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 Incoming links: Surface acoustic impedance

An acoustically rigid boundary has zero perturbation normal velocity. For a reflected acoustic plane wave, pressure amplitudes add in phase and the reflection coefficient is

+ 1

. It is the infinite-surface acoustic impedance limit. A pressure-release boundary instead fixes the pressure perturbation to zero and reflects with coefficient

- 1

The normal velocity is the scalar component of a velocity in a specified unit-normal direction. Its sign depends on the orientation of the normal. At a material interface, no penetration equates the fluid and interface normal velocities; it does not require their tangential components to agree. Surface acoustic impedance uses a fixed inward normal-velocity convention to determine the sign of energy absorbed by a boundary.

Use the prescribed harmonic convention

e^{iω t}

. For a propagating acoustic plane wave, let

k = (ω / c_{0}) cos θ

and

ℓ = (ω / c_{0}) sin θ > 0

. The incident and reflected pressure amplitudes in the upper half-space have vertical factors

e^{i ℓ y}

and

e^{- i ℓ y}

respectively:

p^{'} = e^{iω t - ik x} (A e^{i ℓ y} + R e^{- i ℓ y}) .

iω ρ_{0} v_{y} = - \partial_{y} p^{'}

. At the surface, the normal velocity is therefore

ℓ (R - A) / (ρ_{0} ω) = - V

P = A + R

P = Z V

gives

\frac{A + R}{A - R} = \frac{Z sin θ}{ρ _{0} c _{0}}, \frac{R}{A} = \frac{Z - Z _{f}}{Z + Z _{f}}, Z_{f} = \frac{ρ _{0} c _{0}}{sin θ} .

Here

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} Re (P V^{*}) = \frac{1}{2} Re Z ∣ V ∣^{2} \geq 0

. The four limiting cases have distinct meanings:

If $Z \to 0$ , $R = - A$ . This is a pressure-release boundary: the pressure perturbation vanishes, while the normal velocity is generally nonzero. The reflected pressure has equal amplitude and a phase reversal.
If $Z \to \infty$ , $R = A$ . The boundary is acoustically rigid, with zero normal velocity and doubled total surface pressure. There is no pressure phase reversal.
If $R / A \to 0$ , $Z \to Z_{f}$ . This is a matched boundary, taking up the incoming wave without reflection. Its pressure and normal velocity are those of the incident wave.
Formally, $R / A \to \infty$ means $Z \to - Z_{f}$ . A nonzero outgoing field can then exist with vanishing incoming amplitude. For a passive boundary at a real propagating incidence angle, a negative-real-part surface acoustic impedance cannot describe ordinary absorption: such a scattering pole must be interpreted through an active source or an continued by analytic continuation free-mode resonance. The sheet calculation below identifies the relevant free modes.

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^{iω 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

T

backed by an identical outgoing fluid half-space presents the stated surface acoustic impedance to the other half-space, using upward sheet velocity

iω η = - 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.