Long period and semi-diurnal tidal oscillations.

A brief review is made of Laplace's equations governing tidal oscillations and of the subsequent claims and counter-claims on their validity. The purpose of this study is to investigate these claims further, with regard to long period and semi-diurnal oscillations. As the underlying assumptions are of importance, these are considered first in some depth. A set of equations is thereby formulated which differ from Laplace's equations in that extra terms of the Coriolis force are retained. These equations are taken as the basis from which a comparison is made with the previous findings. Taking the semi-diurnal constituent first, a solution is derived in the Equatorial Canal. Graphs are produced showing the velocity components as functions of canal depth and width. These compare favourably with Laplace's theory. However, whilst the description of the tidal elevation is qualitatively the same as before, there are significant quantitative differences. In particular tides become direct only in a much deeper ocean than previously predicted. Using a similar approach a solution is derived for the long period constituent in a canal-like region near the North Pole. Whereas Laplace's theory for this region gives a solution involving Bessel functions, these become Modified Bessel functions in the derived solution. Arising from this, some different effects are noted in the velocity components.