= Solution
Linearized mass conservation, momentum balance and the isentropic pressure-density relation are $\rho'_t+\rho_0\nabla\cdot\mathbf u=0$, $\rho_0\mathbf u_t=-\nabla\widetilde p$, and $\widetilde p=c_0^2\rho'$. Differentiate mass conservation in time and take the divergence of momentum balance to eliminate velocity. This gives the <acoustic pressure wave equation>
$$
\boxed{\widetilde p_{tt}=c_0^2\nabla^2\widetilde p}.
$$
For the convention $\widetilde p=\operatorname{Re}e^{i(\mathbf k\cdot\mathbf r-\omega t)}$, momentum balance gives the complex velocity amplitude $\mathbf k/(\rho_0\omega)$, with $\omega=c_0|\mathbf k|$. Thus the incident <velocity field> is $\operatorname{Re}[\mathbf k_Ie^{i(\mathbf k_I\cdot\mathbf r-\omega t)}/(\rho_1\omega)]$ in the lower gas.
Let $a=k_I\sin\theta$, $l_1=k_I\cos\theta>0$, $\omega=c_1k_I$, and $l_2=\sqrt{(\omega/c_2)^2-a^2}$ with nonnegative real or imaginary part. This chooses an outgoing propagating wave or an evanescent wave decaying as $y\to+\infty$. Conservation of tangential <wavevector> is the acoustic <Snell law for elastic and acoustic waves>. Write the complex pressures as
$$
p_1=e^{i(ax-\omega t)}(e^{il_1y}+R e^{-il_1y}),\qquad
p_2=T e^{i(ax+l_2y-\omega t)}.
$$
Inviscid kinematic matching requires equal normal velocities at the plate, not equal tangential velocities:
$$
\frac{l_1}{\rho_1\omega}(1-R)=\frac{l_2}{\rho_2\omega}T=-i\omega\eta_0.
$$
Define the <normal acoustic impedance> ratio $\delta=\rho_2l_1/(\rho_1l_2)$, so $T=\delta(1-R)$. For a propagating transmitted wave this is $\rho_2c_2\cos\theta/(\rho_1c_1\cos\theta_2)$. The dynamic condition gives
$$
T-1-R=(m\omega^2-Ba^4)\eta_0=i\beta(1-R),
\quad \beta=\frac{(m\omega^2-Ba^4)l_1}{\rho_1\omega^2},
$$
which is exactly the printed dimensionless parameter. Solving yields the pressure <reflection coefficient> and <transmission coefficient>
$$
\boxed{R=\frac{\delta-1-i\beta}{\delta+1-i\beta},\qquad
T=\frac{2\delta}{\delta+1-i\beta}}.
$$
At $l_2=0$ use the preceding velocity equations, or take their limit, rather than substituting an infinite $\delta$. For real $l_2>0$, the reflected <energy> fraction is $|R|^2$, the transmitted fraction is $|T|^2/\delta$, and these sum to one. An evanescent transmitted wave carries no mean normal <energy> flux.
When the gases match, $\delta=1$, so $R=-i\beta/(2-i\beta)$ and $T=2/(2-i\beta)$. For a nongrazing incident wave of nonzero frequency, perfect transmission means $R=0$, equivalent to
$$
\boxed{\beta=0\quad\Longleftrightarrow\quad m\omega^2=Bk_I^4\sin^4\theta
\quad\Longleftrightarrow\quad mc_0^2=Bk_I^2\sin^4\theta}.
$$
The plate's inertial and bending terms cancel: the incident frequency and tangential <wavevector> coincide with its free <flexural wave> <dispersion relation>. Then $T=1$, including its phase. With positive $m,B$, this condition is possible only at angles and frequencies satisfying that relation; for normal incidence at nonzero frequency it cannot occur unless the mass is zero.
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