Forced vibrations of a rigid circular plate on a semi-infinite elastic space and on an elastic stratum

Abstract

The impedance of a rigid circular plate attached to the free surface of a semi-infinite elastic space or an elastic stratum is determined for its four degrees of freedom. The solution of the dual integral equations arising from this mixed boundary-value problem is avoided by reference to Rayleigh’s reciprocal theorem. This enables the functions of frequency, which determine the in-phase and out-of-phase components of displacement of the plate, to be located between two close bounds and lying much closer to one than to the other. These bounds appear as infinite integrals involving branch functions and are reduced to tractable finite integrals by integration in the complex plane. Dissipation of waves to infinity produces an effective damping, and the added effect of the inclusion of true damping in the medium is discussed. It is to be expected, of course, that the unloaded rigid plate attached to the free surface of a semi-infinite elastic space does not resonate. The change of impedance of the plate with frequency is found to be similar for the two translations and also similar for the two rotations. Resonance occurs in the case of vertical and horizontal translation of the plate attached to the surface of an elastic stratum. However, it does not exist for rotations of the plate on the stratum. Instead, a maximum in the response appears, this maximum being more defined the greater the ratio of plate diameter to stratum depth. The addition of small true damping in the medium changes the characteristics very little. Experimental work substantiating these theoretical results, together with a general discussion of the results and their applications in geophysics and engineering, is being published shortly.