Expansion formulae for flexural-gravity wave propagation during wave blocking

Flexural-gravity wave blocking occurs in the presence of lateral compressive force. The existing expansion formulae for the velocity potential were limited in the case of distinct real roots of the dispersion relation with the multiplicity one. This limitation is overcome, and we obtain the complete solution for the flexural-gravity wave-maker problem at blocking and saddle points by deriving the expansion formula for velocity potential using the Fourier transform corresponding to the roots of the dispersion relation with multiplicity two and three, respectively. Further, we check the convergence of the newly derived expansion formulae by using the spectral representation of the associated eigenfunction. It has been proved that the eigenfunctions related to the expansion formulae are linearly dependent and satisfy the orthogonal mode-coupling relation. Next, we solve a model problem of flexural-gravity wave scattering by a crack using the derived expansion formulae and establish the energy balance relation by applying Green's integral theorem at blocking points for infinite water depth. In addition, we have presented the validation of energy balance relation involving the generalized scattering coefficients.