The motional impedance of simple sources in an isotropic solid

Abstract

This paper analyses the motional impedance of sources of simple geometrical shapes which are either dilating or oscillating in an isotropic solid medium of infinite extent. The equation of wave propagation in an isotropic solid is expressed in cylindrical and spherical polar co-ordinates, assuming that in the cylindrical co-ordinate case there is no motion or variation along the cylinder axis (or in other words the problem is two-dimensional), while in the case of spherical co-ordinates no motion or variation takes place around lines of latitude, the direction of oscillation being regarded as the polar axis. In the case of dilating sources the problem is further simplified, as all movement and variation takes place along radial lines, and an equation can be derived relating the particle displacement and the radial distance from the source. A solution is found which satisfies the boundary conditions on the surface of the source, and which converges to a radiated wave of the form eik1r/r for large values of r, where k1 is the wavelength constant for compressional waves in the surrounding medium. In the case of oscillating sources the wave equation separates into two independent equations, one in ∇.ubar and the other in |∇ × ubar|, the latter being a scalar quantity equal in magnitude to the vector ∇ × ubar. Solutions are obtained which converge at infinity as eik1r/r for ∇.ubar and as eik2r/r for |∇ × ubar|, where k1 is as before the compressional wavelength constant and k2 the shear wavelength constant for the medium. The arbitrary constants are then adjusted to fit the boundary condition on the surface of the source. The total force on the source in the direction of motion is then calculated in each case from the generalized Hooke's law equations relating stress to strain, and hence the motional impedance is obtained by dividing force by velocity of source. Values of motional impedance per unit source area are plotted on an Argand diagram for several values of Poissons ratio of the surrounding medium, showing the variation of impedance with source size, the latter quantity being expressed in units of 1/2π times the length of a compressional wave. This work is intended as a first step towards the analysis of the more complex sources which are encountered in practice, for example, a piston source on the free surface of a semi-infinite medium.