Transcriber’s Note
In what follows, italic text is denoted by _underscores_ while bold
text is denoted by =equal signs=. Small capitals in the original text
have been transcribed as ALL CAPITALS. Superscripted text is prefixed
by the ‘^’ symbol. Braces ‘{}’ may enclose the text when necessary to
avoid ambiguity. Subscripted text is always enclosed in braces ‘{}’ but
will not be preceded by a ‘^’.
—————
See the end of this document for details of corrections and other
changes.
————————————————— Start of Book —————————————————
[Illustration: A CHRISTMAS LECTURE AT THE ROYAL INSTITUTION: “WAVES AND
RIPPLES IN THE AIR.”
FIG. 46 (_see p. 109_).
_Drawn by F. C. Dickinson._] [_From the “Graphic.”_
]
WAVES AND RIPPLES
IN
WATER, AIR, AND ÆTHER
BEING
A COURSE OF CHRISTMAS LECTURES DELIVERED
AT THE ROYAL INSTITUTION OF
GREAT BRITAIN
BY
J. A. FLEMING, M.A., D.Sc., F.R.S.
M. INST. E.E., M.R.I., ETC., ETC.
PROFESSOR OF ELECTRICAL ENGINEERING IN UNIVERSITY COLLEGE, LONDON
_SECOND ISSUE, REVISED_
LONDON
SOCIETY FOR PROMOTING CHRISTIAN KNOWLEDGE
NORTHUMBERLAND AVENUE, W.C.
43, QUEEN VICTORIA STREET, E.C.
BRIGHTON: 129, NORTH STREET
NEW YORK: E. S. GORHAM
1912
[PUBLISHED UNDER THE DIRECTION OF THE GENERAL LITERATURE COMMITTEE]
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 those of maturer years. A subject which admits of
abundant experimental illustrations is accordingly, on these occasions,
a popular one, particularly if it has a bearing upon topics then
attracting public attention. The progress 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 purview of the popular mind. A demand then arises for explanations
which shall dovetail it on to the ordinary experiences of life. The
practical use of æther 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 Royal 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 explanations 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 find 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.
WATER WAVES AND WATER RIPPLES.
PAGE
A visit to the seaside—What is a wave?—Wave-motion on
water—Definition of a wave—Sea waves—Various forms of
wave-motion—Wave length, velocity, and frequency—Atlantic
waves—Rules for speed of sea waves—Illustrations of
wave-motion—A stone falling on water—Production of a
wave-train—Wave-energy—Conditions for the production
of wave-motion—Distinction between wave-velocity and
wave-train velocity—Why a wave breaks—Waves in canals—Rule
for speed of a canal wave—Falling bodies—A “bore”—Tidal
waves—Ripples—Distinction between waves and ripples—Surface
tension on liquids—A needle floating on water—Experimental
production of ripples—Reflection and refraction of ripples
and waves—Interference of waves and ripples—Photography of
waves and ripples 1
CHAPTER II.
WAVES AND RIPPLES MADE BY SHIPS.
Ship-waves—The viscosity of liquids—How it is
demonstrated—Rotational and irrotational motion in fluids—Eddies
and whirls—Smoke rings—Vortex motion—Professor Hele-Shaw’s
experiments—Irrotational or stream-line motion in water—The
motion of water round a ship—The motion of water along a
pipe—Flow in uniform pipes and non-uniform pipes—Relation
between 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.
Air necessary for the production of sound—A 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 waves—A 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—Stationary
waves—A 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—Helmholtz’s theory of discords—Musical
instruments—Pipes—Strings and plates—A pan-pipe—An
organ-pipe—Open and closed organ-pipes—The distribution of
air pressure and velocity in a sounding organ-pipe—Singing
flames—Stringed instruments—The violin—The Stroh violin—The
structure of the ear—The ear a wonderful air-wave detector
and analyzer 147
CHAPTER V.
ELECTRIC OSCILLATIONS AND ELECTRIC WAVES.
The conception of an æther—The phenomena of light require the
assumption of an æther—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—Electromotive force and electric
strain—A 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 ÆTHER.
The experiments of Heinrich Hertz—Electric radiation—Lecture
apparatus for producing and detecting electric radiation—Electric
transparency and opacity—Why 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 æther waves—The
gamut of æther waves—The eye an æther-wave detector of limited
power—The electro-magnetic theory of light—Artificial production
of light—Use 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 232
APPENDIX 287
INDEX 293
USEFUL MEMORANDA.
One statute mile is 5280 feet.
One nautical mile is 6086 feet = 1¹⁄₆ statute mile.
A knot is a speed of 1 nautical mile per hour.
Hence the following rules:—
To convert
Knots to miles per hour—multiply by 1¹⁄₆.
Miles per hour to knots—multiply by ⁶⁄₇.
Feet per second to miles per hour—multiply by ²⁄₃. ⁄
Feet per second to knots—multiply by ⁶⁄₁₀.
Knots to feet per minute—multiply by 100.
WAVES AND RIPPLES IN WATER,
AIR, AND ÆTHER.
—⋄—
CHAPTER I.
WATER WAVES AND WATER RIPPLES.
We have all stood many times by the seashore, watching the _waves_,
crested with white foam, roll in and break upon the rocks or beach.
Every one has more than once cast a stone upon still water in a lake
or pond, and noticed the expanding rings of _ripples_; and some have
voyaged over stormy seas, whereon great ships are tossed by mighty
_billows_ with no more seeming effort than the rocking of a cradle. In
all these things we have been spectators of a _wave-motion_, as it is
called, taking place 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, produced in a medium called the _æther_, 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 waves in æther—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 æther, 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 æther waves,
created in the all-pervading æther, 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 and let down again, or, at most, has a small movement to and
fro. We are led, therefore, to infer that, even when agitated by waves,
each particle of water never moves far from its position when at rest,
and that the real movement of the water is something very different
from its apparent motion. If we place on the surface of water a number
of corks 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 after the other, and not all together. A
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
up, then pushed forward, next let down, and, lastly, pulled back; 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 represent the
paths described by corks floating on the sea-surface when waves are
travelling over it.
[Illustration: FIG. 1.]
Accordingly, we conclude that we have to distinguish clearly between
the actual individual motion of each water particle 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 along any line
must perform this same motion one after the other, that is, lagging
behind each other, and not simultaneously. We 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 separate 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 is 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
of its neighbour on the other side. The result will be to produce a
wave-motion, and, looking at the general effect of the moving knobs, we
shall see that it resembles a hump moving along, just as in the case of
a water wave.
[Illustration: FIG. 2.]
[Illustration: FIG. 3.]
The motion of the particles of the water in a deep-sea wave resembles
that of the white knobs in the model described. Those who swim will
recall to mind their sensations as a sea wave surges over them. The
wave 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.
[Illustration: FIG. 4.]
Again, another form of wave-motion may be illustrated by the model
shown in Fig. 4. In this case a number of 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 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 called a _longitudinal_
wave.
At this stage it may be well to define the meaning of some other
expressions which will be much used in these lectures. We 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 stage 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 defined 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 waves_; but if we make
these pleats very far apart, they would represent _long waves_.
Another phrase much used is the term _wave-velocity_. 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 wave-tossed 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 exaggeration. 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
37¹⁄₂ 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, the
eye of the inexperienced voyager is apt to be deceived.
In all cases of wave-motion there is a very close connection between
the wave-velocity, or speed, the wave-length, 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 first 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_ × _wave-frequency_.
This relation, which we shall have frequent occasion to recall, may be
stated in another manner. We call the _period_ of a wave the time taken
to make one complete movement. The periodic time is therefore inversely
proportional 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 wave-velocity is determined
by the wave-length. This is not the case, as we shall see, with waves
in air or waves in æther. 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 2¹⁄₄ 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 2¹⁄₄ 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.[1]
The above rule for the speed of deep-sea waves, viz. _wave-velocity_ =
_square root of 2¹⁄₄ 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 racing 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.[2]
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 50 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 with 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 circular 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 the
circle which is described by each water-particle is only ¹⁄₅₃₅ of that
at the surface.[3] 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
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 movement in turn. We 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.
[Illustration: FIG. 5.]
[Illustration: FIG. 6.]
[Illustration: FIG. 7.]
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, 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 paper. Then turn it round like a screw. We shall see that
the shadow of the spiral is a wavy line, and that, as 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 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 rise and fall. If the movement is
rapid enough, the appearance of a wave moving along will be seen.[4]
In all these exhibitions of wave-motion the movement of the particles
is due to a common cause, but the moving particles 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 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 displaced, 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 threads,
which resist stretching. When 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. 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 again to return on its steps. We see, therefore,
that each ball must oscillate, or swing to and fro, and that its
movement is gradually communicated to its neighbours. A wave-motion is
thus started, and a true wave is propagated along the line of balls,
in consequence of the presence of _elasticity_ and _inertia_. The
necessary conditions for the production of a true wave in a medium of
any kind are therefore: (1) that the medium must elastically 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 consequence of _inertia_, or something equivalent to it.
Briefly speaking, any material or medium in or on 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, that, for one thing, it resists being made
unlevel. A 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 process. We can
dig a hole in sand, or heap up sand to form a hillock, but we know
full well we cannot do the same thing with water. If, for instance,
some water is placed in a glass tube shaped like the letter ⋃,
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 possesses 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 dispersion 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 interesting work on this
subject is that of Professor A. M. Worthington, 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.[5] 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. Some of Professor Worthington’s results
for a drop of water falling into milk are reproduced in the appended
diagrams. In the first place (Fig. 8) the drop is seen just entering
the water. As it plunges down, it leaves behind it a cavity, or, as it
may be called, a hole in the water (see Fig. 9).
[Illustration: Time after contact = ·0262 sec.
FIG. 8.]
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. 10). No one could anticipate such an
extraordinary 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 up-lifting, force. The effect of this is exactly
analogous to that of giving a blow to the interconnected string of
balls in the model shown in Fig. 7—it propagates a wave. In Fig. 10
is illustrated the next stage, in which this outward-moving initial
wave-crest is shown.
[Illustration: Time after contact = ·0391 sec.
FIG. 9.]
[Illustration: Time after contact = ·101 sec.
FIG. 10.]
So much for the events revealed by the flash-light of an electric
spark; but succeeding these there is a long train of interesting
wave-making performance which can be watched with the eye, or the
stages photographed with 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 once spent a
morning throwing stones into a pond. A friend reproved him for his
idleness. “No,” said the painter, “I have not been idle; I have learnt
how to paint a 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.
[Illustration: FIG. 11.—Ripples on a lake (Sierre), produced by
throwing in a stone.]
The photograph in Fig. 11 will, however, show one stage in the event.
As soon as the first wave-crest, the origin of which we have already
explained, is formed, it 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 _wave-group 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 inner edge, and dying away on its outer 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 subject
in 1876, and subsequently it was elucidated by Professor Osborne
Reynolds[6] and Lord Rayleigh.
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 vibrating balls is displaced may be
called the velocity of the wave-train. The velocity of the wave is,
however, something 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 conveying energy. It is difficult
to give in a compact form any simple definition of what is meant in
modern scientific writings by the word _Energy_.
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, 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 Conservation 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.
In addition to various material substances we find that 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, magnetization, 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 equivalent_, 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 an equivalent of energy in some other form. We 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 ourselves in the presence of inscrutable mysteries. We
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 Energy. We 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.
Returning, 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
transferred 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 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 is strained or deformed, energy
has to be expended on it to create the deformation. Thus to wind up a
watch-spring, 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 Energy 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, therefore, 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 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.
[Illustration: FIG. 12.]
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 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 which slightly
squeezes both out of shape. This pressure is just sufficient to bring
the first ball to rest. The second ball, in turn, expands after the
blow and squeezes the 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 wave-train is ever extending in length and the
oscillatory motion 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 wave-velocity 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 the individual boy, because the former is
3 miles an hour and the latter is 6 miles an hour.
Before 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 that, as it nears the shore, it gets steeper on the shore
side, and gradually 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 the water at its highest position. An 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 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 fluid, being saturated and filled with
water, but the movement of this bottom sand-logged water is hindered
by the sand, and hence the layer of overlying water moves over it 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 be moulded into a
wave-form by the wind, and very curious effects of this kind have been
noticed and described by Dr. Vaughan Cornish, who has made a great
study of the science of waves.[7]
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.
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 63¹⁄₂ lbs.,
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 which 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 _bran_ in the water, or floating in it some glass
balls which have been adjusted so as to just float anywhere in the
water. When this is done, and a wave started in the tank, it runs up
and down, being reflected at each end (see Fig. 13).
[Illustration: FIG. 13.—Water-wave produced in a tank.]
From the motion of the bran we can see that the water swings backwards
and forwards in a horizontal line with a pendulum-like motion, but its
up-and-down or vertical motion is much more restricted. A wave of this
kind travels 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,[8]
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 6 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
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 on the deeper water has travelled along the tank more quickly than
the wave on the shallower water.
[Illustration: FIG. 14.]
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
velocity to be 40 feet per second, or about 26 miles an hour.
The force of the blow which a body administers and suffers 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. When 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.
[Illustration: 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 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 is on a slope and part is horizontal. A 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 performed 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, 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 before the mind what is meant
by saying, in scientific language, that one thing “varies as the
square root” of another. We 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 the 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, however, one important fact connected with wave-propagation 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 12¹⁄₂ hours. The cause
of this change of level in the water-surface 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 effects,
however, are much complicated by the fact 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, there 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.[9] 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 jambed 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, travels
faster than the front part and overtakes it, producing thus a flat or
straight-fronted wave which goes forward with tremendous speed.[10]
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 contracts under
existing conditions so as to become as small as possible. We 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 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[11] that on the free
surface of a liquid, like water, what are 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 (0·78 foot) per second.
More strictly speaking, the matter should be 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 ripples round the wire which moves with it.
These ripples are smaller and closer together the faster the wire is
moved.
Ripples on water are formed in circular expanding rings when rain-drops
fall upon the still surface of a lake or pond, or when drops of water
formed in any other way fall in the same manner. On the other hand, a
stone flung into quiet and deep water will, in general, create waves
of wave-length greater 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 lantern.
A 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 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 follows: A 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 intermittent. 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 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 instance, 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 reflection of the original ripples 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 of the ripple P is in front
of it (see Fig. 17). In the diagram the dotted curves represent the
reflected ripple-crests.
[Illustration: FIG. 16.]
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 above-mentioned rule. We here
obtain a glimpse of a principle which will come before us again in
speaking of æther waves, and furnishes an explanation of the familiar
optical fact that when we view our own reflection in a looking-glass,
the image appears to be as far behind the glass as we are in front of
it.
[Illustration: FIG. 17.—Reflection of circular ripples.]
[Illustration: FIG. 18.]
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 employed 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). The positions of the
two pins A 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 tangent, 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 focus A is in
front of it. Accordingly, it follows that circular ripples diverging
from one focus A of an ellipse must, after reflection at the elliptical
boundary, be converged to the other focus B. This can be shown by the
use of the above described apparatus in a pretty manner.
[Illustration: FIG. 19.]
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 one 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 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, and
each behave as if the other did not exist. A 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.
[Illustration: FIG. 20.—Intersecting ripples produced on a lake by
throwing in simultaneously two stones.]
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 mutual
destruction of wave by wave is called _interference_, and it is a very
important fact in connection with wave-motion. It is not too much to
say that whenever we can 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 points, A 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 it is that the
difference of the distances AP 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 there are 100¹⁄₂ waves, or 101¹⁄₂ or 103¹⁄₂, etc., waves,
then the crest of a wave from A will reach P at the same time as the
hollow of a wave from B, and there will be no wave at all at the point
P. This is true for all such positions of P that the difference of its
distances from A and B are constant.
[Illustration: FIG. 21.]
But again, we may choose a point, Q, such that the difference of
its distances from A and B is equal to an _even_ number of half
wave-lengths, so that whilst in the length AQ there are, say, 100
waves, in the distance BQ there are 101, 102, 103, etc., waves. When
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 any two
points, A 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).
[Illustration: FIG. 22.]
[Illustration: FIG. 23.—Interfering ripples on a mercury surface,
showing interference along hyperbolic lines (Vincent).]
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 to create and adjust two sets of similar water ripples from
origins not far apart, and on looking at the complicated shadow-pattern
due to the interference of the waves, we shall be able to trace out
certain white lines along which the waves are annulled, these lines
being hyperbolic curves (see Fig. 23). With the same appliances
another characteristic of wave-motion, which is equally important, can
be well shown.
[Illustration: FIG. 24.]
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 portion.
We 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 boundary 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 characteristic of wave-motion,
and we shall have brought before us the analogous facts in speaking of
waves in air and waves in æther.
[Illustration: FIG. 25.]
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 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 that the direction of the line of
soldiers will be swung round because, whilst the soldier on the extreme
left marches, say, 300 feet, the one on the extreme right will have
gone 400 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
retardation 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 _parabolic_ 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 he
has photographed the ripples so formed, and given examples of their
reflection and refraction, which are well worth study.[12]
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_, _b_, _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′_, _b′_, _c′_, _d′_, etc., lying behind the boundary,
each of which is the _image_ of the corresponding real source-points,
and lies as far behind the boundary as the real point lies in front of
it.
[Illustration: FIG. 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 seaside, 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 seaside pools, the surface
of which is corrugated with trains of ripples by the breeze.