Solving the Helmholtz equation in terms of amplitude and phase; revisited

In a paper presented at the ISMA conference in 2008 [2], we proposed to solve the Helmholtz equation in terms of the pressure amplitude and phase instead of the complex pressure itself. The major advantage of this approach is the large reduction of the number of degrees of freedom, needed to describe predominantly propagating waves at large wavenumbers. This is due to the fact that, for propagating waves, both amplitude and phase are smooth functions for any wavenumber. A drawback of the method is the resulting non-linear, coupled set of differential equations, as well as the resulting non-linear boundary equations (as opposed to boundary conditions). Despite this increased complexity, in 2008, we did present 2D solutions on meshes of approximately 3 wavelengths per element. Solutions for higher wavenumber values could not be obtained (the non-linear solver did not converge). In this paper, we give an explanation and a solution for this non-convergence problem. First, we present accurate finite element solutions for a 1D benchmark problem, obtained on meshes having an arbitrary number of waves per element. In addition, we present solutions for the 2D problem of 2008, also for meshes having an arbitrary number of waves per element. As a prelude to more practical problems, we show the solution for a vibrating cylinder and a preliminary solution for a plane wave scattering on a rigid cylinder. Unluckily, the scattering problem revealed convergence problems which need further investigation. We will discuss these problems and end with conclusions.