Scattering of long flexural gravity wave due to structural heterogeneity in the framework of wave blocking

Abstract

Scattering of flexural gravity waves due to an abrupt change in structural heterogeneity is studied within the framework of blocking dynamics in time-space under shallow water approximations. As a special case, wave scattering due to a crack in the ice sheet is obtained. In the presence of compressive force, primary and secondary wave blocking occur at two different time-periods where group velocities vanish. Moreover, for each time-period within the primary and secondary blocking points, three propagating wave modes exist of which two are related with positive group velocities and the other one corresponds to negative group velocity. Subsequently, the energy relation of the scattering problem is derived in the case of multiple propagating wave modes using the conservation of energy flux. Out of the two propagating wave modes having positive group velocities, the energy transfer rate associated with the highest wavenumber is contributing significantly to the energy relation in the vicinity of the primary blocking point, whilst the other mode behaves in a similar manner near the secondary blocking point. Moreover, between the primary and secondary blocking points the incident wave mode changes to account for law of conservation of energy flux. Further, the study reveals that the energy transfer rate and the transmission coefficient are inversely proportional to each other. In case of wave scattering due to heterogeneity in ice sheet, the reflection and transmission coefficients pattern can have more than two removable discontinuities at the blocking points, whilst in the case of wave scattering due to a crack in the ice sheet, only two removable discontinuities appear at the primary and secondary blocking points. However, a jump discontinuity occurs at the point where incident wave mode conversion takes place in both the cases of wave scattering due to heterogeneity/crack in the ice sheet. Besides, irregular behaviour in plate deflection is observed due to the superposition of multiple propagating wave modes for a certain time-period within the blocking limits.