Elastodynamic Green tensor ($G_{ik}(\mathbf{x},\mathbf{y},t)$)

The causal Green function for isotropic linear elasticity is tensor-valued: column $k$ is the displacement caused by a point impulsive body force in direction $k$. Its radiative terms are longitudinal P waves at $t=R/\alpha$ and transverse S waves at $t=R/\beta$, proportional to $R^{-1}$. A term supported between the arrivals carries the intermediate/near-field response. Convolution in time and space constructs the displacement for a general source.

In a homogeneous isotropic solid, the far-field P wave is the radial projection of the force and the S wave is its transverse projection. For a unit force axis $d$, the magnitudes are $|\cos \theta |$ and $\sin \theta$. The approximation requires distance large compared with both reduced wavelengths; angular nodes can still be controlled by a smaller near-field contribution.

Choose a surface normal $\mathbf{n}$ from its minus to its plus side and jumps $[u]=u^{+}-u^{-}$, $[t]=({\sigma }^{+}-{\sigma }^{-})\mathbf{n}$. The equivalent force density is $-[t]$ and the dipole density is $M_{jk}=c_{jk\ell m}[u_{\ell }]n_m$. This follows by applying the elastic wave operator to a piecewise smooth displacement field: its distributional source is $-[t_i]{\delta }_{\Sigma }-{\partial }_j(c_{ijkl}[u_k]n_l{\delta }_{\Sigma })$. Integration by parts against the elastodynamic Green tensor gives the displayed signs. At a material interface the dipole term is interpreted through the continuous reciprocal Green traction, rather than an undefined product of a discontinuous stiffness with a surface delta.

Integrating the equivalent dipole body force against the elastodynamic Green tensor by parts gives the displayed time convolution. For a fixed seismic moment tensor shape $M_{kj}=K_{kj}V(t)$, its $R^{-1}$ terms are $[\mathbf{e}(\mathbf{e}\cdot K\mathbf{e})\dot{V}(t-R/\alpha )/{\alpha }^3+(K\mathbf{e}-\mathbf{e}(\mathbf{e}\cdot K\mathbf{e}))\dot{V}(t-R/\beta )/{\beta }^3]/(4\pi \rho R)$. Derivatives of the angular projectors and the retarded near-field integral additionally produce $R^{-2}$ and $R^{-4}$ coefficients, yielding the final static field after the source stops.