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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 , and zero phase angle , that have the same wavenumber , but are moving in opposite directions. In other words,

 (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 , which we shall assume to be the case. Making use of the trigonometric identity (see Appendix B), the previous expression can also be written

 (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 , where 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 , 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,

 (381)

where . In this case, the trigonometric identities and (see Appendix B) yield

 (382)

Thus, the two waves interfere destructively at [i.e., at points where and ] to produce a minimum wave amplitude , and interfere constructively at [i.e., at points where and ] to produce a maximum wave amplitude . It can be seen that the destructive interference is incomplete unless . Incidentally, it is a general result that if two waves of amplitude and interfere then the maximum and minimum possible values of the resulting wave amplitude are and , respectively.

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