The spheroidal wave equation is shown to play a fundamental role in Laplace's tidal equations. For standing waves and inertial waves the first exact analytical solutions of the tidal equations are obtained. The asymptotics of spheroidal wave functions provide an estimate for the domain of validity of the $\beta$-plane approximation in physical and wave number space. The wave number space of the tidal equations is separated at the inertial frequency into a low-frequency domain, where modes are governed by the $\beta$-plane mode number and high-frequency domain, where gravity waves are no longer labeled by the $\beta$-plane mode number.

• Received 29 November 1993

DOI:https://doi.org/10.1103/PhysRevLett.73.1557