A canonical problem for fluid-solid interfacial wave coupling

Wave-coupling involving defects or obstacles on fluid-solid interfaces is of recurrent interest in geophysics, transducer devices, structural acoustics, the acoustic microscope and related problems in non-destructive testing. Most theoretical analysis to date has, in effect, involved stress or pressure loadings along the interface, or scattering from surface inhomogeneities, that ultimately result in unmixed boundary value problems. A more complicated situation occurs if a displacement is prescribed over a region of the interface, and the rest of the interface is unloaded (or stress/pressure loaded); the resulting boundary value problem is mixed. This occurs if a rigid strip lies along part of the interface and will introduce several complications due to the presence of the edge. For simplicity a lubricated rigid strip is considered, i.e. it is smoothly bonded to the elastic substrate. To consider such mixed problems, e.g. the vibration of a finite rigid strip or diffraction by a finite strip, a canonical semi-infinite problem must be solved. It is the aim of this paper to solve the canonical problem associated with the vibrating strip exactly, and extract the form of the near strip edge stress, and displacement fields, and the far-field directivities associated with the radiated waves. The near strip edge results are checked using an invariant integral based upon a pseudo-energy momentum tensor and gives a useful independent check upon this piece of the analysis. The directivities will be useful in formulating a ray theory approach to finite strip problems. An asymptotic analysis of the solution, in the far field, is performed using a steepest descents approach and the far-field directivities are found explicitly. In the light fluid limit a transition analysis is required to determine uniform asymptotic solutions in an intermediate region over which the leaky Rayleigh wave will have an influence. In a similar manner to related work on the acoustic behaviour of fluid-loaded membranes and plates, there is beam formation in this intermediate region along critical rays.