On the partition of energy between elastic waves in a semi-infinite solid

Abstract

Expressions are derived for the power radiated in the compressional, shear and surface waves set up by a circular disk vibrating normally to the free surface of a semi-infinite isotropic solid. The total radiated power is also calculated independently by integrating the displacement velocity over the area of the source. The theory is extended to a general type of multi-element radiator in the form of an array of elements on the circumference of a circle. The calculation of the total power here involves a ‘mutual admittance’ function, a table of which is given for the case when the Poisson’s ratio of the medium is equal to ¼. The theory is applied to a three-element radiator of a type used in a recent geophysical investigation, and it is shown that the efficiency of radiation in the compressional mode can be varied between wide limits by varying the distance between the elements. Finally, an approach is suggested for problems in which the most suitable idealized boundary condition is one of known displacement under the radiator, the stress being zero elsewhere on the free surface. It is shown that the stress under the radiator satisfies an integral equation whose kernel is derived from the mutual admittance function.