Free tidal oscillations in rotating flat basins of the form of rectangles and of sectors of circles

Abstract

A study is made of the periods of free tidal oscillations and of the corresponding wave patterns m rotating flat basins which have the form of rectangles or of sectors of circles. The analysis is based on a variational principle for tidal oscillations. It is shown that, if denotes the tide height and its complex conjugate, the sign of the integral ij(d/ds) ds, which is real, taken around the periphery of the basin, determines whether the tidal wave propagates around the basin m the direction of rotation (positive wave), or opposite to it (negative wave). The sense of propagation can also be told from the sign of dk 2 (r),where k 2 = (or 2 -4w 2 ) r = 2w/o denoting the speed of rotation, and o the frequency. A discussion is given of the removal by rotation of the degeneracy that exists in some modes in the absence of rotation. The method (A) of expansion of in terms of the eigenfunctions for no rotation (T = 0) was found to converge well only for T < 1. Our calculations were carried out by an adaptation of Trefftz’s method, in which the variation of the surface integral is reduced to a variation of a line-integral taken along the boundary. This method (B) was found to be effective for all ranges of rotation. The solutions obtained illustrate that in some modes the tides are always positive, while in others they start out being negative at slow rotation and turn positive as the rotation is increased. A theory is developed, for basins of general shape, showing that as the speed of rotation is increased indefinitely a Kelvin regime sets in, in which the tide concentrates near the periphery, decreasing exponentially towards the interior. The Kelvin wave is positive and the characteristic frequencies o are given by a o n = 2nn (gh)/p, p denoting the perimeter of the basin. It is shown that near a blunt corner of the coast the tide has a singularity like that in potential flow.