Effective Boundary Conditions for Rough Surfaces, and, Acoustics of Oceanic Bubbles.

This thesis consists of two independent parts --one on an analytical treatment of field and flow involving a rough surface, the other on the effect of bubbles in underwater scattering phenomena. Underlying these two different topics, are some common mathematical themes which lead to similar analytical tools--e.g., ensemble averaging, Green's function formulation--being applied to both of them. In the first part a simple model of a rough surface, amenable to an average treatment, is investigated. Fields satisfying various homogeneous boundary conditions on such a surface are analyzed to derive average conditions on an effective surface. The cases of Laplace and Helmholtz equations with Dirichlet or Neumann condition, and of Stokes flow are treated for "dilute" rough surfaces. The dependence of the result on the shape of the roughness elements is investigated for the Laplace equation. For the Laplace -Dirichlet problem, the treatment is further extended to a "dense" rough surface. The second part^1 of the thesis starts with an investigation of low-frequency scattering from bubble clouds. The effect of various cloud shapes are analyzed. The accuracy of a Born approximation, used by other researchers, is also investigated. The remainder of the thesis is devoted to some theoretical aspects of the physics of bubble scattering. An attempt is made to reconcile the effective media formulation, on which the bubble cloud model is founded, with the more traditional theory of incoherent scattering applicable at higher frequencies. It is shown that, underlying the incoherent results, is a WKB solution of the effective equations. The validity of various approximations frequently used by researchers for scattering as well as acoustical imaging of bubble assemblies are examined with the help of a more accurate theory. ftn^1A slightly different version of this work has appeared in print (Sarkar & Prosperetti (1993, 1994)).