I. On the initial value problems of radiation and scattering of water waves by immersed obstacles. II. Gravity waves due to a point disturbance in a stratified flow
<p>I. Some initial value problems are studied regarding the radiation and scattering of gravity waves by finite bodies in an infinitely deep ocean. Emphasis is placed on the case where a finite number of thin plates lie on a vertical line, for which the general solution is obtained by transfor...
| Summary: | <p>I. Some initial value problems are studied regarding the radiation and scattering of gravity waves by finite bodies in an infinitely deep ocean. Emphasis is placed on the case where a finite number of thin plates lie on a vertical line, for which the general solution is obtained by transforming the boundary value problem to one of the Riemann-Hilbert type. Explicit investigations are made for the large time behavior of the free surface elevation for the case of a rolling plate, and for the Cauchy-Poisson problems in the presence of a stationary plate. By taking the limit as t → ∞, the steady state solution is derived for a harmonic point pressure acting on the free surface near a vertical barrier. Finally a formal asymptotic representation of the free surface elevation is given for large time when the geometry of the submerged bodies is arbitrary.</p>
<p>II. The subject gravity waves in the two dimensional flow of a vertically stratified fluid is investigated with regard to the dynamic effects of a submerged singularity. Love's linearized equations are adopted as the basis for the theory. Two specific cases are treated according as the parameter N^2 being a constant or a function of depth, where</p>
<p>N^2 = g/P_o dP_o/dy</p>
<p>characterizes the density variation in the fluid. The first example of constant N^2 is physically a hypothetical case but can be given an exact mathematical analysis; it is found that in a deep ocean with such a density variation the interval waves are local in nature, i.e., their amplitudes diminish to zero as the distance from the singularity becomes very large. In the second example an asymptotic theory for small Froude number, U^2/gL « 1, is developed when N^2(y) assumes the profile roughly resembling the actual situation in an ocean where a pronounced maximum called a seasonal thermocline occurs. Internal waves are now progagated to the downstream infinity in a manner analogous to the channel propagation of sound in an inhomogenous medium.</p> |
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