## Abstract

This paper investigates the low-frequency solutions to the tidal equations on a sphere under the conditions that the wavenumber n is fairly large and that 4$\Omega^2/ghn^2 \leqslant$ O(1), where $\Omega$ is the angular rotation velocity of the sphere, g the gravitational constant, h the depth (assumed constant), and the radius of the sphere is taken as unity. It is shown that the equations reduce to the spheroidal wave equation the solutions of which, as the result of recent investigations, can be considered as known functions. In the special case when 4$\Omega^2/ghn^2 \ll$ 1 one obtains the non-divergent planetary waves, studied in a previous paper (Longuet-Higgins 1964a). On the other hand, near to the equator the solutions can be represented approximately by parabolic cylinder functions, a special case considered by Rattray (1964) and Bretherton (1964). The present paper thus unifies the work of these authors. Next the $\beta$-plane approximation is studied. It is shown that whereas the spheroidal wave approximation is valid to order n$^{-2}$ the $\beta$-plane approximation is valid only to order n$^{-1}$ except near the equator, where it is valid to order n$^{-2}$. The $\beta$-plane approximation is used to interpret the local character of solutions in an unbounded region on the sphere, and to obtain new solutions in closed basins of rectangular, triangular or circular shape. By means of the spheroidal wave approximation, the trajectories of wave packets on the surface of a sphere are calculated. The trajectories always lie between two critical latitude circles (one north and one south of the equator). Polewards of these latitudes the disturbance in any given mode is small. The critical latitudes themselves form ray caustics, where the amplitude is particularly large. The width of the caustic zone is of order n$^{-\frac{2}{3}}$, and cannot be adequately described by the $\beta$-plane approximation with constant $\beta$. This explains some difficulties in a previous attempt by Arons & Stommel (1956) to describe waves travelling due east or due west, using a $\beta$-plane approximation. At a given latitude there is a minimum period for waves of planetary type. Waves of this period are easily excited and may be prominent in the oceans. The theory is readily extended to internal (or baroclinic) waves. It is suggested that some deep currents observed in the Atlantic by J. C. Swallow (1961) were planetary waves at the cutoff period.