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Standing Waves

Figure: Traveling vs. standing waves.

A particularly interesting example of superposition is provided by the case where   $A_1 = A_2 = A_{_0}$,  $k_1 = k_2 = k$  and   $\omega_1 = - \omega_2 = \omega$. That is, two otherwise identical waves propagating in opposite directions. The algebra is simple:

$$A(x,t) = A_{_0} \left[ e^{i(k x - \omega t)} + e^{i(k x + \omega t)} \right] \cr$$ (14.18)

The real part of this (which is all we ever actually use) describes a sinusoidal waveform of wavelength   $\lambda = 2\pi/k$  whose amplitude   $2 A_{_0} \cos(\omega t)$  oscillates in time but which does not propagate in the $x$ direction - i.e. the lower half of Fig. 14.3. Standing waves are very common, especially in situations where a traveling wave is reflected from a boundary, since this automatically creates a second wave of similar amplitude and wavelength propagating back in the opposite direction - the very condition assumed at the beginning of this discussion.