= Transmitted velocity potential from a piston through an acoustic interface
For a finite duct segment of phase length $\lambda$ joined to a semi-infinite medium, continuity of pressure and velocity gives the outgoing potential amplitude
$$
T=\epsilon c_-
\frac{i(\rho_+/\rho_-)\sin\lambda-(c_-/c_+)\cos\lambda}
{(\rho_+/\rho_-)^2\sin^2\lambda+(c_-/c_+)^2\cos^2\lambda}.
$$
At half-integer $\lambda/\pi$, a large impedance mismatch can strongly enhance transmitted flux for a prescribed piston displacement.
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