= Solution
If two harmonic solutions have the same <body force> and prescribed boundary <traction>, their difference has no <body force> and is traction-free. A nonzero difference would therefore be a free <eigenfunction> at that <frequency>. For any <frequency> outside the countable exceptional set found above, \b[the difference must vanish], so the forced boundary-value solution, when it exists, is unique. At an <eigenfrequency>, a free <eigenfunction> can be added to any existing solution, and a resonant forcing may fail to admit a steady harmonic response. This distinguishes uniqueness from existence.
For the full time-dependent problem, let $\mathbf w$ be the difference between two fields with identical forces, <tractions> and initial <displacement field> and <velocity>. It obeys $\rho\mathbf w_{tt}=\nabla\cdot\boldsymbol\sigma(\mathbf w)$, zero boundary <traction>, and zero initial data. Its <energy uniqueness for traction-driven elasticity> follows by direct <integration>:
$$
E(t)=\frac12\int_{\mathcal D}\left[\rho|\mathbf w_t|^2+c_{ijpq}e_{ij}(\mathbf w)e_{pq}(\mathbf w)\right]dV,\qquad e_{ij}=\frac12(\partial_iw_j+\partial_jw_i).
$$
Multiply the equation by $\mathbf w_t$ and integrate by parts. Major symmetry turns the <strain> term into the <derivative> of <elastic energy>, giving $E'(t)=\int_{\partial\mathcal D}\mathbf w_t\cdot\mathbf t(\mathbf w)dS=0$. Stable <elasticity> makes $E\geq0$, and $E(0)=0$ gives $\mathbf w_t=0$. Thus $\mathbf w$ equals its initial value zero. Even rigid zero-strain motions cannot survive these <initial conditions>.
The PDF correctly specifies zero initial <displacement field> and zero initial <velocity>; the TeX duplicates <displacement field> in the second condition. \b[The complete causal <displacement field> is unique], including when its Fourier representation contains resonant modal contributions fixed by the initial data.
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