In the 20th century, an important contribution to the modeling of wave propagation prediction is the Parabolic Equation (PE) approximation method. This did not happen until the last quarter of the century. During that time period, a number of contributions were made in the first 15 years, then, contributions were made continuously until the end of the 20th century. Contributions to the PE method are still going on. During the second half of the 1970 when the PE was introduced to the field of underwater acoustics, its main purpose was to predict long-range, low-frequency acoustic propagations under range-dependent environments in fluid medium; thus, there were a number of limitations. As time progresses these limitations were somewhat relaxed a great deal gradually due to many useful contributions. Up to this date, all contributions are aimed at the enhancement of the capability of the PE method. A few review papers of the PE have been presented in open archived literatures. This paper gives a complete survey of what has been done from the start until the end of 1999. Contributions up to the year 1994 were already reported in some detail in archived literatures; those contributions will be outlined. Surveying the contributions from 1995 to 1999 is a new addition to the already published survey papers. In recent years, the PE method has been used for many real applications; some results are included in this paper. A discussion will be given at the end of this paper on looking ahead how much more the PE can do in order to stimulate future research, development, and applications.

Parabolic equation development in the twentieth century

A three-dimensional (3D) parabolic equation (PE) model for sound propagation in a seismo-acoustic waveguide is developed in Cartesian coordinates, with x, y, and z representing the marching direction, the longitudinal direction, and the depth direction, respectively. Two sets of 3D PEs for horizontally homogenous media are derived by rewriting the 3D elastic motion equations and simultaneously choosing proper dependent variables. The numerical scheme is for now restricted to the y-independent bathymetry. Accuracy of the numerical scheme is validated, and its azimuthal limitation is analyzed. In addition, effects of horizontal refraction in a wedge-shaped waveguide and another waveguide with a polyline bottom are illustrated. Great efforts should be made in future to provide this model with the ability to handle arbitrarily irregular fluid–elastic interfaces.

Three-dimensional parabolic equation model for seismo-acoustic propagation: Theoretical development and preliminary numerical implementation

A coupled three-dimensional fluid-elastic wave propagation mathematical model has been developed to handle environmental interactions in the ocean between a fluid medium and an elastic bottom. The existing model combines three-dimensional fluid and elastic wave propagation models with the incorporation of a set of horizontal fluid-elastic interface conditions. This paper extends the above model to consider an irregular fluid-elastic interface. The theoretical development of the irregular fluid-elastic interface equations is presented. Comparison of the fluid-elastic irregular interface to the horizontal case is one of the major objectives of this paper. Another major objective of this paper is the construction of a complete set of three-dimensional fluid-elastic wave equations, including the irregular interface, in a form suitable for stable numerical solution.

Coupled 3d wave equations with irregular fluid-elastic interface: theoretical development

In a realistic three-dimensional ocean environment involving both fluid and elastic media, a treatment of an irregular fluid-elastic interface is needed for the analysis of wave propagation. This paper introduces a model whcih possesses the necessary cpability. A complete mathematical decelopment of the model is presented with adaptable to the application of the parabolic equation method

An irregular interface model for coupled fluid-elastic parabolic equations

We present in this paper partial differential equations which govern three-dimensional acoustic wave propagation in fluid-elastic media. Working equations are parabolized so as to allow the analysis to be conducted in a plane-by-plane fashion. This simplification, while permitting only outgoing wave propagation, facilitates the analysis and cuts down on computing time and disk storage. To couple working equations in fluid and elastic layers, we impose physically relevant conditions on the interface. On the horizontal interface we demand continuity of the normal displacement and normal stress. In addition, physical reasoning requires that shear stresses vanish on the interface for the present analysis, which is formulated under the inviscid flow assumption. We approximate spatial derivatives with respect to θ and z using the second-order accurate centered scheme. The resulting ordinary differential equation is solved using the implicit scheme to render also second-order prediction accuracy in r. With a numerical scheme, it is highly desirable to be able to check its prediction against suitable test problems, preferably ones for which an exact solution is available. In this three-dimensional study, test problems were chosen to demonstrate the applicability of the code to the individual fluid and elastic layer. We have also verified that the code is applicable to analysis of wave propagation in water and elastic layers, across which there is an interface.

/pdf/a-space-marching-scheme-for-underwater-acoustic-wave-1e2l12qteh.pdf

A space marching scheme for underwater acoustic wave propagation in fluid-elastic media

A three-dimensional mathematical model is developed for handling environmental interactions in the ocean between a fluid medium and an elastic bottom. The model is a combination of three-dimensional wave propagaion models in both the fluid and the elastic media, together with the incorporation of a set of fluid-elastic interface conditons. A complete theory with regard to the mathematical formulation of the fluid and elastic models is discussed, along with an error analyiss, and a description of the derivation of the fluid-elactic interface equations is given. A ordinary differential equation (ODE) method in conjunction with a finite difference marching scheme to obtain the numerical solution of the complete representative system of coupled fluid-elastic wave equations is outlined; moreover, the theory with regard to the stability and consistency of the marching scheme is discussed. The major development of this paper is the combination of the fluid equations, the elastic equations and the set of fluid-el...

A coupled 3D fluid-elastic wave propagation model: mathematical formulation and analysis

Forcing of slow density waves in the C ring by Saturn's quasi-toroidal normal modes

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A coupled 3D fluid-elastic wave propagation model: mathematical formulation and analysis

Forcing of slow density waves in the C ring by Saturn's quasi-toroidal normal modes