 Spherical elastic ray invariant

For an isotropic radial wave speed $\alpha (r)$, the elastic ray equations imply conservation of $\mathbf{x}\times \mathbf{y}$, where $\mathbf{y}$ is the slowness vector. Indeed, its derivative is $\alpha \mathbf{y}\times \mathbf{y}+\mathbf{x}\times \nabla (1/\alpha )=0$. Thus each nonradial elastic ray lies in a fixed plane through the centre and $p=|\mathbf{x}\times \mathbf{y}|$ is constant. The angle $i$ is measured from the radial direction; tangential slowness continuity preserves $p$ across spherical interfaces.