Donald F. St. Mary

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.

Stable marching schemes based on elliptic models of wave propagation

A marching numerical scheme is applied to a far‐field elliptic model of underwater wave propagation. General stability conditions are derived for the scheme in the case of varying parameter functions. Several examples are presented to demonstrate the capacity of the method to detect backscattered energy. In these examples the elliptic equation is cast as an initial value problem with assumed correct initial data. The ill‐posed nature of initial value problems for elliptic problems is discussed.

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.

The Bohl transformation for second order linear differential systems

Abstract The Bohl transformation has been a useful method for studying the oscillation of scalar self-adjoint linear differential equations. In this paper we define and develop a corresponding transformation which is applicable to self-adjoint linear differential systems, and use it to establish criteria for oscillation of these systems. In the process of this development we establish criteria for the global existence of solutions of several nonlinear systems of differential equations.

A modified wide angle parabolic wave equation

Abstract We demonstrate the implicit finite difference discretization of a higher order parabolic-like partial differential equation approximating the reduced wave equation in the far field and show that the discretization is unconditionally stable. We discuss a method of associating an angle of dispersion with parabolic approximations, present an example which can be used to compare methods on the basis of dispersion angle, and make comparisons among well-known methods and the new method.

The Bohl transformation and oscillation of linear differential systems

The Bohl transformation is a nonlinear transformation that, like the Riccati transformation, relates linear and nonlinear differential equations. Recently, we have extended that transformation to differential systems. In this paper the utility of the transformation for extending scalar oscillation results to linear differential systems is demonstrated.

Application of a three‐level explicit finite difference scheme to the parabolic equation.

Traditional explicit finite difference schemes are unstable for all values of the discrete independent variables, i.e., unconditionally unstable, when applied to Schrodinger type partial differential equations. This creates the need to utilize implicit numerical schemes such as Crank–Nicolson, which require relatively large run times. An existing unconditionally stable three level scheme, which utilizes a dissipative term, is extended to apply to the parabolic equation. The scheme is then exercised on several examples in underwater acoustics. [Work supported in part by ONR.]

A marching method for some elliptic models of wave propagation

Parabolic equation (PE) methods for wave propagation in the farfield are frequently obtained from elliptic equations valid in the farfield. Here, properties of the elliptic models are discussed directly. The equations are discretized and a marching method is obtained. Although the underlying initial value problems are not well posed, in certain parameter ranges, the marching method is stable when the step sizes are restricted in a suitable way. Under such restrictions, propagation loss curves are obtained that show good agreement with exact solutions of some test problems. Both fluid and elastic wave problems are considered. [Work supported in part by ONR.]

Error estimates for 3‐D computational schemes of wave propagation

For several computational schemes in underwater wave propagation, stability and consistency bounds are known that imply corresponding rates of convergence. It is shown, in some cases, how this information can be improved to yield explicit bounds for the truncation error between the exact solution of a continuous problem and the approximating solution generated by the computational scheme. In particular, those bounds are demonstrated for an implicit finite difference scheme in three dimensions.

WIDE ANGLE PARABOLIC APPROXIMATIONS IN UNDERWATER ACOUSTICS

The discussion of the approximation of the Helmholtz equation by a parabolic-like partial differential equation is important because of the significant reduction in numerical computation associated with parabolic problems. But, parabolic approximations are generally only capable of detecting acoustic energy propagating in a narrow band about the horizontal. The “wide angle” equation of Claerbout was created to address this problem, and other authors have recently developed higher order parabolic approximations. In this paper a “very wide angle” parabolic approximating equation is introduced. An implicit finite difference scheme for its solution is presented and analyzed, and some of its computational characteristics are discussed. The relationship between general time-dependent one-way wave equations and parabolic equations of underwater acoustics is explored.