Exact solution and instability for geophysical edge waves

We present an exact solution to the nonlinear governing equations in the \begin{document}$ \beta $\end{document}-plane approximation for geophysical edge waves at an arbitrary latitude. Such an exact solution is derived in the Lagrange framework, which describes trapped waves propagating eastward or westward along a sloping beach with a shoreline parallel to the latitude line. Using the short-wavelength instability method, we establish a criterion for the instability of such waves.