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The Helmholtz Equation (-delta-K(2)n(2))u=0 with a variable index of refraction, n, and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems. A numerical algorithm was developed and a computer code implemented that can effectively solve this equation in the intermediate frequency range. The equation is discretized using the finite element method, thus allowing for the modeling of complicated geometrices (including interfaces) and complicated boundary conditions. A global radiation boundary condition is imposed at the far field boundary that is exact for an arbitrary number of propagating modes. The resulting large, non-selfadjoint system of linear equations with indefinite symmetric part is solved using the preconditioned conjugate gradient method applied to the normal equations. A new preconditioner is developed based on the multigrid method. This preconditioner is vectorizable and is extremely effective over a wide range of frequencies provided the number of grid levels is reduced for large frequencies. A heuristic argument is given that indicates the superior convergence properties of this preconditioner. Bayliss, A. and Goldstein, C. I. and Turkel, E. Unspecified Center NASA-CR-172454, NAS 1.26:172454, ICASE-84-49 NAS1-17130; DE-AC02-76CH-00016; RTOP 505-31-83
This book introduces a comprehensive mathematical formulation of the three-dimensional ocean acoustic propagation problem by means of functional and operator splitting techniques in conjunction with rational function approximations. It presents various numerical solutions of the model equation such as finite difference, alternating direction and preconditioning. The detailed analysis of the concept of 3D, N x 2D and 2D problems is very useful not only mathematically and physically, but also computationally. The inclusion of a complete detailed listing of proven computer codes which have been in use by a number of universities and research organizations worldwide makes this book a valuable reference source. Advanced knowledge of numerical methods, applied mathematics and ocean acoustics is not required to understand this book. It is oriented toward graduate students and research scientists to use for research and application purposes.
This report discusses various aspects of the numerical solution of underwater acoustic wave propagation problems. In the first part of the report, a model propagation problem based on the two-dimensional Helmholtz equation with a variable sound speed is considered. A finite element computer code for solving such problems was implemented at NRL on the VAX 11/780. A distinctive feature of the code is the implementation of a recently developed iterative method for solving the resulting large, sparse, indefinite, non-self-adjoint system of equations. This allowed for the efficient solution of over 35,000 complex equations on a relatively small computer. Some of the results obtained after applying this code to the model problem are described. Furthermore, additional modifications that can be made to the code to improve its efficiency and extend its applicability to more general propagation models are discussed. In the second part of this report, the general situation of the coupled acoustic/elastic wave equation in two and three dimensions is considered. For example, this may correspond to an ocean environment in which there is ice on the surface as well as an irregularly shaped bottom structure. Finite difference and finite element methods for solving both the time harmonic and time dependent models are discussed. Various issues are considered that are important in determining the size of the problem that can be adequately treated. This includes the computer power as well as the mathematical and modeling techniques available. (Author).
Contributions to solving initial boundary value problems for partial differential equations have been made by applying finite-difference methods to solve seismic wave propagation problems. Very little has been done in the area of underwater acoustic wave propagation problems, although a set of properly developed numerical methods could very well solve these problems effectively. These numerical methods can solve not only range-dependent problems but also can handle irregular boundaries with arbitrary boundary conditions. In this report, as a start, two accurate general purpose approaches are presented for the solution of variable coefficient parabolic wave equations. In a finite-difference approach, techniques are derived from both the conventional explicit and implicit schemes, and the associated convergence theory is thoroughly analyzed. The techniques are found to be general purpose and to provide reasonable accuracy. In an ordinary differential equation approach the parabolic equation is treated as a system of equations in which the second partial derivative with respect to the space variable is discretized by means of a second order central difference (also known as the Method of Lines). Nonlinear multistep (NLMS) and linear multistep (LMS) methods are used as predictor-and-corrector for solving this system. A built-in variable step-size technique gives the desired accuracy.
Senior level/graduate level text/reference presenting state-of-the- art numerical techniques to solve the wave equation in heterogeneous fluid-solid media. Numerical models have become standard research tools in acoustic laboratories, and thus computational acoustics is becoming an increasingly important branch of ocean acoustic science. The first edition of this successful book, written by the recognized leaders of the field, was the first to present a comprehensive and modern introduction to computational ocean acoustics accessible to students. This revision, with 100 additional pages, completely updates the material in the first edition and includes new models based on current research. It includes problems and solutions in every chapter, making the book more useful in teaching (the first edition had a separate solutions manual). The book is intended for graduate and advanced undergraduate students of acoustics, geology and geophysics, applied mathematics, ocean engineering or as a reference in computational methods courses, as well as professionals in these fields, particularly those working in government (especially Navy) and industry labs engaged in the development or use of propagating models.
This book introduces parabolic wave equations, their key methods of numerical solution, and applications in seismology and ocean acoustics. The parabolic equation method provides an appealing combination of accuracy and efficiency for many nonseparable wave propagation problems in geophysics. While the parabolic equation method was pioneered in the 1940s by Leontovich and Fock who applied it to radio wave propagation in the atmosphere, it thrived in the 1970s due to its usefulness in seismology and ocean acoustics. The book covers progress made following the parabolic equation’s ascendancy in geophysics. It begins with the necessary preliminaries on the elliptic wave equation and its analysis from which the parabolic wave equation is derived and introduced. Subsequently, the authors demonstrate the use of rational approximation techniques, the Padé solution in particular, to find numerical solutions to the energy-conserving parabolic equation, three-dimensional parabolic equations, and horizontal wave equations. The rest of the book demonstrates applications to seismology, ocean acoustics, and beyond, with coverage of elastic waves, sloping interfaces and boundaries, acousto-gravity waves, and waves in poro-elastic media. Overall, it will be of use to students and researchers in wave propagation, ocean acoustics, geophysical sciences and more.