= Solution
Linearized continuity and Euler equations about a uniform resting inviscid fluid are $\rho'_t+\rho_0\nabla\cdot\mathbf u=0$ and $\rho_0\mathbf u_t=-\nabla p'$. For a velocity potential $\mathbf u=\nabla\phi$ and sound-speed relation $p'=c^2\rho'$, integration of Euler's equation fixes the temporal gauge so $p'=-\rho_0\phi_t$. Substitution into continuity gives
$$
\boxed{\phi_{tt}=c^2\nabla^2\phi,\qquad p'=-\rho_0\phi_t.}
$$
Use complex amplitudes with time factor $e^{i\omega t}$. The outgoing wave in fluid two is $\Phi_2(x)=Ae^{-ik_2x}$, with $k_i=\omega/c_i$. Write $\Phi_1=C\cos(k_1x)+D\sin(k_1x)$. Continuity of pressure and normal velocity at zero gives
$$
C=(\rho_2/\rho_1)A,\qquad D=-i(c_1/c_2)A.
$$
The piston velocity is $i\omega\epsilon$, so $\Phi_1'(-L)=i\omega\epsilon$. With $\theta=\omega L/c_1$, this gives
$$
A\bigl[(\rho_2/\rho_1)\sin\theta-i(c_1/c_2)\cos\theta\bigr]
=i\epsilon c_1,
$$
and hence the requested amplitude is
$$
\boxed{A=-\epsilon c_1\left[(c_1/c_2)\cos\theta
+i(\rho_2/\rho_1)\sin\theta\right]^{-1}.}
$$
For complex pressure and velocity amplitudes, the mean <acoustic energy flux> is $\tfrac12\operatorname{Re}(PU^*)$. Here $P=-i\omega\rho_2\Phi_2$ and $U=-ik_2\Phi_2$, so
$$
\boxed{\langle\mathcal F\rangle=
\frac{\rho_2\omega^2\epsilon^2c_1^2}{2c_2}
\left[(c_1/c_2)^2\cos^2\theta+(\rho_2/\rho_1)^2\sin^2\theta\right]^{-1}.}
$$
For $\rho_2\ll\rho_1$ and comparable sound speeds, weak transmission loads the finite segment. There are narrow large-flux resonances near $\theta=(n+1/2)\pi$, separated by $\Delta L=\pi c_1/\omega$. The off-resonance scale is proportional to $\rho_2$, while peak flux is proportional to $\rho_1^2/\rho_2$; the angular width is of order $(\rho_2/\rho_1)(c_2/c_1)$. These amplified solutions remain within linear acoustics only when the resulting wave amplitudes are small.
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