= Solution
Use complex fields with time factor $e^{i\omega t}$ and put $k_\pm=\omega/c_\pm$. The outgoing potential in $x>0$ is
$$
\Phi_+(x)=T e^{-ik_+x}.
$$
Continuity of <pressure> and <velocity> at $x=0$ gives
$$
\Phi_-(0)=\frac{\rho_+}{\rho_-}T,
\qquad
\Phi_-'(0)=-ik_+T.
$$
Hence
$$
\Phi_-(x)=T\left[
\frac{\rho_+}{\rho_-}\cos(k_-x)
-i\frac{c_-}{c_+}\sin(k_-x)
\right].
$$
The piston condition $\Phi_-'(-L)=i\omega\epsilon$, with $\lambda=k_-L$, gives
$$
\boxed{
T=\epsilon c_-
\frac{i(\rho_+/\rho_-)\sin\lambda-(c_-/c_+)\cos\lambda}
{ (\rho_+/\rho_-)^2\sin^2\lambda+(c_-/c_+)^2\cos^2\lambda}.}
$$
This is the <transmitted velocity potential from a piston through an acoustic interface>.
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