Generation of internal waves from rest: extended use of complex coordinates, for a sphere but not a disk

Abstract

AbstractThe anisotropy created by stratification and or rotation places restrictions, severe if viscosity is present, on the construction of analytic solutions to wave generation and scattering problems. Consequently, much literature is devoted to frequency space and so careful consideration of oscillatory motion generated from rest is advisable. Moreover, the use of complex coordinates in the inviscid case has been incompletely presented. The detailed inversion of the Fourier time transform for a breathing or heaving sphere demonstrates an expanded, perhaps more crucial, role for the complex coordinates and shows that the known phase changes in the energy propagation regions are present throughout the St Andrew’s cross that circumscribes the sphere. However, the different solution structure for the heaving disk requires and allows a more direct calculation. Though the inclusion of rotation does not affect the dynamics, it enables the significance of their relative magnitude to be identified and reference to rotation only results achieved.