MCIVER, M., 2011. Acoustic wave trapping in one-dimensional axisymmetric arrays. Quarterly Journal of Mechanics and Applied Mathematics, 64 (3), pp.401-414.
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.