IX. On the application of harmonic analysis to the dynamical theory of the tides. — Part I. On laplace’s "oscillations of the first species” and the dynamics of ocean currents

Abstract

The earliest attempt to subject the Theory of the Tides to a rigorous dynamical treatment was given by Laplace in the first and fourth books of the ‘Mécanique Céleste.’ The subject has since been treated by Airy, Kelvin, Darwin, Lamb, and other writers, but with the exception of the extension of Laplace’s results to include the theory of the long-period tides, but little practical advance has been made with the subject, in spite of the enormous increase in the power of the mathematical resources at our disposal, and the problem has remained in very much the same condition as it was left by Laplace. This arises no doubt partly from the difficulties inherent to the subject, but partly from the form in which the theory was originally presented by Laplace in the ‘Mécanique Céleste,’ which has been described by Airy as “perhaps on the whole more obscure than any other part of the same extent in that work.” The obscurity complained of does not however seem to have been entirely removed by Laplace’s successors, and it was the fact that every presentment of the theory with which I was acquainted offered some points of difficulty, that in the first instance led me to take up the problem ab initio , partly with the purpose of allaying the doubts which had arisen in my own mind as to the validity of certain approximations employed by Laplace and adopted by his successors, and partly in the hope that I might be able to extend the results of Laplace to meet more fully the case presented by the circumstances actually existent in Nature. Up to the present I have been unable to free the problem to any extent from the limitations which have been imposed by previous writers, and consequently it would be futile to claim that the results I am now able to put forward materially advance our knowledge of the tides as they actually exist; but I venture to hope that these results, as applied to the oscillations of an ideal ocean, considerably simpler in character than the actual ocean, may prove of some interest from the point of view of pure hydrodynamical theory.