Bilateral GermanRussian Project
WAMPUS: WideAngle Mode Parabolic equations in Underwater acoustics: derivation and numerical Solutions
(10/201912/2019)

Summary
The project is aimed to develop the theory of wideangle 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
highlyefficient tool for solving sound propagation
problems in complicated threedimensional shallow and deepwater environments.
So far, however, mostly narrowangle MPEs were extensively tested
and used in practical computations. Although there exist some works dedicated to wideangle MPE theory,
it is neither fullydeveloped 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 narrowangle 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 followup 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 wideangle
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 wideangle 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 wideangle 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 wideangle MPE for different order of Padé
approximation.
 Development of the numerical technique for solving the
derived wideangle MPE based on CrankNicolson method and on
SSF/ETD pseudospectral method. An efficient numerical approach
for solving wideangle MPEs will be developed and validated by comparison with analytical solutions.
 Development of absorbing boundary conditions (ABCs) for the derived wideangle MPEs.
An ABC for MPE based on the rationallinear Padé approximation of the square root is a direct
generalization of an ABC from previous work.
 Validation of the wideangle 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 narrowangle 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.
Lowfrequency (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
"NonDirect Radio Communication", which can be used without satellites and is therefore much less expensive.
German team:
Russian team:
German institution:
Russian institution:
 V.I. Il'ichev Pacific Oceanological Institute, Far Eastern Federal University, Vladivostok.
Publications related to the Project
Talks related to the Project
Posters related to the Project
Related Projects