The coupled mode parabolic equation (PE) is a generalization of the adiabatic mode PE that includes mode coupling terms. It is practical to apply this approach to large-scale problems involving coupling of energy between both modes and azimuths. The solution is expressed in terms of the normal modes and mode coefficients, which satisfy coupled horizontal wave equations. The coupled mode PE may be solved efficiently with the splitting method. The first step is equivalent to solving the adiabatic mode PE over one range step. The second step involves the integration of the coupling term. The coupling mode PE solution conserves energy, which is an important aspect of a range-dependent propagation model. The derivation of the coupled mode PE, which involves completing the square of an operator, is related to the derivation of an adiabatic mode PE that accounts for ambient flow. Examples are presented to illustrate the accuracy of the coupled mode PE.
The equations of motion for pressure and displacement fields in a waveguide have been used to derive an energy-conserving, one-way coupled mode propagation model. This model has three important properties: First, since it is based on the equations of motion, rather than the wave equation, instead of two coupling matrices, it only contains one coupling matrix. Second, the resulting coupling matrix is anti-symmetric, which implies that the energy among modes is conserved. Third, the coupling matrix can be computed using the local modes and their depth derivatives. The model has been applied to two range-dependent cases: Propagation in a wedge, where range dependence is due to variations in water depth and propagation through internal waves, where range dependence is due to variations in water sound speed. In both cases the solutions are compared with those obtained from the parabolic equation (PE) method.
We prove that the eigenvalues of the Laplacian on a sphere with a Dirichlet boundary condition specified on a segment of a great circle lie between an integer and a half-integer and for a Neumann boundary condition they lie between a half integer and an integer. These eigenvalues correspond to the eigenvalues of the angular part of the Laplacian with boundary conditions specified on a plane angular sector, which are relevant in the calculation of scattering amplitude. These eigenvalues can also be used to determine the behavior of the fields near the tip of a plane angular sector as a function of the distance to the tip. The first few eigenvalues for both Dirichlet and Neumann boundary conditions are calculated. The same eigenvalues are also calculated using the Wentzel–Kramers–Brillouin (WKB) method. There is excellent agreement between the exact and the WKB eigenvalues.
The virtual source technique, which is based on the boundary integral method, provides the means to impose boundary conditions on arbitrarily shaped boundaries by replacing them by a collection of sources whose amplitudes are determined from the boundary conditions. In this paper the virtual source technique is used to model propagation of waves in a range-dependent ocean overlying an elastic bottom with arbitrarily shaped ocean-bottom interface. The method is applied to propagation in an elastic Pekeris waveguide, an acoustic wedge, and an elastic wedge. In the case of propagation in an elastic Pekeris waveguide, the results agree very well with those obtained from the wavenumber integral technique, as they do with the solution of the parabolic equation ͑PE͒ technique in the case of propagation in an acoustic wedge. The results for propagation in an elastic wedge qualitatively agree with those obtained from an elastic PE solution.
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