We can look at a slight variant of the problem above. Given the differential equation:

\[\frac{\partial^{2} \delta}{\partial t^{2}}=\frac{\partial}{\partial x}\left(g h \frac{\partial \delta}{\partial x}\right) \quad h=\frac{x^{2}}{2 b}\]

show that the following wave solution satisfies the differential equation:

\[\delta=\frac{A}{\sqrt{x}} \cos \left\{p\left(\sqrt{\frac{2 b}{g}-\frac{1}{4 p^{2}}} \ln x+t\right)+\alpha\right\}\]