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Plane Waves Of Simple Type

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PLANE WAVES OF SIMPLE TYPE. Whatever may be the character of the medium and of its vibration, the analytical expression for an infinite train of plane waves is • A cos -21(Vt a2) + a 1 (1), in which x represents the wave-length, and V the corresponding velocity of propagation. The coefficient A is called the amplitude, and its nature depends upon the medium and must therefore here be left an open question. The phase of the wave at a given time and place is represented by a. The expression retains the same value whatever integral number of wave-lengths be added to or subtracted from x. IL is also periodic with respect to t, and the period is (2) In experimenting upon sound we are able to determine independently T, x, and V ; but on account of its smallness the periodic time of luminous vibrations eludes altogether our means of observation, and is only known indirectly from x and V by means of (2).

There is nothing arbitrary in the use of a circular function to represent the waves. As a general rule this is the only kind of wave which can be propagated without a change of form ; and, even in the exceptional cases where the velocity is independent of wavelength, no generality,is really lost by this procedure, because in accordance with Fourier's theorem any kind of periodic wave may be regarded as compounded of a series of such as (1), with wavelengths in harmonical progression.

A well-known characteristic of waves of type (1) is that any number of trains of various amplitudes and phases, but of the same ware-length, are equivalent to a single train of the same type. Thus • The composition of vibrations of the same period is precisely analogous, as was pointed out by Fresnel, to the composition of forces, or indeed of any other two-dimensional vector quantities. The magnitude of the force corresponds to the amplitude of the vibration, and the inclination of the force corresponds to the phase. A group of forces, of equal intensity, represented by lines drawn from the centre to the angular points of a regular polygon, constitute a system in equilibrium. Consequently, a system of vibrations of equal amplitude and of phases symmetrically distributed round the period has a zero resultant.

According to the phase-relation, determined by (a - a'), the amplitude of the resultant may vary from (A A') to (A+ A'). If A' and A are equal, the minimum resultant is zero, showing that two equal trains of waves may neutralize one another. This happens when the phases are opposite, or differ by half a (complete) period, and the effect is usually- spoken of as the interference of light. From a purely dynamical point of view the word is not very appropriate, the vibrations being simply superposed with as little interference as eau be imagined.

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