Forced Response of Two-Dimensional Homogeneous Media Using the Wave and Finite Element Method

The forced response of two-dimensional homogenous media subjected to time harmonic loading is treated. The approach starts with the wave and finite element (WFE) method where a small segment of the homogeneous medium is modelled using conventional finite element (FE) methods. Here, the full power of commercial and/or in-house FE packages can be harnessed to obtain the mass and stiffness matrices of the segment. These are then used, along with periodicity conditions, to formulate an eigenvalue problem. Unlike the case of waveguides where the knowledge of the frequency of excitation is enough to solve the eigenvalue problem, wave propagation in two-dimensional media results in an eigenvalue problem involving the excitation frequency and the wavenumber (or propagation constant) in two directions. Here, one can (for example) formulate the eigenvalue problem in terms of the wavenumber in the x - direction for a given frequency and wavenumber in the y - direction. This information is then used to obtain the response of the medium to a convected harmonic pressure (CHP). Since the Fourier transform of a general two-dimensional excitation is a linear combination of CHPs, the response to a general excitation is a linear combination of the responses to CHPs. Thus, the response of a two-dimensional medium to a general excitation can be obtained by evaluating an inverse Fourier transform. Mathematically, this is a double integral, one of which can be evaluated analytically using contour integration and the residue theorem. The other integral can be evaluated numerically, and there is a wealth of techniques for doing that. Hence, the approach presented herein enables finding the response of a two-dimensional medium to an arbitrary loading via: a) modelling a small segment of the medium using standard FE methods and post-processing its mass and stiffness matrices to obtain the wave characteristics, b) formulating the Fourier transform of the response to a general loading, and c) computing the inverse of the Fourier transform semi-analytically using contour integration and the residue theorem followed by a numerical integration to find the response at any point in the medium. Numerical examples are presented to illustrate the approach.