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The polarization of an elastic wave in an isotropic solid specifies the direction of its displacement field amplitude relative to its wavevector. The longitudinal wave component is parallel to the wavevector; the transverse wave component is perpendicular. For a real unit propagation vector $n$, $a_L=(a\cdot n)n$ and $a_T=a-a_L$. An evanescent constituent has a complex spatial derivative vector, so the corresponding amplitude ratio can be imaginary. The longitudinal and transverse constituents of a Rayleigh wave combine into ellipse particle motion at the free surface.

Let $d(\mathbf{y},t)=[u^{[n]}]$ denote the signed opening, positive when the faces separate, and let $S$ be the crack surface. Its displacement jump is $\mathbf{b}=d\mathbf{n}$. The singular strain associated with this jump has moment density $c_{ijkl}{b}_k{n}_l$. For the isotropic elastic stiffness tensor, contraction givesThe equivalent distributional body force is the negative divergence of this prescribed moment density:The resulting forced elastic wave in an isotropic solid equation is $\rho u_{i,tt}={\partial }_j{\sigma }_{ij}+f_i$. This is a tensile-crack seismic source; it need not be represented as an ordinary smooth volume force. For a small nearly planar patch of area $S_0$, uniform normal $\mathbf{n}$ and opening $d(t)$, define its volume change $V(t)=S_{0}d(t)$. Its seismic moment tensor and point-source force areThe sign convention makes a positive opening a positive extensional moment. Reversing the convention for the displacement jump changes both signs together; the force is independent of reversing the orientation of the crack normal.

Taking the divergence of the Cauchy momentum equation and commuting constant-coefficient derivatives givesThus the dilatation is a wave of speedTaking the curl, and using $\nabla \times \nabla f=0$, givesso the rotation propagates atThese are respectively the speeds of P-waves and S-waves.