The M Waves Emitted by an Expanding Phase Boundary Create a “Lacuna”

The M waves introduced by Burridge and Willis (1969, “The Self-Similar Problem of the Expanding Crack in an Anisotropic Solid,” Math. Proc. Cambridge Philos. Soc., 66(2), pp. 443–468) are emitted by the surface of a self-similarly expanding elliptical crack, and they give Rayleigh waves at the corresponding crack speed. In the analysis for the self-similarly expanding spherical inclusion with phase change (dynamic Eshelby problem), the M waves are related to the waves obtained on the basis of the dynamic Green’s function containing the contribution from the latest wavelets emitted by the expanding boundary of phase discontinuity, and they satisfy the Hadamard jump conditions for compatibility and linear momentum across the moving phase boundary of discontinuity. In the interior of the expanding inclusion, they create a “lacuna” with zero particle velocity by canceling the effect of the P and S. It is shown that the “lacuna” and Eshelby properties are also valid for a Newtonian fluid undergoing phase change in a self-similarly expanding ellipsoidal region of a fluid with different viscosity.