= Solution
Let $\Omega$ be the solid exterior to the cavity, and choose $\mathbf n$ outward from the solid, hence into the cavity on its boundary $S$. Subtraction of the incident solution from the total solution gives the scattered <displacement field> equations
$$
\rho\ddot u_i^s=\partial_j\sigma_{ij}^s,\qquad
\sigma_{ij}^s=c_{ijkl}e_{kl}^s,\qquad
e_{kl}^s=\tfrac12(u_{k,l}^s+u_{l,k}^s),\qquad \mathbf x\in\Omega.
$$
The traction-free cavity condition for the total field gives \b[$\sigma_{ij}^s n_j=-\sigma_{ij}^0 n_j$ on $S$]. The scattered field has zero <initial displacement> and <initial velocity> before the pulse reaches the cavity and satisfies an outgoing <radiation condition>, excluding an additional incoming <elastic wave>. Equivalently one seeks the causal response to this prescribed boundary <traction>. Although the incident plane pulse extends across an infinite plane, the causal scattered field from the bounded cavity has finite propagation distance at each finite time.
To prove <energy uniqueness for traction-driven elasticity>, take the difference $\mathbf w$ of two solutions with the same <initial conditions>, <body force>, boundary <traction> and outgoing condition. It satisfies
$$
\rho\ddot w_i=\partial_j\sigma_{ij}(w),\quad
\sigma_{ij}(w)n_j=0\text{ on }S,\quad
\mathbf w(\cdot,t_0)=\dot{\mathbf w}(\cdot,t_0)=0.
$$
For a large enclosing ball $B_L$, multiply by $\dot w_i$ and integrate over $\Omega\cap B_L$. <Integration by parts> gives
$$
\int_{\Omega\cap B_L}\rho\ddot w_i\dot w_i\,dV
=\int_{\partial(\Omega\cap B_L)}\dot w_i\sigma_{ij}(w)n_j\,dS
-\int_{\Omega\cap B_L}\sigma_{ij}(w)\partial_j\dot w_i\,dV.
$$
Symmetry of <stress> replaces the last derivative by $\dot e_{ij}(w)$, and major symmetry of the <elastic stiffness tensor> makes that integral the derivative of the <strain> <energy>. Therefore
$$
\frac{d}{dt}\frac12\int_{\Omega\cap B_L}
[\rho|\dot{\mathbf w}|^2+c_{ijkl}e_{ij}(w)e_{kl}(w)]\,dV
=\int_{\partial(\Omega\cap B_L)}\dot w_i\sigma_{ij}(w)n_j\,dS.
$$
The cavity contribution is zero. On any fixed finite time interval choose $L$ beyond the causal support of the scattered difference; the outer contribution is then zero as well. This gives conservation of its nonnegative <elastic energy>. Positive <mass density> and positive <strain> <energy> imply that zero initial <energy> forces $\dot{\mathbf w}=0$ everywhere; the zero initial <displacement field> then gives $\mathbf w=0$. The same argument works for finite-energy outgoing solutions by taking the limit of enclosing surfaces and excluding incoming <energy>. Hence \b[the causal scattered field $\mathbf u^s$ is unique]. Specifying the incident field and cavity <traction> alone without the initial and outgoing conditions would not establish uniqueness.
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