= Elastodynamic surface-jump representation
{title2=$u_i=\int_0^t\int_\Sigma[-G_i^{\ j}[t_j]+M_{jk}\partial_{\xi_k}G_i^{\ j}]\,dS\,d\tau$}
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.
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