CHAPTER II

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.

+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.

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.

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.

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+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.

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.

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+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.

Thus the simple wave of length OX has the greatest amount of energy in it.

Fig. 15 wall represent a simple wave of wave-length OX, the energy in all the other waves being zero.

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The three curves given in Fig. 16 give a comparison of the waves from the sun, an arc lamp, and an ordinary gas-burner.

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