= Solution
Write the in-plane <displacement field> as $(u,v)$. The <infinitesimal strain tensor> gives $2e_{XY}=u_Y+v_X$, $e_{YY}=v_Y$, and $\operatorname{tr}e=u_X+v_Y$. The two nontrivial zero-<traction> conditions become
$$
\boxed{u_Y+v_X=0,\qquad\lambda u_X+(\lambda+2\mu)v_Y=0\quad(Y=0).}
$$
Equivalently for the <complex amplitudes> they are $U'(0)+ikV(0)=0$ and $ik\lambda U(0)+(\lambda+2\mu)V'(0)=0$. Because the <constitutive equation> is real and linear, vanishing of these complex <amplitudes> enforces the conditions at every position and time.
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