OF THE FORM OF RECTANGLES AND OF SECTORS OF CIRCLES

A study is made of the periods of free tidal oscillations and of the corresponding wave pattcrns in rotating flat basins which havc 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 C denotes the tide height and * its complex conjugate, the sign of the integral if<'*(d /ds) ds, which is real, taken around the periphery of the basin, deteirmiines whether the tidal wave propagates around the basin in the direction of rotation (positive wave), or opposite to it (negativc wave). The sense of propagation can also be told from the sign of dk2(r)/dr, where k2 = (a-2 -4 )2)/gIh and T- 2w/, (J) denoting the specd of rotation, and C the frequency. A discussion is given of the removal by rotation of the degeneracy that cxists irl some modes in the absence of rotation. The method (A) of expansion of g in terms of the eigenfunctions for no rotation (7 = 0) was found to converge well only for r < 1. Our calculations were carried out by an adaptation of Trefftz's metlhod, in which the variation of the surface integral is reduced to a variation of a linc-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 indefinitcly a Kelvin regime sets in, in which the tidc concentrates near the periphery, decreasing exponentially towards the interior. The Kelvin wave is positive and the characteristic frequencics ,, are given by o, = 2rn1\/(gh)/p, p denoting the perimeter of the basin. It is shown that ncar a blunt corner of the coast the tide has a singularity like that in potential flow. In this investigation we present a gcneral theory of the free tidal oscillations of rotating flat basins having the form of rectangles and of circular sectors. Our aim has been to gain insight into the dynamics of tides in these simple basins so as to serve as a guide in the interpretation of tidal studies in the more complicated real world oceans. In the latter, the approach is necessarily numerical, requiring the adoption of a rectangular grid for the finite difference method. The resulting jagged representation of the coastline was found to rctard the convergence of the solution, and the question arises as to the theoretical distribution of the tide around a blunt corner such as is shown in figures 16 to 21. Our analysis shows that near a corner of angular opening lr/,u (,u< 1), the tide height C is singular, as in potential flow, and is approximated by equation (95), wherc r 2/oa. Here co denotes tlhe angular rotation f tthe basin and fa the frequency of tidal oscillation. The degree of approxi_ mation attained by representation (95) is shown in table 1. In the spectral analysis of the tidal oscillations, where the characteristic frequencies x,n (?) are determined as functions of the angular rotation o, we found that a more meaningful