= Solution
For the $b_1$ root the <elastic wave polarization> is parallel to the complex <derivative> vector $(ik,-b_1)$, giving
$$
\boxed{B_1=\frac{i b_1}{k}A_1.}
$$
For the $b_2$ root the field has zero <divergence>, $ikA_2-b_2B_2=0$, giving
$$
\boxed{B_2=\frac{ik}{b_2}A_2.}
$$
These relations also follow by inserting each decay root into either row of the <matrix>. The complete decaying <displacement field> has complex components
$$
U(Y)=A_1e^{-b_1Y}+A_2e^{-b_2Y},\qquad
V(Y)=\frac{ib_1}{k}A_1e^{-b_1Y}+\frac{ik}{b_2}A_2e^{-b_2Y},
$$
multiplied by $e^{ik(X-ct)}$, followed by the <real part>. The first constituent is irrotational and the second is divergence-free; both are needed to cancel the two surface <tractions>.
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