Modelling of progressive short waves using wave envelopes

We consider progressive waves such that the time independent potential satis es the Helmholtz equation, for example, the travelling wave di racted from a body. In order to model the wave potential using nite 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 in terms of the real wave envelope A and the real phase p such that =Aeip, and expect that in most regions the functions A and p vary much more gradually over the domain than does the oscillatory potential . 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 rst 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 di raction e ects. This approach shall therefore not be considered. (We note though that Kellerʹs theory of geometrical di raction,3 an extension to geometrical optics, does allow for di raction e ects 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.4) An iterative procedure is described whereby an estimate of the phase is rst given and from the resulting nite 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 rst given the iterated values converge to the exact values but very slowl