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 ρ0(y) and the gravity effect are characterized by two flow parameters σ = −(dρ0/dy)/ρ0 and λ = gL/U2, where λ−½ may be regarded as the internal Froude number if L assumes a characteristic value of σ−1. Two special cases of σ and λ are treated in this paper. In the first case of constant σ (and arbitrary λ) an exact mathematical analysis is carried out. It is shown that the flow is subcritical or supercritical according as λ> or <½, in analogy to the corresponding states of channel flows. In addition to a potential surface wave, which exists only for λ>½, there arises an internal wave which is attenuated at large distances for λ>¼ and decays exponentially for λ<¼. In the second example an asymptotic theory for large λ is developed while σ(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.