Non-linear resistance of a micro-perforate

At high sound levels the flow in the holes of a micro-perforate separates at the hole edges and the plate resistance grows with the hole velocity. When this is on, the plate impedance of every face of the wall is computed from its own peak hole velocity, at the level the inlets are excited with.

Last updated 2026-11-06

Theory

The linear micro-perforate impedance holds while the acoustic velocity in the holes is small. Above it the jets formed at each hole end lose their kinetic energy, which adds a resistance proportional to the peak hole velocity, and the end-correction mass is partly blown away Allam and Abom (2011). The same resistance term is used for slits by Kabral et al. (2014); a slit has no end correction to reduce.

The hole velocity depends on the impedance itself, so it is found by iteration at every frequency: solve with the current impedance, take the pressure jump across each face of the wall, compute the hole velocity from it, update that face's impedance, and repeat (under-relaxed) until the hole velocities change by less than 0.1 %.

The excitation is the one the transmission loss is computed with: the amplitude (or power) and phase of every excited inlet. The result is therefore only meaningful when the inlet ports carry the real source level. The default amplitude of 1 Pa is an incident sound pressure level of about 91 dB, where the term is negligible.

Equations

z(\hat u_h) = z_{\mathrm{lin}} + \frac{|\hat u_h|}{\sigma c} + i\,\frac{x_e}{\sigma}\left[\left(1 + \frac{|\hat u_h|}{\sigma c}\right)^{-1} - 1\right]

(1)

where	meaning	unit
$z$	Normalised transfer impedance of the plate	-
$z_{\mathrm{lin}}$	Linear plate impedance of the chosen micro-perforate model	-
$\hat u_h$	Peak acoustic velocity in the holes of the face	m/s
$\sigma$	Open-area ratio	-
$c$	Speed of sound of the gas at the wall	m/s
$x_e$	End-correction reactance of one hole, $\omega\,0.85\,d\,F_\delta / c$ (0 for slits)	-

|\hat u_h| = \frac{1}{\sigma}\,\frac{|\Delta \hat p|}{\rho c\,|z|}

(2)

where	meaning	unit
$\Delta \hat p$	Peak pressure difference across the face	Pa
$\rho$	Density of the gas at the wall	kg/m³

L = 20\log_{10}\frac{A}{\sqrt{2}\cdot 20\,\mu\mathrm{Pa}}

(3)

where	meaning	unit
$L$	Incident sound pressure level of an inlet (reported in the log)	dB
$A$	Inlet amplitude (peak)	Pa

Default and rule

Default: Off (linear impedance)
Rule: Off: the linear impedance, which is what all earlier results used.
Source: Allam and Abom (2011), eqs. 4 to 6 (non-linear resistance and end correction) with eqs. 23 and 24; Kabral et al. (2014), eq. 2 (slits).

Valid range On / off; only for the micro-perforated impedance models

Scope

Applies only to the micro-perforated impedance models. The sound field is computed one frequency at a time, so harmonics generated by the non-linearity are not included, as in the cited papers.

Theory

Equations

Default and rule

Valid range On / off; only for the micro-perforated impedance models

Related 4

References 2