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Modelling of progressive short waves using wave envelopes
Author(s) -
Chadwick Edmund,
Bettess Peter
Publication year - 1997
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/(sici)1097-0207(19970915)40:17<3229::aid-nme209>3.0.co;2-8
Subject(s) - helmholtz equation , eikonal equation , diffraction , discretization , envelope (radar) , geometrical optics , physical optics , mathematical analysis , wave equation , physics , frequency domain , domain (mathematical analysis) , wave propagation , mathematics , optics , boundary value problem , computer science , radar , telecommunications
We consider progressive waves such that the time independent potential satisfies the Helmholtz equation, for example, the travelling wave diffracted from a body. In order to model the wave potential using finite elements it is usual to discretize the domain such that there are about ten nodal points per wavelength. However, such a procedure is computationally expensive and impractical if the waves are short. The goal is to be able to model accurately with few elements problems such as sonar and radar. Therefore we seek a new method in which the discretization of the domain is more economical. To do so, we express the complex potential ϕ in terms of the real wave envelope A and the real phase p such that ϕ= A e i p , and expect that in most regions the functions A and p vary much more gradually over the domain than does the oscillatory potential ϕ. Therefore instead of modelling the potential we model the wave envelope and the phase. The usual approach then uses the well known geometrical optics approximation (see p. 109 of Reference 1) : if the wave number k is large then the potential can be expanded in decreasing powers of k . The first two terms give the eikonal equation for the phase and the transport equation for the wave envelope respectively (see p. 149 of Reference 2). However, using the geometrical optics approximation (or ray theory) gives no diffraction effects. This approach shall therefore not be considered. (We note though that Keller's theory of geometrical diffraction, an extension to geometrical optics, does allow for diffraction effects and this may be considered at a later date.) We shall consider a new method which shall be described in the present paper and apply it to two‐dimensional problems, although the method is equally valid for arbitary three‐dimensional problems. (The method has already been validated for the case of one‐dimensional problems.) An iterative procedure is described whereby an estimate of the phase is first given and from the resulting finite element calculation for the wave envelope a better estimate for the phase is obtained. The iterated values for the phase and wave envelope converge to the expected values for the test progressive wave examples considered. Even if a very poor estimate for the phase is first given the iterated values converge to the exact values but very slowly. © 1997 John Wiley & Sons, Ltd.