Theory

In the acoustic model a perforated wall is a cut through the mesh whose two sides are coupled by a transfer impedance $Z_p$ : the normal particle velocity $v_n$ is continuous through the wall, and the pressure jumps by $Z_p v_n$ , equation (1). It enters the finite-element system as $\frac{i\omega}{Z_p}$ times the jump matrix of the cut face (a resistance dissipates sound, a mass reactance adds inertia). The time convention is $e^{+i\omega t}$ , so a mass reactance is positive imaginary and a resistance non-negative.

Every model computes the normalised impedance of ONE hole, $\zeta_h$ ; the wall impedance is the hole impedance divided by the open-area ratio $\sigma$ , equation (2) (Lee and Ih, 2003). The models:

Elnady 2010 (Elnady et al., 2010): Crandall's oscillating viscous flow in a short tube with an effective viscosity $2.179\,\mu$ (viscous plus thermal losses), the discharge coefficient $C_D$ (effective open area $\sigma C_D$ ), a resistive end correction $\delta_{re}$ , a reactive end correction $0.5\,d$ reduced by the hole-interaction factor $f_{int}$ , the radiation resistance, and terms for grazing flow $0.3\,M_g$ and through-hole (bias) flow $1.15\,M_b/C_D$ (the paper quotes the last two from Bauer 1977), equations (3)-(5). Recommended: no flow, grazing flow and through-hole flow.
Kirby & Cummings 1998 (Kirby and Cummings, 1998): flat plates under fully developed turbulent grazing flow, with the friction velocity $u_*$ ; when a packing lies on ONE side of the wall the end correction of that side is made complex (their porous-backing correction), equation (6). For dissipative straight-through silencers.
Cross-flow: the whole flow forced THROUGH the holes (perpendicular baffles and cross-flow mufflers): resistance $2.57\,M_h$ from the mean through-flow Mach number in the holes, reactance $0.95\,k(t + 0.75\,d)$ , equation (7); without flow the resistance $6\times10^{-3}$ and the reactance $k(t + 0.75\,d)$ (Sullivan, 1979) (Munjal, 2014). No grazing term.
Lee & Ih 2003 (Lee and Ih, 2003): a regression of measured circular orifices in grazing flow, equation (8), with the reactance clamped to zero below $1/|b_4| = 151$ Hz, where the printed fit changes sign.
Physics baseline (thin plate): the exact viscous impedance of the hole as a short tube plus Rayleigh's end correction, equation (9); options Thin plate: use C_D, Hole interaction (thin plate), End-correction factor (thin plate). No flow dependence: an independent no-flow cross-check.
Sullivan & Crocker 1978 (legacy) (Sullivan and Crocker, 1978): the measured single-hole resistance of one perforate (Single-hole resistance (Sullivan & Crocker)) and the reactance $k(t + 0.75\,d)$ . Offered only to a wall that already uses it.

With Include bias flow on, the bias resistance $1.15\,M_b/C_D$ is added to the hole impedance of every model except Elnady and cross-flow, which carry their own through-flow term. On a micro-perforate this term is borrowed from ordinary perforates and is not validated (a warning is shown).

Equations

p_1 - p_2 = Z_p\,v_n, \qquad Z_p = \rho c\,\zeta_p

(1)

\zeta_p = \frac{\zeta_h}{\sigma}

(2)

where	meaning	unit
$p_1, p_2$	Acoustic pressure on the two sides of the wall	Pa
$v_n$	Normal particle velocity through the wall	m/s
$\rho, c$	Density and speed of sound of the local gas	kg/m³, m/s
$\zeta_h$	Normalised impedance of one hole	-
$\sigma$	Open-area ratio used by the acoustics	-

\theta = \mathrm{Re}\left\{\frac{ik}{C_D}\left[\frac{t}{F(K'd)} + \frac{\delta_{re} f_{int}}{F(Kd)}\right]\right\} + 1 - \frac{2J_1(kd)}{kd} + 0.3\,M_g + \frac{1.15\,M_b}{C_D}

(3)

\chi = \mathrm{Im}\left\{\frac{ik}{C_D}\left[\frac{t}{F(K'd)} + \frac{0.5\,d\,f_{int}}{F(Kd)}\right]\right\}, \qquad \zeta_h = \theta + i\chi

(4)

F(x) = 1 - \frac{2J_1(x/2)}{(x/2)J_0(x/2)}, \quad K = \sqrt{-i\omega/\nu}, \quad K' = \sqrt{-i\omega\rho/(2.179\,\mu)}, \quad \delta_{re} = 0.2d + 200d^2 + 16000d^3,\quad f_{int} = 1 - 1.47\sqrt{\sigma} + 0.47\sqrt{\sigma}^{3}

(5)

where	meaning	unit
$k$	Wavenumber $\omega/c$	1/m
$d, t$	Hole diameter and wall thickness (d in m in $\delta_{re}$ )	m
$C_D$	Discharge coefficient	-
$M_g$	Grazing Mach number	-
$M_b$	Bias (through-hole) Mach number	-
$\mu, \nu$	Dynamic and kinematic viscosity of the gas	Pa s, m²/s
$J_0, J_1$	Bessel functions of the first kind	-

\zeta_h = \theta_f + \frac{t}{c\,d}\sqrt{8\nu\omega} + ik\left(t + \frac{\delta}{\delta_0}\,0.85\,d\right) - i\,0.425\,kd\left(1 - \frac{\rho_{eq}}{\rho}\right), \qquad \theta_f = \max\left(0,\ \frac{\left[26.16\,(t/d)^{-0.169} - 20\right]u_* - 4.055\,f d}{c}\right)

(6)

\zeta_h = 2.57\,M_h + 0.95\,ik\,(t + 0.75\,d)

(7)

\zeta_h = R_n + i\max(I_n, 0), \qquad R_n = a_0(1 + a_1|f - f_0|)(1 + a_2 M)(1 + a_3 d)(1 + a_4 t), \qquad I_n = b_0(1 + b_1 d)(1 + b_2 t)(1 + b_3 M)(1 + b_4 f)

(8)

\zeta_h = \frac{1}{\rho c}\left[\frac{i\omega\rho t}{F(2k_s a)} + i\omega\rho\,\epsilon\,f_{int}\,d\right], \qquad k_s^2 = -i\omega\rho/\mu,\ a = d/2

(9)

where	meaning	unit
$\theta_f$	Flow-induced resistance of Kirby & Cummings	-
$u_*$	Friction velocity of the grazing flow	m/s
$\delta/\delta_0$	Reduction of the end correction by the flow (Kirby & Cummings eq. 13)	-
$\rho_{eq}$	Equivalent density of a packing on one side (only then the last term)	kg/m³
$M_h$	Mean through-flow Mach number in the holes	-
$a_i, b_i, f_0$	Lee & Ih regression constants; $f_0 = 412(1 + 104M)/(1 + 274d)$	-
$\epsilon$	End-correction factor (End-correction factor (thin plate), 0.85)	-

Default and rule

Default: Elnady 2010 (new walls); old case files keep the global model they stored
Rule: New perforated walls use Elnady 2010; a micro-perforated wall uses the circular-hole micro-perforate model (Maa 1998). A case file written before the model became a per-wall setting has no entry for the wall and uses the global perforate model it stored.
Source: The equations and constants are those of the original papers (Elnady et al., 2010) (Kirby and Cummings, 1998) (Lee and Ih, 2003) (Sullivan and Crocker, 1978). The program lists Elnady 2010 first as the recommended model; Sullivan & Crocker is no longer offered for new walls. Cross-flow: Sullivan's resistance $\theta_0 = 2.57\,U_0/c$ with $U_0$ the mean hole speed (his eqs. 16-18) and Munjal's reactance $0.95\,k(t + 0.75\,d_h)$ . Munjal writes the resistance as $0.514\,d_1 M/(l\sigma)$ , equation (3.128) of the 2nd edition = (3.153) of the 1st, with $M$ the mid-stream Mach number of the pipe ( $0.514 = 2.57 \times 0.8/4$ ; 0.8 = mean over mid-stream speed). Before 2026-11-07 the program used $4 \times 0.514\,M_h$ with the MEAN hole Mach number (resistance 20 % low) and no factor 0.95. Validation of the jump formulation: a flat perforated sheet in a duct agrees with the exact $TL = 20\lg|1 + \zeta/2|$ within 0.016 dB.

Valid range Perforated (mm holes): Elnady 2010, Kirby & Cummings 1998, cross-flow, Lee & Ih 2003, physics baseline (thin plate), Sullivan & Crocker 1978 only on a wall that already uses it. Micro-perforated (sub-mm holes): circular holes (Maa 1998), slits (Guo, Allam & Abom 2008)

Micro-perforated plates

Sub-millimetre holes or slits with an open-area ratio of about 0.2-5 %: circular holes after Maa (1998) with the edge factors and grazing-flow terms of Allam and Abom (2011) (Hole edges of a micro-perforate); slits after Guo et al. (2008) as given by Kabral et al. (2014) (Slit width, Grazing factor of a slit micro-perforate). The flow solution of such a wall uses the viscous resistance of the same papers (Micro-perforate flow resistance).

Micro-perforates: flow regime

Both micro-perforate models are for no flow or GRAZING flow along the wall, the regime they were derived and measured in: the flow term is $K M/\sigma$ (circular holes, $K = 0.15$ , with the end correction times $F$ ) or $\beta M/\sigma$ (slits). Allam and Abom (2011) (sec. 3.1.3) state that, the steady pressure-drop coefficient of a micro-perforate being high, only grazing flow is of interest. They are not models of flow THROUGH the plate (a micro-perforated baffle across a duct, bias flow).

Micro-perforates: validation

Circular holes against measured transmission loss: Allam and Abom (2011) two-chamber muffler without flow (rms 1.1 dB), three chambers with grazing flow up to Mach 0.1 (the measured trend with flow is reproduced; the level is 5-7 dB high at the maximum; the cause is not settled, a small leak between the chambers reproduces the measurement), Lee et al. (2016) single chamber with a film lining (rms 1.8 dB below 3 kHz), Maa (1998) absorption. Slits: Kabral et al. (2014) without flow (rms 1.8 dB) and with grazing flow at Mach 0.05 (rms 0.6 dB below 2.4 kHz); above the Cremer peak the model is about 5 dB high.

Use when

Elnady: no flow, grazing or through-hole flow; $C_D$ measured for d 0.6-2.4 mm and t 0.8-2.4 mm, not for packing touching the holes. Kirby & Cummings: grazing flow with packing on the perforate, fitted for σ 0.20-0.27, t/d 0.29-0.54. Lee & Ih: d 2-9 mm, t 1-5 mm, σ 3-22 %, M ≤ 0.2, 60-4000 Hz. Cross-flow: flow through the holes only. Baseline: no flow.

Validity gate

Frequencies and parameters outside a model's published range are listed by the acoustic QA gate (a warning; the run continues). A negative resistance fails the gate: it would inject energy.

Not modelled

Mean-flow convection in the duct is not part of these impedances; the amplitude (non-linear) dependence of the resistance is not modelled; Kirby & Cummings' porous-backing correction is applied only when a packing touches ONE side of the wall.

Acoustic impedance model of a perforated wall

Theory

Equations

Default and rule

Valid range Perforated (mm holes): Elnady 2010, Kirby & Cummings 1998, cross-flow, Lee & Ih 2003, physics baseline (thin plate), Sullivan & Crocker 1978 only on a wall that already uses it. Micro-perforated (sub-mm holes): circular holes (Maa 1998), slits (Guo, Allam & Abom 2008)

References 11

Theory

Equations

Default and rule

Valid range Perforated (mm holes): Elnady 2010, Kirby & Cummings 1998, cross-flow, Lee & Ih 2003, physics baseline (thin plate), Sullivan & Crocker 1978 only on a wall that already uses it. Micro-perforated (sub-mm holes): circular holes (Maa 1998), slits (Guo, Allam & Abom 2008)

Related 17

References 11