Gravity waves in a stratified fluid

A unified treatment of wave motion in a stratified fluid, with or without density discontinuities, is achieved by reducing the governing differential system to a Sturm-Liouville system. With the aid of Sturm's comparison theorem, it is found (without detailed calculations) that, for any stratification, the phase velocity increases as the wave-number decreases and that, for the same wave-number, the phase velocity increases as the density gradient is increased everywhere and decreases as the density is increased everywhere by a constant amount. Sturm's oscillation theorem provides upper and lower bounds for the phase velocity for a given stratification, a given wave-number, and a given number of zeros of the eigenfunction (or a given number of stationary surfaces in the fluid). The inequalities giving these bounds are used to explain the well-known tendency for surfaces of density discontinuities to behave as rigid boundaries when the stratification in each layer is slight. The rigid-boundary behaviour of interfaces in such cases enables one to obtain the approximate eigenvalue spectrum by superimposing the spectra of the individual layers (with the interfaces treated as rigid) on the spectrum of the interfacial (or free surface) waves, obtained by ignoring the slight continuous stratification in each layer. It is pointed out that the Ritz method can be used for calculating the eigenvalues even when the density is discontinuous, and examples are given to show the accuracy of the Ritz method. The nature of the spectrum when the depth is infinite is also clarified. In the course of the development of the theory, the effects of compressibility and of three-dimensionality are determined and given explicitly, the rate of growth of unstable stratifications is related to the phase velocity of waves in stable ones, and equipartition of energy is proved. Motion due to a wave-maker is discussed in order to bring out the connexion between the type of the governing partial differential equation and the nature (local or not local) of the disturbances. The effect of surface tension and the stability of a stratified fluid under vertical oscillation are also discussed.