GRAPHIC REPRESENTATION OF WAVES
A system of ripples on the surface of water appears in vertical section
at any instant somewhat as in Fig. 10. The dotted line AB represents
the undisturbed surface of the wafer, and the solid line the actual
surface. If the disturbance which is causing the ripples is an
oscillation of perfectly regular period the individual ripples will be
all alike, except they will get shallower as they become more remote
from the disturbance.
[Illustration: FIG. 10.]
+Wave-length.+--The distance between two successive crests will be the
same everywhere, and this distance or the distance between any two
corresponding points on two successive ripples is called the
wave-length. Evidently, the wave-length is the distance in which the
whole wave repeats itself.
+Phase.+--The position of a point in the wave is called the phase of
the point. Thus the difference of phase between the two points A and C
is a quarter {23} of a wave-length. As the waves move on along the
surface it is evident that each drop of water executes an up and down
oscillation, and at the points C, C the drop has reached its highest
position and at the points T, T its lowest.
+Amplitude.+--The largest displacement of the drop, _i.e._ the distance
from the dotted line to C or to T, is called the amplitude of the wave.
The time taken for a drop to complete one whole oscillation, _i.e._ the
time taken for a wave to travel one whole wave-length forward, is
called the period of the wave. The number of oscillations in one
second, _i.e._ the number of wave-lengths travelled in one second, is
called the frequency.
[Illustration: FIG. 11.]
Although there is no visible displacement in the waves of light and
heat, yet we may represent them in much the same way. Thus if AB, Fig.
10, represents the line along which a ray of light is travelling, the
length NP is drawn to scale to represent the value of the electric
field at the point N, and is drawn upwards from the line AB when the
field is in one direction and downwards when it is in the opposite
direction.
Thus the direction of the field at different points in the wave XY,
Fig. 11, is shown by the dotted arrows as if due to electrified rods of
quartz and ebonite placed above and below XY.
In the case of the electromagnetic wave, the {24} amplitude will be the
maximum value to which the electric field attains in either direction,
and the other terms--wave-length, phase, period and frequency--will
have the same meaning as for water ripples.
+Wave Form.+--Waves not only differ in amplitude, wave-length, and
frequency, but also in wave form. Waves may have any form, _e.g._ Fig
12. Or we may have a solitary irregular disturbance such as is caused
by the splash of a stone in water.
[Illustration: FIG. 12.]
But there is one form of motion of a particle in a wave which is looked
upon as the simplest and fundamental form. It is that form which is
executed by the bob of a pendulum, the balance wheel of a watch, the
prong of a tuning-fork, and most other vibrations where the controlling
force is provided by a spring or by some other elastic solid.
It is called "Simple Harmonic Motion" or "Simple Periodic Motion," and
the essential feature of it is that the force restoring the displaced
particle to its undisturbed position is proportional to its
displacement from the undisturbed position. A wave in which all the
particles execute simple harmonic motion has the form in Fig. 10 or
Fig. 11, which is therefore looked upon as the fundamental wave form or
simple wave form.
Simple waves will vary only in amplitude, wave-length, and frequency,
and the energy in the wave will depend upon these quantities.
{25}
+Energy in a Simple Wave.+--If the velocity is the same for all
wave-lengths, then the frequency will evidently be inversely
proportional to the wave-length and the energy will depend upon the
amplitude and the wave-length. The kinetic energy of any moving body,
_i.e._ the energy due to its motion, is proportional to the square of
its velocity, and we may apply this to the motion of the particles in a
wave and to show how the energy depends upon the amplitude and
wave-length.
Since the distance travelled by a particle in a single period of the
wave will be equal to four times the amplitude, the velocity at any
point in the wave must be proportional to the amplitude and therefore
the kinetic energy is proportional to the square of the amplitude.
With the same amplitude but with different wave-lengths, we see that
the time in which the oscillation is completed is proportional to the
wave-length and that the velocity is therefore inversely proportional
to the wave-length. The kinetic energy is therefore inversely
proportional to the square of the wave-length.
+Addition of Waves.+--The superposition of two waves so as to obtain
the effect of both waves at the same place is carried out very simply.
The displacements at any point due to the two waves separately are
algebraically added together, and this sum is the actual displacement.
In Fig. 13 the dotted lines represent two simple waves, one of which
has double the wave-length of the other. At any point P on the solid
line, the displacement PN is equal to {26} the algebraic sum of the
displacement NQ due to one of the waves and NR due to the other. The
solid line, therefore, represents the resulting wave. We may repeat
this process for any number of simple waves, and by suitably choosing
the wave-length and amplitude of the simple waves we may build up any
desired form of wave. The mathematician Fourier has shown that any
form of wave, even the single irregular disturbance, can thus be
expressed as the sum of a series of simple waves and that the
wave-lengths of these simple waves are equal to the original
wave-length, one-half of it, one-third, one-quarter, one-fifth, and so
on in an infinite series. Fourier has also shown that only one such
series is possible for any particular form of wave.
[Illustration: FIG. 13.]
The importance of this mathematical expression lies in the fact that in
a number of ways Fourier's series of simple waves is manufactured from
the original wave and the different members of the series become
separated. Thus the most useful way in which we can represent any wave
is, not to draw the actual form of a wave, but to represent what simple
waves go to form it and to show how much energy there is in each
particular simple wave.
{27}
+Energy--Wave-length Curve.+--This can be done quite simply as in Fig.
14. The distance PN from the line OA being drawn to scale to represent
the energy in the simple wave whose length is represented by ON.
[Illustration: FIG. 14.]
Thus the simple wave of length OX has the greatest amount of energy in
it.
[Illustration: FIG. 15.]
Fig. 15 wall represent a simple wave of wave-length OX, the energy in
all the other waves being zero.
{28}
The three curves given in Fig. 16 give a comparison of the waves from
the sun, an arc lamp, and an ordinary gas-burner.
[Illustration: FIG. 16.]
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