Trapped modes due to narrow cracks in thin simply-supported elastic plates

We consider an elastic plate of infinite length and constant width supported
simply along its two parallel edges and having a finite length crack along its
centreline. In particular, we look for and find trapped modes (localised oscillations)
in the presence of the crack. An explicit wide-spacing approximation
based on the Wiener-Hopf technique applied to incident wave scattering by
semi-infinite cracks is complemented by an exact formulation of the problem
in the form of integro-differential equations. An application of a Galerkin
method for the numerical calculation of results from the latter method leads
to a novel explicit ‘small-spacing’ approximation. In combination with the
wide-spacing results this is shown to provide accurate results for all lengths
of crack.