Acoustic impedance ($Z=\rho c$)

The acoustic impedance of a fluid is $Z=\rho c$. For a progressive plane acoustic wave, pressure amplitude $P$ and velocity amplitude $U$ in its propagation direction satisfy $P=ZU$. At an oblique planar interface, zero reflection requires a matching condition involving the normal impedances $Z/\cos \theta$ or, equivalently for displacement amplitudes, $Z\cos \theta$.

Equality of the relevant wave impedances across an interface. In the elementary normal-incidence model this makes the reflection coefficient vanish, even when other material properties differ.

The product $Z_s=\rho V_s$ of bulk density and S wave speed. At a normal-incidence boundary it controls the corresponding stress-amplitude reflection coefficient.

The normal acoustic impedance of a progressive acoustic wave at a planar boundary is the pressure amplitude divided by the normal-velocity amplitude. For propagation angle $\theta$ from the normal,The distinction from the acoustic impedance $\rho c$ matters at oblique incidence: interface reflection depends on the ratio of normal impedances. For an evanescent outgoing wave $k_n$ is imaginary and the normal impedance is reactive, giving no mean normal acoustic energy flux.

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.

With pressure amplitude $P=ZV$ and velocity $V$ directed into a boundary, the mean absorbed acoustic energy flux is $\frac{1}{2}\mathrm{Re}(P{V}^{\ast })=\frac{1}{2}\mathrm{Re}Z|V{|}^2$. A passive boundary cannot supply net energy, so $\mathrm{Re}Z\ge 0$ at a real frequency. Its reflection from a propagating acoustic plane wave has magnitude at most one. A pole continued to a complex incidence angle or frequency must be interpreted separately from real-frequency passive scattering.

A layer of impedance $Z_1$, wavenumber $k_1$, and thickness $L$ between identical media of impedance $Z_0$ has pressure-amplitude transmission coefficientIt transmits perfectly at the layer resonances $k_{1}L=n\pi$, irrespective of the impedance mismatch.

In a coherent specular model, let $r_b$ be the effective lower pressure reflection coefficient referenced to $z=0$, and let the planar upper interface lie at $z=d$. With layer and upper normal impedances $Z_1,Z_2$, $r_{12}=(Z_2-Z_1)/(Z_2+Z_1)$ and $t_{21}=2Z_1/(Z_1+Z_2)$. Summing repeated round trips gives the displayed downward amplitude at the upper interface. The mean layer field is $D{e}^{i\alpha x}[e^{-i\beta (z-d)}+r_b{e}^{2i\beta d}{e}^{i\beta (z-d)}]$. Feedback through diffuse roughness channels must be neglected in this scalar model or included in a context-dependent effective reflection coefficient.

For a P-wave incident at angle $\theta$ from a liquid $(\rho ,c,\lambda )$ onto $({\rho }^{\prime },c^{\prime },{\lambda }^{\prime })$, with transmitted angle ${\theta }^{\prime }$ obeying Snell law for elastic and acoustic waves,No reflection is equivalent to ${\rho }^{\prime }{c}^{\prime }\cos \theta =\rho c\cos {\theta }^{\prime }$.