# Banded waveguide synthesis

In the case of the standard one-dimensional wave equation ${\displaystyle y_{tt}=c^{2}y_{xx}}$ disturbances of all frequencies travel with the same constant speed ${\displaystyle c}$. In dispersive media, the traveling speed of disturbances depends on their frequency and we get ${\displaystyle c(\omega )}$ where ${\displaystyle \omega }$ is the frequency of the disturbance. Many physical systems are dispersive, for example the elastic beams described by the Euler-Bernoulli beam equation ${\displaystyle y_{tt}=ky_{xxxx}}$ where ${\displaystyle k}$ is a material constant.