, HMFA'.s

l;ll'l'LES.

51

to create and adjust two sets of similar water ripples from is not far apart, and on looking at the complicated

*h:idn\v-].;tu,-ni duo to the interference of the waves, we shall be aMr tn trace out certain white lines along which

FIG. 23. Interfering ripplrs on a mnvury surt'uiv. showing iuter-
ferenco along hyperbolic lines (Vincent).
the waves are annulled, these lines being hyperbolic curves (see Fig. 23). With the same appliances another

52

WAVES AND RIPPLES.

characteristic of wave-motion, which is equally important, can be well shown.
We make one half of the circular tank in which the

ripples are generated much more shallow than the other

half, by placing in it a thick semicircular plate of glass. It has already been explained that the speed with which long waves travel in a canal increases with the depth of the water in the canal. The same is true, with certain

restrictions, of ripples produced in a confined space or
tank, one part of which is much shallower than the rest. If waves are made by dropping water on to the water-

surface in the deeper part of the tank, they will travel

more quickly in this deeper part than in the shallower

We portion.

can then adjust the water-dropping jet in

such a position that it creates circular ripples which originate in deep water, but at certain places pass over a

WATER WAVES AND WATER RIPPL/
Imiindary into a region of shallower water (see Fig. 24). The left-hand side of the circular tank represented in the diagram is more shallow than the right-hand side.
When this is done, we notice two interesting facts,
viz. that the wave-lines are bent, or refracted, where they pass over the boundary, and that the waves are shorter or nearer together in the shallower region. This bending, or refraction, of a wave-front in passing the boundary line between two districts in which the wave has different velocities is an exceedingly important charactistic of wave-
motion, and we shall have brought before us the analogous facts in speaking of waves in air and waves in aether.
It is necessary to explain a little more in detail how it
comes to pass that the wave-line is thus bent. Imagine a row of soldiers, ab, marching over smooth grass, but going towards a very rough field, the line of separation SS between the smooth and the rough field being oblique to
a
4 .'

Q1

^ -'5

*-'*'

FIG 25.

the line of the soldiers (see Fig. 25). Furthermore, suppose the soldiers can march 4 miles an hour over the smooth
grass, but only 3 miles an hour over the rough field.
Then let the man on the extreme left of the line be the
first to step over the boundary. Immediately he passes into a region where his speed of marching is diminished, but his comrade on the extreme right of the row is still going easily on smooth grass. It is accordingly clear

54

WAVES AND RIPPLES.

that the direction of the line of soldiers will be swuno round because, whilst the soldier on the extreme left marches, say, 300 feet, the one on the extreme right will have gone 40tf' feet forward ; and hence by the time all the men have stepped over the boundary, the row of soldiers will no longer be going in the same direction as before
it will have become bent, or refracted.

This same action takes place with waves. If a wave meets obliquely a boundary separating two regions, in one of which it moves slower than in the other, then, for the same reason that the direction of the row of soldiers

in the above illustration is bent by reason of the retar-

dation of velocity experienced by each man in turn as he

steps over the dividing line, so the wave-line or wave-

front is bent by passing from a place where it moves

quickly to a place where it moves more slowly. The

ratio of the velocities or speeds of the wave in the two

regions is called the index of refraction.
We can, by arranging suitably curved reflecting surfaces

or properly shaped shallow places in a tank of water,

illustrate all the facts connected with the change in wave-

fronts produced by reflection and refraction.
We can generate circular waves or ripples diverging

from a point, and convert them, by reflection from a para-

bolic

reflector,

into

plane

waves ;

and again,

by means

of

refraction at a curved or lens-shaped shallow, converge

these waves to a focus.

Interesting experiments of this kind have been made by means of capillary ripples on a mercury surface by Mr. J. H. Vincent, and lie has photographed the ripples so formed, and given examples of their reflection and refraction, which are well worth study.*

* " Oil the Photography of Eipples," by J. H. Vincent, Philosophical

WATER WAVES AND WATER RIPPLES. 55
We do not need, however, elaborate apparatus to see
these effects when we know what to look for.
A stone thrown into a lake will create a ripple or
wave-train, which moves outwards at the rate of a few feet a second. If it should happen that the pond or lake has
an immersed wall as part of its boundary, this may form
an effective reflecting surface, and as each circular wave meets the wall it will be turned back upon itself as a reflected wave. At the edge of an absolutely calm sea, at low tide, the author once observed little parallel plane waves advancing obliquely to the coast; the edge of the water was by chance just against a rather steep ledge of hard sand, and each wavelet, as it met this reflecting surface, was turned back and reflected at an angle of
reflection equal to that of incidence. It is well to notice that a plane wave, or one in which
the wave front or line is a straight line, may be considered as made up out of a number of circular waves diverging
from points arranged closely together along a straight line.
Thus, if we suppose that a, I, c, d, etc. (see Fig. 26), are
source-points, or origins, of independent sets of circular waves, represented by the firm semicircular lines, if they send out simultaneous waves equal in all directions, the effect will be nearly equivalent to a plane wave, represented by the straight thick black line, provided that the source-points are very numerous and close together.
Supposing, then, we have a boundary against which this plane wave impinges obliquely, it will be reflected and its subsequent course will be exactly as if it had proceeded
from a series of closely adjacent source-points, a, V, c',d',etc., lying behind the boundary, each of which is the image of
Magazine, vol. 43, 1897, p. 411, and also vol. 48, 1899. These photographs of ripples have been reproduced as lantern slides by Messrs. Newton and Co., of Fleet Street, London.

56

WAVES AND RIPPLES.

the corresponding real source-points, and lies as far behind the boundary as the real point lies in front of it.

a
Fio. 26.
An immediate consequence of this is that the plane
reflected wave-front makes the same angle with the plane
reflecting surface as does the incident or arriving wave,
and we thus establish the law, so familiar in optics, that
the angle of incidence is equal to the angle of reflection
when a plane wave meets a plane reflecting surface. At the sea-side, when the tide is low and the sea calm or
ruffled only by wavelets due to a slight wind, one may often
notice trains of small waves, which are reflected at sharp edges of sand, or refracted on passing into sudden shallows, or interfering after passing round the two sides of a rock.
A careful observer can in this school of Nature instruct
himself in all the laws of wave-motion, and gather a fund of knowledge on this subject during an hour's dalliance at low tide on some sandy coast, or in the quiet study of sea- side pools, the surface of which is corrugated with trains of ripples by the breeze.

CHAPTER II.

WAVES AND RIPPLES MADE BY SHIPS.

is impossible for the most careless spectator to look
IT at a steam-vessel making her way along a lake, a
boy's boat skimming across a pond, or even a duck puddling on a stream, without noticing that the moving body is accompanied in all cases by a trail of waves or ripples, which diverge from it and extend behind. In the
of a steamer there is an additional irregular wave-

motion of the water caused by the paddle-wheels or screw,

which churn it up, and leave a line of rough water in the

steamer's wake. This, however, is not included in the

We true ship-wave effect now to be discussed.

can best

observe the proper ship-wave disturbance of the water in
the case of a yacht running freely before the wind when

the sea is fairly smooth. The study of these ship-waves

has led to most important and practical improvements in

the art of ship-designing and shipbuilding, and no treat-

ment of the subject of waves and ripples on water would

ht complete in which all mention of ship-waves was

omitted.

In order that we may explain the manner in which

these waves are formed, and their effect upon the motion of the ship, and the power required to move it forward, we must begin by a little discussion of some fundamental

facts concerning liquids in motion.

58

WAVES AND RIPPLES.

Every one is aware that certain liquids are, as we say

A sticky,, or, to use the scientific term, viscous.

request to

mention sticky liquids would call up the names of such

fluids as tar, treacle, gum-water, glycerine, and honey.

Very few people would think of including pure water, far

less spirits of wine, in a list of sticky, or viscous liquids ;
and yet it is quite easy to show by experiment that even

these fluids possess some degree of stickiness, or viscosity.

We An illustration may be afforded as follows :

provide

several very large glass tubes, nearly filled respectively
A with quicksilver, water, alcohol, glycerine, and oil.

small space is left in each tube containing a little air, and

the tubes are closed by corks. If we suddenly turn all

the tubes upside down, these bubbles of air begin to climb

We up from the bottom of the tube to the top.

notice

that in the quicksilver tube it arrives at the top in a

second or two, in the water tube it takes a little longer,

in the oil tube longer still, and in the tube filled with

glycerine it is quite a minute or more before the bubble of

air has completed its journey up the tube. This experiment,

properly interpreted, shows us that water possesses in some

degree the quality of viscosity. It can, however, be more

forcibly proved by another experiment.

To a whirling- table is fixed a glass vessel half full of
water. On this water a round disc of wood, to which is

attached a long wire carrying a paper flag, is made to float. If we set the basin of water slowly in rotation,

at first the paper flag does not move. The basin rotates

without setting the contained water in rotation, and so

to speak slips round it. Presently, however, the flag

begins to turn slowly, and this shows us that the water

has been gradually set in rotation. This happens because

the water sticks slightly to the inner surface of the basin,

WAVES AND RIPPLES MADE JIY SHIPS. r,0

and tlio layers of water likewise stick to one another. Hrii.v, as the glass vessel slides round the water it
i dually forces the outer layer of water to move with it, and this again the inner layers of water one by one, until at last the floating block of wood partakes of the motion, and the basin and its contents turn round as one mass.

This effect could not take place unless the water possessed some degree of viscosity, and also unless so-called skin friction existed between the inside of a glass vessel and ilic water it contains.

\\V, may say, however, at once that no real liquid

with which we are acquainted is entirely destitute of

We stickiness, or viscosity.

can nevertheless imagine a

liquid absolutely free from any trace of this property, and

ih is hypothetical substance is called & perfect fluid.

It is clear that this ideal perfect liquid must neces-

sarily differ in several important respects from any real

fluid, such as water, and some of these differences we pro-

We ceed to examine.

must point out that in any liquid

there may be two kinds of motion, one called irrotational

motion, and the other called rotational or vortex motion.

Consider any mass of water, such as a river, in motion
in any way ; we may in imagination fix our attention upon some small portion of it, which at any instant we will

consider to be of a spherical shape. If, as this sphere of
liquid moves along embedded in the rest of the liquid, it is turning round an axis in any direction as well as being distorted in shape, the motion of that part of the fluid is
called rotational. If, however, our little sphere of liquid is merely being stretched or pulled into an ovoid or ellipsoidal shape without any rotation or spinning motion, then the motion of the liquid is said to be irrotational.
We might compare these small portions of the liquid to

60

WAVES AND RIPPLES.

a crowd of people moving along a street. If each person moves in such a way as always to keep his face in the same direction, that movement would be an irrotational

movement. If, however, they were to move like couples

dancing in a ball-room, not only moving along but turning

round, their motion would be called rotational. Examples

of rotational, or vortex motion are seen whenever we

We empty a wash-basin by pulling up the plug.

see the

water swirl round, or rotate, forming what is called an

eMy, or whirlpool. Also eddies are seen near the margin

of a swiftly flowing river, since the water is set in rotation

by friction against objects on the banks. Eddies are

likewise created when two streams of water flow over

A each other with different speeds.

beautiful instance

of this may be viewed at an interesting place a mile or

two out of the city of Geneva. The Ehone, a rapid river, emerges as a clear blue stream from the Lake of Geneva.

At a point called Junction d'eaux it meets the river Arve, a more sluggish and turbid glacier stream, and the two then run together in the same channel. The waters of the Ehone and Arve do not at once mix, but the line

of separation is marked by a series of whirlpools or eddies set up by the flow of the rapid Ehone water against the

slower Arve water in contact with it.

Again, it is impossible to move a solid body through a liquid without setting up eddy-motion. The movement

of an oar through the water, or even of a teaspoon through

tea, is seen to be accompanied by little whirls which

detach themselves from the oar or spoon, and are really the ends of vortices set up in the liquid. The two facts

to notice particularly are that the production of eddies

in liquids always involves the expenditure of energy, or, in mechanical language, it necessitates doinf/ work. To

WAVES AND RIPPLES MADE HY 811 !!'.<. Gl

set in rotation a mass of any liquid requires the delivery

to it of <//</////, just as is the case \\hen a heavy wheel

is made to rotate or a heavy train set in movement.

Tliis energy must be supplied l>y or absorbed from the

moving solid or liquid which creates tliIn the next place, we must note that eddies or \ or

up in an imperfect fluid, such as water, are ultimately

destroyed by fluid friction. Their energy is frittered down into heat, and a mass of water in which eddies.have

been created by moving through it a paddle, is warmer

after the eddies have subsided than before. It is obvious,

from what has been said, that if a really perfect thud did

exist, it would be impossible by mechanical means to

make

eddies

in

it ;

but

if

they were

created, they would

continue for ever, and have something of the permanence

of material substances.

A vortex motion in water may be either a terminated
vortex, in which case its ends are on the surface, and are

seen

as

eddies,

or whirls ;

or

it

may

be

an

endless

vortex,

in which case it is called a vortcM ritiy. Such a ring

FIG. 27. The production of a vortex ring in air.

A is very easily made in the air as follows:

cubical

wooden box about 18 inches in the side has a hole

6 inches in diameter made in the bottom (see Fig. 27).

62

WAVES AND EIPPLES.

The open top of the box is covered tightly with elastic

cloth. The box is then filled with the white vapour of
ammonium chloride, by leading into it at the same time dry hydrochloric acid gas and dry ammonia gas. When quite full of dense white fumes, we give the cloth cover

of the box a sharp blow with the fist, and from the

round hole a white smoke ring leaps out and slides

through the air. The experiment may be made on a

smaller scale by using a cardboard box and filling it with the smoke of brown paper or tobacco.* If we look closely at the smoke ring as it glides through the air, we shall

see that the motion of the air or smoke particles com-

posing the ring is like that of an indiarubber umbrella-

ring fitted tightly on a round ruler and pushed along.

The ring turns itself continually over and over, the

rotation being round the circular ring axis line. This

rotatory motion is set up by the friction of the smoky

air against the edge of the hole in the box, as the puff

of air emerges from it when the back of the box is

A thumped.

simple but striking experiment may be made

without filling the box with smoke. Place a lighted

candle at a few feet away from the opening of the above-
described box, and strike the back. An invisible vortex

ring of air is formed and blows out the candle as it passes over it. Although it is quite easy to make a rotational

motion in an imperfect fluid, and in fact difficult not

to do it, yet of late years a very interesting and valuable

discovery has been made by Professor Hele-Shaw, of a

method of creating and rendering visible a motion in an

imperfect liquid like water, which is irrotational. This

* Some smokers can blow these sinoke rings from their mouth, and they may sometimes be seen when a gun is fired with blaek old-fashioned gunpowder, or from engine-funnels.

WAVES AND RIPPLES MADE BY 8HIP8. 63

li covery was that, if water is made to flow in a thin

t between two plates, say of flat glass, not more than

a fiftieth of an inch or so apart, the motion of the water

is exactly that of a perfect fluid, and is it-rotational. No

matter what objects may be placed in the path of tin:

water, it then flows round them just as if all fluid friction

or viscosity was absent.

This interesting fact can be shown by means of an

apparatus designed by Professor Hele-Shaw.* Two glass

plates are held in a frame, and separated by a very small

distance. By means of an inlet-pipe water is caused to

A How between the plates.

metal block pierced with

small holes is attached to the end of one plate, and this

serves to introduce several small jets of coloured water

into the main sheet. In constructing the apparatus great

care has to be exercised to make the holes in the above-

mentioned

block

very

small

(not

more

than

-
t J^-

inch

in

diameter) and placed exactly at the right slope.

The main water inlet-pipe is connected by a rubber

tube with a cistern of water placed about 4 feet above

the level of the apparatus. The frame and glass plates

are held vertically in the field of an optical lantern so as

to project an image of the plates upon the screen. The

side inlet- pipe leading to the pierced metal block is con-

nected to another reservoir of water, coloured purple with permanganate of potash (Condy's fluid), and the flow of

* For details and illustrations of these researches, the reader is

referred to papers by Professor H. S. Hele-Shaw, entitled, "Investigation

of the Nature of Surface-resistance of Water, and of Stream-line Motion

under Experimental Conditions," Proceedings of the Institution of Naval

A Architects, July, 1897, and March, 1898.

convenient apparatus for

exhibiting these experiments in lectures has been designed by Professor

Hole-Shaw, and is manufactured by the Imperial Engineering Company.

Pembroke Place, Liverpool.

64

WAVES AND KIPPLES.

both streams of water controlled by taps. The clear water is first allowed to flow down between the plates, so as to exclude all air-bubbles, and create a thin film of flowing water between two glass plates. The jets of coloured water are then iDtroduced, and, after a little adjustment, we shall see that the coloured water flows down in narrow, parallel streams, not mixing with the clear water, and not showing any trace of eddies. The regularity of these streams of coloured water, and their sharp definition, shows that the liquid flow between the plates is altogether
irrotational.
The lines marked out by the coloured water are called stream-lines, and they cut up the whole space into uniform tiibes of flow. The characteristic of this flow of liquid is that the clear water in the space between two coloured streams of water never passes over into an adjacent tube. Hence we can divide up the whole sheet of liquid into tubular spaces called tubes of flow, by lines called stream-
lines.
If now we dismount the apparatus and place between
the glass a thin piece of indiarubber sheet -cut, say, into
the shape of a ship, and of such thickness that it fills up the space between the glass plates we shall be able to
observe how the water flows round such an obstacle.
If the air is first driven out by the flow of the clear water, and then if the jets of coloured water are introduced, we see that the lines of liquid flow are delineated by coloured streams or narrow bands, and that these streamlines bend round and enclose the obstructing object.
The space all round the ship-shaped solid body is thus cut up into tubes of flow by stream-lines, but these tubes of flow are now no longer straight, and no longer of equal width at all points.

WAVES AND RIPPLES MADE BY SHIPS. 65
They are narrower opposite the middle part of the
obstruction than near either end.
At this point we must make a digression to explain a
fundaiiu'iital law roinvrning fluid flow in tubes. Suppose
we have a uniform horizontal metal tube, through which -r is flowing (see Fig. 28). At various points along

Fio. 28.

the tube let vertical glass pipes be inserted to act as
gauge or pressure-tubes. Then when the fluid flows along

the horizontal pipe it will stand up a certain height in

each pressure-tube, and this height will be a measure of

the pressure in the horizontal pipe at the point where the

We pressure-tube is inserted.

shall notice that when

the water flows in the horizontal pipe, the water in the

gauge-pipes stands at different heights, indicating a fall

We in pressure along the horizontal pipe.

also notice

that a line joining the tops of all the liquid columns in

the pressure-pipes is a straight, sloping line, which is

called the hydraulic gradient. This experiment proves to
us that when fluid flows along a uniform-sectioned pipe

there is a uniform fall or decrease in pressure along the pipe. The force which is driving the liquid along the

horizontal pipe is measured by the difference between

the pressures at its extreme ends, and the same is true of

any selected length of the horizontal pipe.

It will also be clear that, since water is not compres-

sible to any but the very slightest extent, the quantity of

F

66

WAVES AND E1PFLES.

water, reckoned, say in gallons, which passes per minute across any section of the pipe must be the same.
In the next place, suppose we cause water to flow through a tube which is narrower in some places than in
others (see Fig. 29). It will be readily admitted that

FIG. 29.
in this tube also the same quantity of water will flow across every section, wide or narrow, of the tube. If, however, we ask Where, in this case, will there be the greatest pressure ? it is certain that most persons would reply In the narrow portions of the tube. They would think that the water-particles passing through the tube resemble a crowd of people passing along a street which is constricted in some places like the Strand. The crowd would be most tightly squeezed together, and the pressure of people would therefore be greater, in the narrow portions of the street. In the case of the water flowing through the tube of variable section this, however, is not the case. So far from the pressure being greatest in the narrow portions of the tube, it can be shown experimentally that
it is precisely at those places it is least.
This can be demonstrated by the tube shown in Fig. 29. If water is allowed to flow through a tube constricted in some places, and provided with glass gauge-pipes at
various points to indicate the pressure in the pipe at those

WAVES AND RIPPLES MADE BY SHIPS. 07

places, it is found that the pressure, as indicated by the

height of the water in the gauge-glasses at the narrow

PHI -ts of the tube, is less than that which it would have at

those places if the tube were of uniform section and length,

We an I passed the same quantity of water.

can formulate

this fact under a general law which controls fluid motion

also in other cases, viz. that where the velocity of the liquid

is greatest, there the pressure is least. It is evident, since
the tube is wider in some places than in others, and as a

practically incompressible liquid is being passed through it, that the speed of the liquid must be greater in the narrow portions of the tube than in the wider ones. But
-xperiment shows that after allowing for what may be

called the proper hydraulic gradient of the tube, the

pressure is least in those places, viz. the constricted portions, where the velocity of the liquid is greatest. This general principle is of wide application in the science
of hydraulics, and it serves to enable us to interpret aright
many perplexing facts met with in physics.
We can, in the next place, gather together the various
facts concerning fluid flow which have been explained above, and apply them to elucidate the problems raised by the passage through water of a ship or a fish.
Let us consider, in the first place, a body totally submerged, such as a fish, a torpedo or a submarine boat,
and discuss the question why a resistance is experienced when an attempt is made to drag or push such a body through water. The old-fashioned notion was that the water has to be pushed out of the way to make room for the fish to move forward, and also has to be sucked in to fill up the cavity left behind. Most persons who have not been instructed in the subject, perhaps even now have the
idea that this so-called "head resistance" is the chief

68

WAVES AND EIPPLES.

cause of the resistance experienced when we make a body

A of any shape move through water.

common assumption

is also that the object of making a ship's bows sharp is
that they may cut into the water like a wedge, and more

easily push it out of the way. Scientific investigation has,

however, shown that both of these notions are erroneous.

The resistance felt in pulling or pushing a boat through

the water is not due to resistance offered by the water in

virtue of its inertia. No part of this resistance arises from

the exertion required to displace the water or push it out

of the way.

The Schoolmen of the Middle Ages used to discuss the

question how it was that a fish could move through the water. They said the fish could not move until the water

got out of the way, and the water could not get out of the

way until the fish moved. This and similar perplexities

were not removed until the true theory of the motion of a

solid through a liquid had been developed.
Briefly it may be said that there are three causes, and only three, for the resistance which we feel and have to

overcome when we attempt to drag a boat or ship through

the water. These are : First, skin friction, due to the

friction

between

the

ship-surface

and

the

water ;

secondly,

eddy-resistance, due to the energy lost or taken up in

making water

eddies ;

and

thirdly, wave-resistance,

due to

energy taken up in making surface-waves. The skin

friction and the eddy-resistance both arise from the fact

that water is not a perfect fluid. The wave-resistance

arises, as we shall show, from the unavoidable forma-

tion of waves by the motion of the boat through the

water.

In the case of a wholly submerged body, like a fish, the only resistance it has to overcome is due to the first

WAVES AND RIPPLES MADE BY SHIPS. 69
two causes. The fish, progressing through the water wholly under the surface, makes no waves, but the water adlito its skin, and there is friction between them as he moves. Also 1m creates eddies in the water, which require energy to produce them, and whenever mechanical work has to le done, as energy drawn off from a moving body, thiimplies the existence of a resistance to its motion whirl i
has to be overcome.
Accordingly Nature, economical on all occasions in nu'i'gy expenditure, has fashioned the fish so as to reduce the power it has to expend in moving through water as
much as possible. The fish has a smooth slippery skin. (We say "as slippery as an eel.") It is not covered either
with fur or feathers, but with shiny scales, so as to reduce
to a minimum the skin friction. The fish also is regular
and smooth in outline. It has no long ears, square shoulders, or projecting limbs or organs, which by giving it an irregular outline, would tend to produce eddies in the water as it moves along. Hence, when we wish to design a body to move quickly under the water, we must imitate in these respects the structure of a fish. Accordingly, a Whiteliead torpedo, that deadly instrument employed in naval warfare, is made smooth and fishshaped, and a submarine boat is made cigar-shaped and as smooth as possible, for the same reason.
If the floating object is partly above the surface, yet nevertheless, as far as concerns the portion submerged, there is skin friction, and the production of eddy-resistance. Hence, in the construction of a racing-yacht, the
greatest care has to be taken to make its surface below
water of polished metal or varnished wood, or other very smooth material, to diminish as far as possible the skin friction. In the case of bodies as regular in outline as a

70

WAVES AND HIPPIES.

ship or fish, the proportion of the driving power taken up in making eddies in the water is not large, and we may, without sensible error, say that in their case the whole resistance to motion is comprised under the two heads of skin friction and wave-making resistance. The proportion which these two causes bear to each other will depend upon the nature of the surface of the body which moves over the water, and its shape and speed.
At this point we may pause to notice that, if we could
obtain a perfect fluid in practice, it would be found that an object of any shape wholly submerged in the fluid could be moved about in any way without experiencing the least resistance. This theoretical deduction is, at
first sight, so opposed to ordinary preconceived notions on the subject, that it deserves a little attention. It is
difficult, as already remarked, for most people who have not carefully studied the subject, to rid their minds of
the idea that there is a resistance to the motion of a solid
through a liquid arising from the effort required to push the liquid out of the way. But this notion is, as already
explained, entirely erroneous.
In the light of the stream-line theory of liquid motion, it is easy to prove, however, the truth of the above statement.
Let us begin by supposing that a solid body of regular and symmetrical shape, say of an oval form (see Fig. 30), is moved through a fluid destitute of all stickiness or viscosity, which therefore does not adhere to the solid. Then, if the solid is wholly submerged in this fluid, the mutual action of the liquid and the solid will be the same, whether we suppose the liquid to be at rest and the solid to move through it, or the solid body to be at rest and
the liquid to flow past it.

WAVES AND RIPPLES MADE BY SHIPS. 71
If, then, wo suppose the perfect fluid to How round
the obstacle, it will distribute itself in a certain manner,
FIG. 30. Stream-lines round an ovoid.
and its motion can be delineated by stream-lines. There will be no eddies or rotations, because the liquid is by
FIG. 31. Tube of flow in a liquid.
assumption perfect. Consider now any two adjacent
stream-lines (see Fig. 31). These define a tube of flow, represented by the shaded portion, which is narrower in

72

WAVES AND RIPPLES.

the middle than at the ends. Hence the liquid, which we shall suppose also to be incompressible, must flow faster when going past the middle of the obstacle where the stream-tubes are narrow, than at the ends where the
stream-tubes are wider.

By the principle already explained, it will be clear

that the pressure of the fluid will therefore be less in

the narrow portion of the stream-tube, and from the

perfect symmetry of the stream-lines it is evident there

will be greater and equal pressures at the two ends of

the immersed solid. The flow of the liquid past the solid subjects it, in fact, to a number of equal and balanced

pressures at the two ends which exactly equilibrate each

other. It is not quite so easy to see at once that if the

solid body is not symmetrical in shape the same thing is

true, but it can be established by a strict line of reasoning. The result is to show that when a solid of any shape is

immersed in a perfect liquid, it cannot be moved by the

liquid flowing past it, and correspondingly would net require any force to move it against and through the

liquid. In short, there is no resistance to the motion of
a solid of any shape when pulled through a perfect or
frictionless liquid. When dealing with real liquids not

entirely free from viscosity, such resistance as does exist

is due, as already mentioned, to skin friction and eddy

formation. In the next place, leaving the consideration

of the movement of wholly submerged bodies through liquids whether perfect or imperfect, we shall proceed to

discuss the important question of the resistance offered

by water to the motion through it of a floating object,

We such as a ship or swan.

have in this case to take

into consideration the wave-making properties of the

floating solid.

WAVES AND RIPPLES MADE BY SHIPS. 73

We have already pointed out that to make Ji wave on

\\airr ivijuiivs iin expenditure of energy or the performa:

of mechanical work. If a wave is made and travels away

<>vrr water, it carries with it energy, and hence it can only
1)0 created if we have a store of energy to draw upon. If we suppose that skin friction is absent, and that the

ship floats upon a perfect fluid, it would nevertheless be

true that, if the moving object creates waves, it will thereby

reduce its own movement and require the application of

We force to it to keep it going.

may say therefore that

if any floating object creates waves on a liquid over

which it moves, these waves rob the floating body of

some of its energy of motion. The creation of the waves

will bring it to rest in time, unless it is continually

urged forward by some external and impressed force,

and wave-generation is a reason for a part at least of
the resistance we experience when we attempt to push

it along.
Accordingly, one element in the problem of designing
a ship is that of finding a form which will make as little wave-disturbance as possible in moving over the liquid.
It is comparatively easy to tiud a shape for a floating
solid which shall make a considerable wave-disturbance

on the water when it is pulled over it, but it is not quite so easy to design a shape which will not make waves, or make but very small ones.
If we look carefully at a yacht gliding along before a fresh breeze on a sea or lake surface which is not much

ruffled by other waves, it is possible to discover that a ship, when going through the water, creates four distinct
systems of waves. Two of these are very easy to see, and
two are more difficult to identify. These wave-systems are called respectively the oblique bow and stern waves,

74

WAVES AND BIPPLES.

We and the transverse and rear waves.

shall examine

each system in turn. The most important and easily observed of the four
sets of waves is the oblique bow wave. It is most easily seen when a boy's boat skims over the surface of a pond, and readily observed whenever we see a duck paddling along on the water. Let any one look, for instance, at a
duck swimming on a pond. He will see two trains of
little waves or ripples, which are inclined at an angle to

FIG. 32. Echelon waves made by a cluck.
the direction of the duck's line of motion. Both trains are made up of a number of short waves, each of which extends beyond or overlaps its neighbour (see Fig. 32).
Hence, from a common French word, these waves have
FIG. 33. Echelon waves made by a model yacht.
been called echelon waves* and we shall so speak of them. On looking at a boy's model yacht in motion on the
* The French word Echelon means a step-ladder-like arrangement ; but it is usually applied to an arrangement of rows of objects when each row extends a little beyond its neighbour. Soldiers are said to march in echelon when the ranks of men are so ordered.

WAVES AND HIPPIES MADE B7 SHIPS. 75

water, the same system of waves will be seen; and on looking at any real yacht or steamer in motion on smooth

watrr, they are quite easily identified (see Fig. 33).
The complete explanation of the formation of these bow or echelon waves is difficult to follow, but in a

general way their formation can be thus explained: Suppose we have a fiat piece of wood, which is held upright in water, and to which we give a sudden push.
We shall notice that, in consequence of the inertia of the
liquid, it starts a wave which travels away at a certain speed over the surface of the water. The sudden movement of the wood elevates the water just in front of it, and this displacement forms the crest of a wave which is

then handed on or propagated

along the surrounding watersurface. If two pieces of
wood are fastened together obliquely, as in Fig. 34, and

held in water partly sub-
merged, we shall find that when this wood is suddenly

thrust forward like a wedge, it starts two oblique waves
which move off parallel to the inclined wooden sides.

The bows of a ship, roughly speaking, form such a wedge.

FIG. 34.

Hence, if we consider this wedge or the bows of a ship

to be placed in still water and then pushed suddenly for-

ward, they will start two inclined waves, which will move

off parallel to themselves.
If we then consider the wedge to leap forward and repeat the process, two more inclined waves will be

76

WAVES AND RIPPLES.

formed

in

front

of

the

first ;

and

again

we

may suppose

the process repeated, and a third pair of waves formed.

The different positions of the ship's bows are shown in

the

diagram

at 1, 2, and

3

in

Fig.

35 ;

and c, e, and/ are

the three corresponding sets of echeloned waves. For the

sake of simplicity, the waves are shown on one side only.

If, then, we imagine the ship to move uniformly forwards,

its bows are always producing new inclined waves, which

move with it, and it is always, so to speak, leaving the

FIG 35.

old ones behind. All these echelon waves produced by the bow of the ship are included within two sloping lines which each make with the direction of the ship's line of
movement, an angle of 19 28'.* This angle can be thus
set off: Draw a circle (see Fig. 36), and produce the diameter BC of this circle for a distance, CA, equal to its
A own length. From the end of the produced diameter
draw a pair of lines, AD, AD', called tangents, to touch the circle. Then each of these lines will make an angle of 19 28' with the diameter. If we suppose a ship to be
A placed at the point marked in the diagram (see Fig. 36),
all the echelon waves it makes will be included within
these lines AD, AD'.

*

See Lord Kelvin on

" Ship Waves," Popular Lectures,

vol.

iii.

p.

482.

WAVES AND RIPPLES MADE 71Y SIJ/

77

Moreover, the angle of the lines will not alter, whether the ship goes fast or slow. This is easily seen in the case of a duck swimming on a lake. Throw bits of bread to a duck so as to induce it to swim faster or slower, and
notice the system of inclined or echelon ripples made by

the duck's body as it swims. It will be seen that the angle at which the two lines, including both the trains of echelon ripples meet each other is not altered as the duck

changes its speed.

This echelon system of inclined waves is really only a

part of a system of waves which is completed by a trans-

A verse group in the rear of the vessel.

drawing has been

given

by

Lord

Kelvin,

in

his

lecture

on

"
Ship

Waves,"

of

the complete system of these waves, part of which is as

represented by the firm lines in Fig. 37. This complete

system is difficult to see in the case of a real ship moving over the water. The inclined rear system of waves can sometimes be well seen from the deck of a lake steamer,

such as those on the large Swiss or Italian lakes, and may

sometimes be photographed in a snap-shot taken of a boy's yacht skimming along on a pond,

78

WAVES AND RIPPLES.

In addition to the inclined bow waves, there is a similar system produced by the stern of a vessel, which is, however, much more difficult to detect. The other two
wave-systems produced by a ship are generally called the

FIG. 37.
transverse waves. There is a system of waves whose
crest-lines are at right angles to the ship, and they may
be seen in profile against the side of any ship or yacht as it moves along. These transverse waves are really due to the unequal pressures resulting from the distribution of the stream-lines delineating the movement of the water
past the ship.
If we return again to the consideration of the flow of a perfect fluid round an ovoid body, it will be remembered that it was shown that, in consequence of the fact that the stream-lines are wider apart near the bow and stern than
they are opposite the middle part of the body, the pres-
sure in the fluid was greater near the bow and stern than
at the middle. When a body is not wholly submerged,
but floats on the surface as does a ship, these excess pres-
sures at the bow and stern reveal themselves by forcing up the water- surface opposite the ends of the vessel and lowering it opposite the middle. This may be seen on

WAVES AND NIPPLES MADE BY SHI

TO

looking at any yacht in profile as it sails. The yacht appears to rest on two cross-waves, one at the bow and

FIG. 38.

one at the stern, and midships the water is depressed (see

Fig. 38).
These waves move with the yacht. If the ship is a

long one, then each of these waves gives rise to a wave-

train ;

and on looking

at

a

long

ship in motion, it will be

seen that, in addition to the inclined bow wave-system,

there is a series of waves which are seen in profile against

the hull.

When a ship goes at a very high speed, as in the case
of torpedo-boat destroyers, the bow of the vessel is gene-

rally forced right up on to the top of the front transverse

waves, and the boat moves along with its nose entirely

out of water (see Fig. 39). In fact, the boat is, so to speak, always going uphill, with its bows resting on the side of a wave which advances with it, and its stern

80

WAVES AND RIPPLES.

followed by another wave, whilst behind it is left a con-
tinually lengthening trail of waves, which are produced by those which move with the boat.

The best way to see all these different groups of shipwaves is to tow a rather large model ship without masts or sails in fact, a mere hulk over smooth water in a

canal or lake. Let one person carry a rather long pole, to

the end

of

which

a

string

is

tied ;

and

by means of

the

string let the model ship be pulled through the water.

Let this person run along the banks of the canal or lake,

and tow the ship steadily through the water as far as

possible at a constant speed. Let another person, provided with a hand camera, be rowed in a boat after the

model, and keep a few yards behind. The second observer will be able to photograph the system of ship-waves made by the model, and secure various photographs when the model ship is towed at different rates. The echelon and transverse waves should then be clearly visible, and if the water is smooth and the light good, it is not difficult to

secure many useful photographs. By throwing bits of bread to ducks and swans disport-
ing themselves on still water, they also may be induced
to take active exercise in the right direction, and expose themselves and the waves or ripples that they make to the
lens of a hand camera or pocket kodak. From a collection

of snap-shot photographs of these objects the young in-
vestigator will learn much about the form of the waves made by ships, and will see that they are a necessary accompaniment of the movement of every floating object on water. By conducting experiments of the above
kind under such conditions as will enable the exact

speed of the model to be determined, and the resistance it experiences in moving through the water, information

WAVES AND RIPPLES MADE B7 SHIPS. 81

!M been accumulated of the utmost value to ship-

builders.

Our scientific knowledge of the laws of ship-iwe o\\r chiefly to the labours of two great rn-incers, Mi. Scott llussell and Mr. William Froude. Mr. Frond

wurk was begun privately at Torquay about the year 1870,

and was subsequently continued by him for the British

Admiialty. ,Mr. Froude was the first to show the value

and utility <>!' experiments made with model ships dragged

through the water. He constructed at Torquay an experi-

ment tank about 200 feet in length, which was a sort <f

covered swimming-bath, and he employed for his experi-

ments model ships made of wood or paraffin wax, the

latter being chosen because the model could be so easily

cut to the desired shape, and all the chips and the model

itself could be melted up and used over again for subse-

quent experiments. Without detailing in historic order

his discoveries, suffice it to say that, as the outcome of hi-

work, Mr. Froude was able to state two very important

laws which relate to the relative resistance experienced

when two models of different sizes are dragged through

the water at different speeds.

The first of these relates

to

what

is

called

the

"
01

*l>'>nding speeds" Suppose we have a real ship 250 feet long, and we make an exact model of this ship 10 feet
long, then the ship is twenty-five times longer than the model. Mr. Fronde's law of corresponding speeds is as follows :

If the above model and the ship are both made to move over still water, the ship going five times as fast as the model, the system of waves made by the model will exactly reproduce on a smaller scale the system of waves made by the ship. In other words, if we were to take a couple of
a

82

WAVES AND RIPPLES.

photographs, one of the ship going at 20 miles an Tiour, and one of the model one twenty-fifth of its size going at 4 miles an hour, and reduce the two photographs to the same size, they would be exactly alike in every
detail.
Expressed in more precise language, the first law of
Fronde is as follows : When a ship and a model of it
move through smooth water at such speeds that the speed of the ship is to the speed of the model as the square root
of the length of the ship is to the square root of the length of the model, then these speeds are called " corre-
sponding speeds." At corresponding speeds the wavemaking power of the model resembles that of the ship on
a reduced scale. If we call L and I the lengths of the ship and the model, and S and s the speeds of the ship and
the model, then we have

V S = /L

5

7

where S and s are called corresponding speeds.

Mr. Froude then established a second law of equal

importance, relating to that part of the whole resistance

due to wave-making experienced by a ship and a model,

or by two models when moving at corresponding speeds.

Mr. Froude's second law is as follows : If a ship and

a model

are

moving

at

" corresponding

speeds,"

then

the

resistances to motion due to wave-making are proportional

to the cube of their lengths. To employ the example

given above, let the ship be 250 feet long and the model

10 feet long, then, as we have seen, the corresponding

speeds are as 5 to 1, since the lengths are as 25 to 1. If,
therefore, the ship is made to move at 20 miles an

hour, and the model at 4 miles an hour, the resistance

WAVES AND RIPPLES MADE tiY SHIPS. 83

erieiiccd by the ship due to wave-making is to that

experienced hy the model as the cube of 2.~ is in the cube

of 1, or in ratio oi

to I ~>, <'>-."

1.

In symbols the second law

may In- expressed thus: Let Ji be the resistance due to

wave-making experienced by the ship, and r that of tin;

model when moving at corresponding speeds, and let L

and / be their lengths as before; then

2 ~ ff

r

I*

I'M -fore these laws could be applied in the design of
real ships, it was necessary to make experiments to ascertain the skin friction of different kinds of surfaces when moving
through water at various speeds. Mr. Fronde's experiments on this point were very
extensive. For example, he showed that the skin friction of a clean copper surface such as forms the sheathing of a
ship may be taken to be about one quarter of a pound per square foot of whetted surface when moving at 600 feet a
minute. This is equivalent to saying that a surface of
4 square feet of copper moved through water at the rate
of 10 feet a second experiences a resisting force equal to the weight of 1 Ib. due entirely to skin friction. Very roughly speaking, this skin resistance increases as the square of the speed.* Thus at 20 feet per second the skin friction of a surface of 4 square feet of copper would be 4 Ibs., and at 30 feet per second it would
be 9 Ibs. Any roughness of the copper surface, however,
greatly increases the skin friction, and in the case of a ship the accumulation of barnacles on the copper sheathing has an immense effect in lowering the speed of the vessel by increasing the skin friction. Hence the necessity
* More accurately, as the 1'83 power of the speed.

84

WAVES AND RIPPLES.

for periodically cleaning the ship's bottom by scraping off these clinging growths of seaweed and barnacles.
Mr. Froude also made many experiments on surfaces of
paraffin wax, because of this material his ship models were
made. It may suffice to say that the skin friction in this
case, in fresh water, is such that a surface of 6 square feet of paraffin wax, moving at a speed of 400 feet per minute, would experience resistance equal to the weight of 1 Ib. There are, however, certain corrections which have to be applied in practice to these rules, depending upon
the length of the immersed surface. The mean speed of the water past the model or ship-surface depends
on the form of the stream-lines next to it, and it has already been shown that the velocity of the water next to the ship is not the same at all points of the ship-surface. It is greater near the centre than at the ends. Hence the longer the model, the less is the mean resistance per square
foot of wetted surface due to skin friction when the model
is moved at some constant speed through the water. The above explanations will, however, be sufficient to
enable the reader to understand in a general way the
problem to be solved in designing a ship, especially one
intended to be moved by steam-power.
If a shipbuilder accepts a contract to build a steamer say a passenger-steamer for cross-Channel services he is put under obligation to provide a ship capable of travel-
ling at a stated speed. Thus, for instance, he may under-
take to guarantee that the steamer shall be able to do 20 knots in smooth water. In order to fulfil this contract
he must be able to ascertain beforehand what enginepower to provide. For, if the engine -power is insufficient,
he may fail to carry out his contract, and the ship may
be returned on his hands. Or if he goes to the opposite