In this paper we investigate the propagating solutions of the acoustic equation in a stratified shallow ocean with a poroelastic, semi‐infinite seabed. The ocean‐seabed system is first Hankel transformed. Then the method of transmutation is used to generate the transformed solutions in the water column; whereas, the modal seabed equations are solved using the computer algebra Macsyma. In the water column the transformed solutions are first decomposed using the Mittag‐Leffler expansion for the discrete spectrum. Both the discrete and continuous spectrum are then used to give a spectral representation of the solution, from which we develop a numerical scheme. Some numerical examples are given to illustrate the method.

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In this paper the boundary integral equation method is used to solve a scattering problem in a shallow ocean with an elastic seabed. The Hankel transformation and Mittag–Leffler decomposition were used to construct the propagating solution for both far-field and near-field. In particular, necessary and sufficient conditions are found for the existence of the propagating solution. Using the propagating solution, the scattering problem is recast as a boundary integral equation. A numerical algorithm is developed for solving this boundary integral equation and its implementation on a T3D parallel computer is used to compute an illustrative example.

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This paper which is Part I of a sequence deals with the problem of determining a radially dependent coefficient n (r) in the equation ∆ u − n2 (r) u = 0, in the unit disk Ω from the Dirichlet–Neumann data pair [Formula: see text]. We prove that the sufficiency condition for uniqueness established in Ref. 2 is, in some instances, also a necessity for uniqueness. We also discuss the solvability of this inverse problem. In Part II numerical experiments will be presented which illustrate the theory developed here.