The slowness surfaces of incompressible and nearly incompressible elastic materials

The slowness surface of a compressible elastic material has three sheets whilst that of an incompressible elastic material has only two sheets. The explanation for this qualitative difference is found to be that as the material approaches an incompressible limit the inmost sheet becomes a small sphere collapsing to the origin whilst the other two sheets tend to the two sheets of the limiting incompressible solid. The theory of nearly incompressible materials is developed here because of its important applications to rubberlike solids. Some results on the wave polarisations and on the convexity of the slowness surfaces are also given.