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128.

Problem 19.36 (RHK)

We consider two point sources and , which emit waves of the same frequency and amplitude. The waves start in the same phase, and this phase relation at the sources is maintained throughout time. We consider points P at which is nearly equal to . We have to show (a) that the superposition of these two waves gives a wave whose magnitude varies with the position P approximately according to

in which . (b) We then have to show that total cancellation occurs when , n being any integer, and the total re-enforcement occurs when The locus of points whose difference in distance from two fixed points is a constant is a hyperbola, the fixed points being the foci. Hence each value of n gives a hyperbolic line of constructive interference and a hyperbolic line of destructive interference. At points at which and are not approximately equal (as near the sources), the amplitudes of the waves from and differ and cancellations are only partial.

Solution:             Click For PDF Version

We will write functions representing spherical waves of the same frequency and amplitude emitted by sources and . We assume that the waves from and start in the same phase and this relation is maintained throughout time. These functions are

where and are distances measured from and , respectively. The resultant wave at any point in space will be given by the superposition of these two spherical waves. If at some point P the distances and are approximately equal, we may approximate and by

In this approximation,

Or,

We rewrite the resultant wave function in the form

Amplitude varies with position P approximately as

From this expression we note that = 0, for

In this situation there is total cancellation of the waves reaching at P from sources and .

And, at points where

the two waves reinforce each other and there is constructive interference.

The locus of points whose difference in distance from two fixed points is a constant describes a hyperbola, the fixed points being foci. Hence each value of n gives a hyperbolic line of constructive interference and a hyperbolic line of destructive interference.