The elastic wave equation with variable cross section is:
$E\frac{\partial}{\partial x}\left[A(x) \frac{\partial u(x,t)}{\partial x}\right] = \rho A(x) \frac{\partial^2 u(x,t)}{\partial t^2}$,
where $E$ is the constant Young's modulus, $\rho$ is the constant mass density and $A(x)$ is the variable cross section. Various references mention that the phase velocity of a wave is $\sqrt{E/\rho}$, but do not give more details. How do they get this result?