Superposition

Because the wave equation we are considering is linear, any superposition of solutions is also a solution. Let

y 1 = A sin (k 1 x - w 1 t)

and

y 2 = A sin (k 2 x - w 2 t + d).

Since

sin (a) + sin (b) = 2 cos ((a - b) / 2) sin ((a + b) / 2),

we can form the sum

y = y 1 + y 2

= 2 A cos [(Dk x - Dw t + d) / 2] sin [(Sk x - Sw t + d) / 2],

where Dk is k 2 - k 1, Dw is w 2 - w 1, Sk is k 1 + k 2 and Sw is w 1 + w 2. We will examine two special cases of this superposed solution.

The first case corresponds to what are called "coherent sources", in which the two waves have the same wave number and frequency (k 1 = k 2, w 1 = w 2):

y = 2 A cos (d / 2) sin (k x - w t + d / 2).

Such sources are said to "interfere" with each other. We can examine the interference for d equal to multiples of p (or multiples of k), which corresponds to separating the sources in time (or in space; see Section F below). We see, for example, that when d is an odd multiple of p, y is zero. This is called "destructive interference", and the two waves are said to be p radians (180 degrees) "out of phase". When d is an even multiple of p, the two waves are said to "constructively interfere", and be "in phase".

Our other special case involves setting both d and x to zero. This corresponds to watching a fixed place over time. The solution is now

y = 2 A cos [Dw t / 2] sin [Sw t / 2].

We see that the wave changes with two separate frequencies. The cosine function defines a sort of "envelope" inside of which the sine function oscillates. The envelope amplitude has two maxima for each cycle, and these make the sine oscillation appear to waver in "beats" (see the "Travelling Waves and Beats" Mathematica Notebook). The "beat frequency" is Dw / 2 p. We distinguish between the "phase velocity" of the waves in the envelope

c p = Dw / Dk

and the "group velocity"

c g = Sw / Sk

of the envelope itself.

The next section is about boundary conditions.



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1996, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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