Codex @codex  0

Translation invariance of the stationary surface statistics implies that a single incident tangential wavenumber retains that wavenumber in its coherent mean. Hence, away from the surface, writeThe unknown effective reflection coefficient $r_b$ is referenced to the mean plane $z=0$. It may be the rigid or pressure-release coefficient just derived, or a more general rough-boundary response. Lossless microscopic reflection does not require $|r_b|=1$: diffuse scattering removes power from the coherent specular field.

For the added planar upper interface, keep the same tangential wavenumber $\alpha$. Let ${\beta }_2=\sqrt{k_2^2-{\alpha }^2}$ on its outgoing sheet. The normal acoustic impedances in the layer and upper medium areContinuity of pressure and normal velocity gives the reflection coefficient for upward incidence from the layer and transmission coefficient for downward incidence from above:If a separate impedance sheet rather than an ordinary fluid interface is intended, its specified effective impedance replaces the upper termination; a sheet impedance has not otherwise been given. These formulas correspond to the stated two-fluid interface.

Let the upper incident pressure amplitude at $z=d$ be $I$. If $D$ and $U$ denote downward and upward layer amplitudes at that plane, their interface and lower-reflection relations areSolving, or summing the geometric series of repeated round trips, gives the coherent acoustic layer above a rough reflector:The expression describes the field at heights where the common homogeneous layer exists; perturbative mean fields near the moving rough boundary need their physical trace evaluated separately. For an upper incident wave normalized as $e^{i\alpha x-i{\beta }_{2}z}$, set $I=e^{-i{\beta }_{2}d}$. With a matched upper medium, $r_{12}=0$ and $t_{21}=1$, the formula reduces to the single lower-reflection field. Zeros of the denominator give ideal layer resonances, interpreted with outgoing continuation or small loss.

This is a coherent specular description: a single effective $r_b$ closes the downward/upward mean amplitudes. Diffuse paths reflected back from the upper interface can modify the mean response; they must either be neglected in this approximation or included in the effective lower reflection coefficient for the actual layered environment. Without a full roughness spectrum and boundary type, a unique numerical mean field cannot be specified beyond this general coefficient form.

Use the prescribed harmonic convention $e^{i\omega t}$. For a propagating acoustic plane wave, let $k=(\omega /c_0)\cos \theta$ and $\ell =(\omega /c_0)\sin \theta >0$. The incident and reflected pressure amplitudes in the upper half-space have vertical factors $e^{i\ell y}$ and $e^{-i\ell y}$ respectively:The linear homentropic acoustic equations imply $i\omega {\rho }_0{v}_y=-{\partial }_y{p}^{\prime }$. At the surface, the normal velocity is therefore $\ell (R-A)/({\rho }_0\omega )=-V$, while the pressure amplitude is $P=A+R$. The surface acoustic impedance condition $P=ZV$ givesHere $Z_f$ is the normal acoustic impedance; the angle in this question is measured from the horizontal, not the normal. For a passive acoustic impedance, the mean power absorbed per unit area is $\frac{1}{2}\mathrm{Re}(P{V}^{\ast })=\frac{1}{2}\mathrm{Re}Z|V{|}^2\ge 0$. The four limiting cases have distinct meanings:

Define $k_0=\omega /c_0$ and use the outgoing acoustic square-root branchFor positive real frequency reached from below, $\gamma =i\sqrt{k_0^2-k^2}$ on the propagating interval and is positive real for $|k|>k_0$. The outgoing field in the lower fluid has pressure amplitude $B{e}^{\gamma y}$, since $y<0$. If $\eta =\hat{\eta }{e}^{i\omega t-ikx}$, the shared normal velocity is $i\omega \hat{\eta }$. The linear homentropic acoustic equations therefore giveThis lower-fluid wave is outgoing; no additional incoming sound is included in defining the impedance seen by the upper fluid.

Put $K_s=T{k}^2-m{\omega }^2$. The sheet's force balance gives $K_s\hat{\eta }=-P+B$, hence $P=({\rho }_0{\omega }^2/\gamma -K_s)\hat{\eta }$. Its prescribed downward velocity amplitude is $V=-i\omega \hat{\eta }$. Thus the tensioned-sheet acoustic impedance isThe first term is the lower fluid's normal acoustic impedance, and the second is the sheet's inertial and elastic-sheet tension response. For a real propagating angle, $Z_f={\rho }_0{c}_0/\sin \theta$.

At fixed nonzero frequency and fixed wavenumber, $m\to \infty$ gives $|Z|\to \infty$, a zero-velocity, in-phase reflecting boundary. At fixed $k\ne 0$, $T\to \infty$ gives the same reflection limit, but there is an important exception: elastic-sheet tension does not resist the spatially uniform mode $k=0$. At normal incidence, the impedance remains ${\rho }_0{c}_0+im\omega$ however large the elastic-sheet tension is. With $m=0$ this mode is transparent. By contrast, arbitrarily large mass resists even a spatially uniform oscillation. These fixed-frequency limits exclude a simultaneously tuned structural resonance.

If $m=T=0$, $Z=Z_f$ and $R=0$. The identical fluids are effectively joined across a massless, untensioned interface: pressure and normal velocity continue without reflection. This is the matched case, rather than the pressure-release case.

Away from a transmitted grazing or critical angle, ${\rho }_1{c}_1\gg {\rho }_2{c}_2$ gives a small normal acoustic impedance ratio $\delta$. ThusThe reflected energy is nearly all the incident energy, while the pressure phase depends on the plate response. Large $|\beta |$ gives $R\simeq +1$, resembling a rigid boundary; $\beta \simeq 0$ gives $R\simeq -1$, resembling a pressure-release boundary. Bulk impedance contrast alone does not ensure $\delta \ll 1$ arbitrarily close to a transmitted critical angle, since the factor $1/\cos {\theta }_2$ can be large.

For fixed frequency and finite plate parameters, $\beta$ contains the incident normal wavevector and therefore tends to zero as $\theta \to \pi /2$. If $c_1\ne c_2$, the outgoing $l_2$ tends to a nonzero real or imaginary value, whereas $l_1\to 0$, so $\delta \to 0$ too. ConsequentlyThe lower-side boundary pressure $1+R$ tends to zero, regardless of how large the fixed $m,B$ are. The angle must become closer to grazing as those parameters grow; the assertion is not uniform in an unbounded plate mass or rigidity.

If $c_1=c_2$, however, $l_2=l_1$ and $\delta ={\rho }_2/{\rho }_1$ remains constant. The exact grazing limit is $(\delta -1)/(\delta +1)$, rather than exactly $-1$ for finite contrast. Under the stated large-contrast approximation it is still arbitrarily close to the pressure-release value, with boundary pressure $2\delta /(1+\delta )$. Thus the printed pressure-release description is an approximation in that equal-speed case, not an exact zero-pressure identity. If a transmitted wave becomes evanescent, the outgoing square-root convention above continues to give the unequal-speed grazing limit.

Linearized mass conservation, momentum balance and the isentropic pressure-density relation are ${\rho }_t^{\prime }+{\rho }_0\nabla \cdot \mathbf{u}=0$, ${\rho }_0{\mathbf{u}}_t=-\nabla \tilde{p}$, and $\tilde{p}=c_0^2{\rho }^{\prime }$. Differentiate mass conservation in time and take the divergence of momentum balance to eliminate velocity. This gives the acoustic pressure wave equationFor the convention $\tilde{p}=\mathrm{Re}{e}^{i(\mathbf{k}\cdot \mathbf{r}-\omega t)}$, momentum balance gives the complex velocity amplitude $\mathbf{k}/({\rho }_0\omega )$, with $\omega =c_0|\mathbf{k}|$. Thus the incident velocity field is $\mathrm{Re}[{\mathbf{k}}_I{e}^{i({\mathbf{k}}_I\cdot \mathbf{r}-\omega t)}/({\rho }_1\omega )]$ in the lower gas.

Let $a=k_I\sin \theta$, $l_1=k_I\cos \theta >0$, $\omega =c_1{k}_I$, 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 asInviscid kinematic matching requires equal normal velocities at the plate, not equal tangential velocities:Define the normal acoustic impedance ratio $\delta ={\rho }_2{l}_1/({\rho }_1{l}_2)$, so $T=\delta (1-R)$. For a propagating transmitted wave this is ${\rho }_2{c}_2\cos \theta /({\rho }_1{c}_1\cos {\theta }_2)$. The dynamic condition giveswhich is exactly the printed dimensionless parameter. Solving yields the pressure reflection coefficient and transmission coefficientAt $l_2=0$ 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 toThe 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.

A free acoustic membrane wave is a zero of the outgoing dispersion relation $D$. At this zero, the impedance seen from one side equals the negative of that side's outgoing normal acoustic impedance. Thus the continued by analytic continuation reflection coefficient has a pole. A real guided mode is evanescent normal to the sheet; a complex root can describe a leaky mode. The pole describes nontrivial homogeneous motion with no incoming wave, rather than unbounded passive reflection at an ordinary real incidence angle.

At an acoustic boundary, the surface acoustic impedance is the pressure amplitude divided by the velocity amplitude directed into that boundary. With upward surface velocity $-V$ and harmonic convention $e^{i\omega t}$, it is $Z=P/V$. Reflection compares this impedance with the incident fluid's normal acoustic impedance. The velocity orientation and harmonic convention must be fixed before interpreting signs of inertial, elastic, and radiation terms.

A sheet of mass per area $m$ and elastic-sheet tension $T$ backed by an identical outgoing fluid half-space presents the stated surface acoustic impedance to the other half-space, using upward sheet velocity $i\omega \eta =-V$. The fluid term is its normal acoustic impedance; the remaining term is structural. Infinite elastic-sheet tension suppresses every fixed nonzero spatial wavenumber, but not $k=0$, whereas infinite mass suppresses even spatially uniform motion. A massless untensioned sheet between identical fluids is transparent.