The normal modes of a uniform, compressible Maxwell half-space

Abstract

The analytical solution for the load-induced deformation of a uniform, compressible, hydrostatically pre-stressed elastic half-space is derived. The solution is correct to first order in the quantity e, which is inversely proportional to the wave number k of the deformation. Usually e is very small compared with unity for Earth deformations on a scale amenable to the halfspace approximation. Since pre-stress advection is included in the analysis, the correspondence principle allows us to solve the field equations governing the deformation of the associated Maxwell half-space. The viscoelastic solution shows that the relaxation of the Maxwell continuum is characterized by a fundamental mode and a rapidly decaying overtone of much smaller amplitude. In the incompressible limit the overtone is not excited. The significance of the results for the relaxation of the Earth's mantle is briefly discussed.