Acoustic wave trapping in one-dimensional axisymmetric arrays

The existence of acoustic, Rayleigh–Bloch modes in the vicinity of a one-dimensional (1D) periodic array of rigid, axisymmetric structures is established with the use of a variational principle. Axisymmetric modes at frequencies below the cut-off frequency are shown to exist for all piecewise smooth structures and non-axisymmetric modes are found for a class of structures whose radial dimension is sufficiently large compared to the structure spacing. The theory is illustrated with numerical calculations of the wave numbers of Rayleigh–Bloch modes for an array of circular plates. An integral equation for the acoustic wave field in the neighbourhood of such an array is obtained and solved with the use of a Galerkin technique, which builds in the singularity in the derivative of the field at the rim of the plate.