Rayleigh-Bloch surface waves along periodic gratings and their
connection with trapped modes in waveguides.
R. Porter & D. V. Evans (submitted to J. Fluid Mech)
Rayleigh-Bloch surface waves are acoustic or electromagnetic waves which
propagate parallel to a two-dimensional diffraction grating and which
are exponentially damped with distance from the grating. In the
water-wave context they describe a localised wave having dominant
wavenumber $\beta$ travelling along an infinite periodic array of
identical bottom-mounted cylinders having uniform cross-section
throughout the water depth. A numerical method is described which
enables the frequencies of the Rayleigh-Bloch waves to be determined
as a function of $\beta$ for an arbitrary cylinder cross-section.
For particular symmetric cylinders, it is shown how a special choice of
$\beta$ produces results for the trapped mode frequencies and mode
shapes in the vicinity of any (finite) number of cylinders spanning
a rectangular waveguide or channel. It is also shown how one particular
choice of $\beta$ gives rise to a new type of trapped mode near an
{\it un}symmetric cylinder contained within a parallel-sided waveguide
with distorted walls. The implications for large forces due to incident
waves on a large but finite number of such cylinders in the ocean is
discussed.
Download gzipped postscript of paper