A possible solution to the problem was shown by Woldemar Voigt (1887), who investigated the Doppler effect for waves propagating in an incompressible elastic medium and deduced transformation relations that left the Wave equation in free space unchanged, and explained the negative result of the Michelson-Morley Experiment.
The phenomenon of lacunas has been extensively investigated in Atiyah, Bott and Gårding (1970, 1973).
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The vibrations of an idealized circular drum head—essentially an elastic membrane of uniform thickness attached to a rigid circular frame—are solutions of the wave equation with zero boundary conditions.
The cnoidal wave solutions were derived by Korteweg and de Vries, in their 1895 paper in which they also propose their dispersive long-wave equation, now known as the Korteweg–de Vries equation.
Experimentally, every light signal can be decomposed into a spectrum of frequencies and wavelengths associated with sinusoidal solutions of the wave equation.