We can transform the wave equation from a partial differential equation to an ordinary differential equation by considering sustained harmonic motion of the type \(\cos (n t+\phi)\). Using the equation for the wave height in a channel of varying cross-section;

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

consider an estuary, like the Hudson river, for which \(b=\beta x / \alpha, h=\gamma x / a\) and \(0

\[\delta_{\lambda}=K \cos (n t+\phi)\]

occurs. Show that the tidal waves of the estuary are given by:

\[\delta_{\lambda}=K \frac{J_{1}(\sqrt{4 \kappa x})}{J_{1}(\sqrt{4 \kappa a})} \sqrt{\frac{a}{x}} \cos (n t+\varepsilon)\]