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Wave Interference

What is the relationship between traveling wave and standing wave solutions to the wave equation, (361), in an infinite medium? To help answer this question, let us form a superposition of two traveling wave solutions of equal amplitude $ A$ , and zero phase angle $ \phi$ , that have the same wavenumber $ k$ , but are moving in opposite directions. In other words,

$\displaystyle \psi(x,t) = A\,\cos(\omega\,t-k\,x) + A\,\cos(\omega\,t+k\,x).$ (379)

Because the wave equation, (361), is linear, the previous superposition is a valid solution provided the two component waves are also valid solutions: that is, provided $ \omega=k\,v$ , which we shall assume to be the case. Making use of the trigonometric identity $ \cos a + \cos b\equiv 2\,\cos[(a+b)/2]\,\cos[(a-b)/2]$ (see Appendix B), the previous expression can also be written

$\displaystyle \psi(x,t) = 2\,A\,\cos(k\,x)\,\cos(\omega\,t),$ (380)

which is a standing wave [cf., Equation (362)]. Evidently, a standing wave is a linear superposition of two, otherwise identical, traveling waves that propagate in opposite directions. The two waves completely cancel one another out at the nodes, which are situated at $ k\,x=(n-1/2)\,\pi$ , where $ n$ is an integer. This process is known as total destructive interference. On the other hand, the waves reinforce one another at the anti-nodes, which are situated at $ k\,x=n\,\pi$ , generating a wave whose amplitude is twice that of the component waves. This process is known as constructive interference.

As a more general example of wave interference, consider a superposition of two traveling waves of unequal amplitudes which again have the same wavenumber and zero phase angle, and are moving in opposite directions: that is,

$\displaystyle \psi(x,t) = A_1\,\cos(\omega\,t-k\,x) + A_2\,\cos(\omega\,t+k\,x),$ (381)

where $ A_1, A_2>0$ . In this case, the trigonometric identities $ \cos(a-b)\equiv \cos a \cos b+\sin a \sin b$ and $ \cos(a+b)\equiv \cos a \cos b-\sin a \sin b$ (see Appendix B) yield

$\displaystyle \psi(x,t)= (A_1+A_2) \cos(k x) \cos(\omega t) + (A_1-A_2) \sin(k x) \sin(\omega t).$ (382)

Thus, the two waves interfere destructively at $ k\,x=(n-1/2)\,\pi$ [i.e., at points where $ \cos(k\,x)=0$ and $ \vert\sin(k x)\vert=1$ ] to produce a minimum wave amplitude $ \vert A_1-A_2\vert$ , and interfere constructively at $ k\,x=n\,\pi$ [i.e., at points where $ \vert\cos(k x)\vert=1$ and $ \sin(k x)=0$ ] to produce a maximum wave amplitude $ A_1+A_2$ . It can be seen that the destructive interference is incomplete unless $ A_1=A_2$ . Incidentally, it is a general result that if two waves of amplitude $ A_1>0$ and $ A_2>0$ interfere then the maximum and minimum possible values of the resulting wave amplitude are $ A_1+A_2$ and $ \vert A_1-A_2\vert$ , respectively.

next up previous
Next: Energy Conservation Up: Traveling Waves Previous: Traveling Waves in an
Richard Fitzpatrick 2013-04-08