Ding Lee

PARABOLIC EQUATION DEVELOPMENT IN RECENT DECADE

Numerous contributions have been made in the enhancement of the Parabolic Equation (PE) approximation method, which has been shown to be a useful tool for solving realistic problems in many different scientific fields. Evidence of its usefulness is the application of PE to solve ocean acoustic propagation problems. In early years, when the PE was introduced to the field of underwater acoustics, its main purpose was to predict long-range, low-frequency acoustic propagations in range-dependent environments; thus, there were certain limitations. In the recent decade, these limitations have been relaxed a great deal due to many useful contributions. The time has come to survey and report these important contributions and to discuss how these contributions enhance the capability of the PE method. This paper gives a brief review of what had been done before 1984 and highlights some important PE developments from 1984 to 1994. Also, some applications of the PE to predict ocean acoustic propagation problems will be presented. We shall call attention to a few important issues related to the PE developments and applications. Looking ahead we will discuss what more a PE can do in order to stimulate future research, development, as well as applications.

Examination of three‐dimensional effects using a propagation model with azimuth‐coupling capability (FOR3D)

A three‐dimensional wave propagation model of parabolic approximation type (FOR3D) is used to examine 3‐D ocean environmental variations. The background theory and characteristics of the model are reviewed briefly. Propagation situations that are classified as 3‐D, N×2‐D, and 2‐D are described in connection with FOR3D and are interpreted in several ways. An analytic exact solution is used to demonstrate the model’s accuracy and its capability for treating fully 3‐D propagation, when coupling exists between solutions in adjacent vertical planes of constant azimuth. It is also employed to illustrate a procedure for using approximate conditions at vertical side boundaries in a 3‐D calculation. An application is made to an Atlantic Ocean shelf‐slope environment with realistic bottom topographic variations and sound‐speed profiles. The occurrence of significant azimuthal coupling is demonstrated in propagation loss versus range curves. It follows that, while the N×2‐D approximation of no azimuthal coupling is ...

A wide‐angle three‐dimensional parabolic wave equation

A third‐order partial differential equation with wide‐angle capability is formulated to predict three‐dimensional underwater sound propagation. The development is based on physical acoustic characteristics and mathematical theory. Both operator and asymptotic analyses are given to thoroughly discuss the validity of the formulation. Physical conditions are indicated when a three‐dimensional approach is needed.

Numerical ocean acoustic propagation in three dimensions

This text 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 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 aims to make this book a valuable reference source. Advance 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.

A higher‐order parabolic wave equation

A higher‐order paraboliclike approximation to the wave equation is presented. An implicit finite‐difference discretization of the equation is implemented and its stability is discussed. The accuracy of the treatment of wide‐angle propagation is examined by comparing computed solutions with exact solutions. Comparisons with other approaches are made to investigate the wide‐angle capability of the method.

Analysis of an implicit finite difference solution to an underwater wave propagation problem

Abstract An implicit finite difference scheme approximating a third order partial differential equation is examined. The scheme is derived, shown to be consistent, and its stability properties are analyzed. The partial differential equation is a parabolic approximating equation to the reduced wave equation.

A mathematical model for the 3-dimensional ocean sound propagation

Abstract A mathematical model is developed to represent sound propagation in a 3-dimensional ocean. The complete development is based on characteristics of the physical environment, mathematical theory, and computational accuracy. An exact solution test is performed to examine the validity of the theoretical development. A real application is included to demonstrate the models capability.

Aspects of three-dimensional parabolic equation computations

Abstract In some problems of interest, sound propagation in the ocean involves significant variations in all three space dimensions. A review is provided of physical situations where three-dimensionality occurs and of physical mechanisms which can cause it. Parabolic approximations to the Helmholtz equation are described which are appropriate for three-dimensional propagation problems. Particular attention is given to estimates of limits of validity for the parabolic equations along with algorithms for their numerical calculation. Some computational examples are discussed for finite-difference approximations. Simple analytical solutions are suggested which are useful for three-dimensional test computations. These solutions illustrate comparisons between the Helmholtz equation and the parabolic approximation. In addition, they can show transition between two- and three-dimensional propagation characteristics and can provide accuracy tests for numerical computations.

THE INFLUENCE OF THE REFERENCE WAVENUMBER IN COMPUTATIONAL OCEAN ACOUSTICS

A class of ocean acoustic propagation problems can be solved efficiently by the Parabolic Equation (PE) approximation method. The application of the PE method for the prediction of wave propagation introduces a new parameter, the reference wavenumber k0. This requires selection of the most appropriate k0, which is related to the reference sound speed c0. The influence on the acoustic field by the choice of c0 is rarely visible under weak range-dependent environments. Even if it is visible, the difference is small and is usually negligible since the present judicious choice of the c0 seems to provide acceptable results. When the environment is not weakly range-dependent, the choice of c0 will likely affect the computation of acoustic results. This paper examines a few different choices of c0 and analyzes how these different choices can influence the acoustic results. An application is given where the farfield wave equation represents a realistic range-dependent environment. Different choices of c0 were made for the computation of the acoustic field; as a consequence, different choices of c0 produce different acoustic results. These numerical results are not in agreement with a known reference exact solution. The differences are not too small and may be considered non-negligible. So, there is a need to make an appropriate choice of c0 in order to produce reasonable results. For the purpose of achieving satisfactory and acceptable acoustic results dealing with a PE-type equation, the requirements of the reference wavenumber will be discussed both mathematically and physically. Then, a number of computational choices of c0 will be examined, especially the k0-formula. An analysis as well as an assessment of the k0-formula will be given.

A numerical marching scheme to compute scattering in the ocean

In the study of underwater propagation of sound in an ocean environment, much effort has been expended in considering energy propagating in a designated direction. In a range-dependent ocean environment, scattering in all directions will occur, but in some ocean environments the all-direction scattering is weak and often can be ignored. For long-range propagation, keeping the cumulative weak scattering can be important. Numerical treatment of this type of scattering in a very long range presents two computational problems: (1) the required memory storage, and (2) the required computation time. In this paper, a marching technique is developed to handle the cumulative scattering, thus alleviating the memory storage problem, and an efficient numerical solution is introduced which reduces the computation time. When using a marching technique to solve this problem, one usually encounters the problem of well-posedness. In the context of the development of the numerical scheme, an approximation is made which suppresses the instability associated with the well-posedness question. Additionally, in the scheme, at large distances from the source a continuation process is employed (essentially a PE) to continue the solution, thereby modeling an actual physical environment without scattering. The theoretical formulation of a representative scattering equation and the development of the scheme for solving this equation will be discussed. Moreover, a realistic problem with weak scattering is presented to demonstrate the validity of this treatment.