The periodic acoustic boundary element method for modelling sound field generated by an infinitely long periodic structure

Prediction of sound field generated by an infinitely long periodic structure is often required in engineering. One of the examples is the sound field created by vibration of the rail of a slab railway track, of which the radiating and scattering boundaries are periodic in the track direction due to the rail fasteners. To provide a proper computational tool for such problems, we develop the periodic acoustic boundary element method (PABEM) which only requires a three-dimensional (3D) boundary element mesh for a single cell. The development of the PABEM involves Floquet-transformation of the acoustic boundary integral equation of the infinite periodic structure, exploration of the Floquet-transformed Green's functions, and solution of the boundary integral equation using the boundary element method (BEM). Although the last step is largely the same as the conventional BEM, differences do exist and need to be carefully treated. Special attentions are given to the singularities of the Floquet-transformed Green's functions and to the numerical treatment of such singularities. The PABEM is shown to be directly applicable to sound radiation from a propagating vibration wave in an infinite periodic structure. This makes the PABEM useful for railway noise study since the response of the rail is often expressed as the sum of propagating waves at particular wavenumbers.