Bilateral German-Russian Project

WAMPUS: Wide-Angle Mode Parabolic equations in Underwater acoustics:
derivation and numerical Solutions

financed by the Heinrich Hertz Stiftung

(10/2019-12/2019)

Summary

The project is aimed to develop the theory of wide-angle mode parabolic equations for solving problems of sound propagation in shallow water. Mode parabolic equations (MPEs) were introduced approximately 25 years ago and were proven to be a highly-efficient tool for solving sound propagation problems in complicated three-dimensional shallow- and deep-water environments. So far, however, mostly narrow-angle MPEs were extensively tested and used in practical computations. Although there exist some works dedicated to wide-angle MPE theory, it is neither fully-developed nor thoroughly validated. In the proposed study we are planning to combine the experience of the Vladivostok group in the modelling of sound propagation using (which includes narrow-angle MPEs as well as other techniques) with Wuppertal's expertise in the field of numerical solution of parabolic equations, including construction and implementation of artificial/absorbing boundary conditions.

This research study will be a natural combination of the ideas emerging from the recent papers of the Vladivostok research group and by M. Ehrhardt and significantly improved in his follow-up works. This combination of ideas can be successfully developed from the close collaboration in the framework of the proposed project.

Scientific Objectives

The main goal of the proposed study is the development of wide-angle parabolic approximation theory for different horizontal refraction equations (HREs). We rely on standard rational approximations (also known as Padé approximations) for the operator square root. It should be possible to derive wide-angle MPEs by following the standard approach. Our main task however is the development of efficient numerical method for solving the obtained equations. In this section we list the project objectives and outline a detailed plan of the research.

Project objectives

Derivation of the wide-angle MPE from the HREs of Collins and Trofimov by the method of Padé approximation of the operator square root. We will obtain a series of wide-angle MPE for different order of Padé approximation.
Development of the numerical technique for solving the derived wide-angle MPE based on Crank-Nicolson method and on SSF/ETD pseudospectral method. An efficient numerical approach for solving wide-angle MPEs will be developed and validated by comparison with analytical solutions.
Development of absorbing boundary conditions (ABCs) for the derived wide-angle MPEs. An ABC for MPE based on the rational-linear Padé approximation of the square root is a direct generalization of an ABC from previous work.
Validation of the wide-angle MPEs by solving practical problems of underwater acoustics. Several benchmark problems will be solved in order to estimate the improvement in accuracy of the solution in comparison with narrow-angle MPEs. In particular, we will consider the problem of sound propagation in a shallow sea with underwater canyon and the problem of sound propagation in a penetrable wedge.
(optional) Development of the MPE theory in the case of an elastic bottom. In this case spectral problem can be solved, e.g., using Kraken or Orca normal mode codes allowing to take bottom elasticity into account.

Connection with the works of Heinrich Hertz

Although underwater acoustics are the main field of application for the envisaged project results, these results can also be used for modelling the radio wave propagation in the atmosphere over inhomogeneous terrain. The theory of these radio waves, or more generally the theory of electromagnetic waves (EM waves), comes originally from the works of Heinrich Hertz (historically they have also been called Hertzian waves), who succeeded in the experimental proof of EM waves in 1888.

Low-frequency (between 30 and 3,000 kHz) vertically polarized radio waves can be the following surface waves in the earth's contour in a adiabatic regime known as ground wave propagation. In this way, the radio wave propagates through interaction with the conductive surface of the Earth. The waves in such an adiabatic regime follow the ground and therefore these "ground waves" can propagate over mountains and far beyond the horizon. In this propagation mode, however, they also exhibit a horizontal refraction, which is caused by terrain inhomogeneities. The MPE theory can also be used for the simulation of ground wave propagation.

During the research on the previously proposed project plan we also took the very first steps in the development of the MPE theory for ground waves. Such a theory can be used to solve many practical problems, including the development of the modern so called "Non-Direct Radio Communication", which can be used without satellites and is therefore much less expensive.

German team:

Matthias Ehrhardt

Russian team:

Pavel Petrov

German institution:

Department of Mathematics and Informatics, Faculty of Mathematics and Natural Sciences, University of Wuppertal.

Russian institution:

V.I. Il'ichev Pacific Oceanological Institute, Far Eastern Federal University, Vladivostok.

Publications related to the Project

A.G. Tyshchenko, P.S. Petrov and M. Ehrhardt, Wide-angle mode parabolic equation with transparent boundary conditions and its applications in shallow water acoustics, Proceedings of the International Conference DAYS on DIFFRACTION 2019, St.Petersburg, Russia, 2019.
P.S. Petrov, M. Ehrhardt, A.G. Tyshchenko and P.N. Petrov, Wide-angle mode parabolic equations for the problems of underwater acoustics and their numerical solution on unbounded domains, accepted: Journal of Sound and Vibration (2020), DOI: 10.1016/j.jsv.2020.115526.
P.S. Petrov, M. Ehrhardt and M.Yu. Trofimov, On the decomposition of the fundamental solution of the Helmholtz equation via solutions of iterative parabolic equations, accepted: Asymptotic Analysis, 2021.

Talks related to the Project

P. Petrov, Horizontal refraction and whispering gallery waves in the vicinity of curvilinear isobaths in a shallow sea, Seminar of Working Group Applied Mathematics and Numerical Analysis at University of Wuppertal, Nov 12, 2019.