I don't think it would be possible. At the very least, if it is possible, it would be difficult to produce them, right? This is because the wave speed will depend on the wavelength & frequency so that finding the conditions for 1:1 constructive interference would be harder?
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2 Answers
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Yes, standing waves are possible in dispersive media.
A standing wave (in one dimension) consists of two traveling waves of the same frequency propagating in opposite directions. Thus, the dispersion relation is not relevant to standing waves because the dispersion relation tells us about waves of different frequencies, but standing waves have a single frequency.
Water waves are dispersive, yet standing waves in water are common place, for example.
answered Oct 12, 2016 at 22:27
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$\begingroup$ yes I accidentally assumed there was damping in a dispersive system. In that case, the speed would get lower and so the frequency & wavelength would change right? $\endgroup$– OblivCommented Oct 12, 2016 at 22:44
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$\begingroup$ I don't follow what you mean by "the speed would get lower". Do you mean that a the wave would slow down over time, or that the speed would be lower in the medium with damping than in the medium without it? $\endgroup$ Commented Oct 12, 2016 at 22:55
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$\begingroup$ The magnitude of the wave speed would drop overtime through damping. This consequently affects the frequency & wavelength in a dispersive system so I imagine timing the waves for 1:1 constructive interference would get difficult. $\endgroup$– OblivCommented Oct 12, 2016 at 23:06
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1$\begingroup$ Damping does not change the speed of a wave over time, except in the case of amplitude dispersion. $\endgroup$ Commented Oct 12, 2016 at 23:10
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A standing wave has a single definite frequency. Therefore, the dispersion, i.e., the change of wavelength with frequency, should not make it more difficult to achieve a standing wave. However, according to the Kramers-Kronig dispersion relations, if you are in a frequency range of very strong dependence of wave length on frequency, you are also in a region of strong absorption of this wave. This might have an influence on resonances due to standing waves because of the strong damping.
answered Oct 12, 2016 at 22:32