Though equation (5.2) is not hyperbolic, it is simple to obtain a dispersion relationship by considering wave-like solutions of the form , where is the frequency variable, and is the spatial wavenumber. The relationship can be written as

which has solutions

The phase and group velocities can then be written, from (3.12), as

These velocities are now dependent on the spatial wavenumbers, and hence wave propagation is

We also mention that a good model for the piano string is based on the wave equation, and complemented by several perturbation terms, among which are a fourth spatial derivative term like the above [25]; such a term models frequency-dependent dispersion in the string.