Gravity Waves Due to a Point Disturbance in a Plane Free Surface Flow of Stratified Fluids

Abstract

The fundamental solution of the gravity waves due to a two-dimensional point singularity submerged in a steady free surface flow of a stratified fluid is investigated. A linearized theory is formulated by using Love's equations. The effect of density stratification p[sub]o(y) and the gravity effect are characterized by two flow parameters [sigma] = -(dp[sub]o/dy)/p[sub]o and [lambda] = gL/U^2, where [lambda]^-1/2 may be regarded as the internal Froude number if L assumes a characteristic value of [sigma]^-1. Two special cases of [sigma] and [lambda] are treated in this paper. In the first case of constant [sigma] (and arbitrary [lambda]) an exact mathematical analysis is carried out. It is shown that the flow is subcritical or supercritical according as [lambda] > or < 1/2, in analogy to the corresponding states of channel flows. In addition to a potential surface wave, which exists only for [lambda]>1/2, there arises an internal wave which is attenuated at large distances for [lambda] > 1/4 and decays exponentially for [lambda] < 1/4. In the second example an asymptotic theory for large [lambda] is developed while [sigma](y) may assume the profile roughly resembling the actual situation in an ocean where a pronounced maximum called a seasonal thermocline occurs. Internal waves are now propagated to the downstream infinity in a manner analogous to the channel propagation of sound in an inhomogeneous medium.