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An explicit solution is provided for the scattering of an obliquely incident flexural-gravity wave by a narrow straight-line crack separating two semi-infinite thin elastic plates floating on water of finite depth. By first separating the solution into the sum of symmetric and antisymmetric parts it is shown that a simple form for each part can be derived in terms of a rapidly convergent infinite series multiplied by a fundamental constant of the problem. This constant is simply determined by applying an appropriate edge condition. Curves of reflection and transmission coefficients are presented, showing how they vary with plate properties and angle of incidence. It is also shown that in the absence of incident waves and for certain relations between their wavelength and frequency, symmetric edge waves exist which travel along the crack and decay in a direction normal to the crack.

The subject of this paper, the scattering of flexural waves by constrained elastic plates floating on water is relatively new and not an area that Professor Newman has worked in, as far as the authors are aware. However, in two respects there are connections to his own work. The first is the reference to his work with H. Maniar on the exciting forces on the elements of a long line of fixed vertical bottom-mounted cylinders in waves. In their paper (J Fluid Mech 339 (1997) 309-329) they pointed out the remarkable connection between the large forces on cylinders near the centre of the array at frequencies close to certain trapped-mode frequencies, which had been discovered earlier, and showed that there was another type of previously unknown trapped mode, which gave rise to large forces. In Sect. 6 of this paper the ideas described by Maniar and Newman are returned to and it is shown how the phenomenon of large forces is related to trapped, or standing Rayleigh-Bloch waves, in the present context of elastic waves. But there is a more general way in which the paper relates to Professor Newman and that is in the flavour and style of the mathematics that are employed. Thus extensive use has been made of classical mathematical methods including integral-transform techniques, complex-function theory and the use of special functions in a manner which reflects that used by Professor Newman in many of his important papers on ship hydrodynamics and related fields.

Scattering of waves by vertical barriers in infinite-depth water has received much attention due to the ability to solve many of these problems exactly. However, the same problems in finite depth require the use of approximation methods. In this paper we present an accurate method of solving these problems based on a Galerkin approximation. We will show how highly accurate complementary bounds can be computed with relative ease for many scattering problems involving vertical barriers in finite depth and also for a sloshing problem involving a vertical barrier in a rectangular tank.

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