Port results: noise reduction, reflection, absorption, impedance, transfer functions, four-pole

The frequency-response results of the Results step that are computed from the stored scattering matrix of the sweep, without a new solve: noise reduction (NR), reflection coefficient, absorption, transmission and dissipation coefficients, input impedance, pressure transfer functions and, for a model with one inlet and one outlet, the four-pole (transfer matrix). They appear as soon as the acoustic run has finished.

Last updated 2026-11-11

Theory

For every frequency the run stores the waves leaving every port for a unit wave arriving at every port, for all port modes, with every port anechoic. The port modes are normalised to a unit mean square over the port face, so the plane mode is $\psi = 1$ and the area-mean pressure of a port is exactly its plane-mode amplitude $a + b$ : the higher modes have zero mean over the face. Every result below is therefore an area-mean, plane-mode quantity of the port planes; above the first non-plane cut-on of a port (table "Port modes and cut-on") its higher modes still carry power but are not in the pressure and velocity used here.

The time convention is $e^{+i\omega t}$ : a wave travelling in $+x$ is $e^{-ikx}$ , and a phase that falls with frequency is a delay.

Two port states are offered (Port results: ports anechoic or outlets radiating). Anechoic: every port absorbs what reaches it, the state of the transmission loss; the results describe the model alone. Outlets radiating: every outlet ends in the radiation impedance the insertion loss uses (unflanged, flanged or the radiation domain, radiating into the ambient air: Insertion loss: radiation into the ambient air), the inlets stay anechoic; the reflected waves are folded in exactly from the stored matrix. Noise reduction is normally quoted in this state, because with anechoic ports a resonance of the model is not excited by the outlet's reflection.

The reflection coefficient, impedance and absorption of a port are for that port excited alone. For the noise reduction and the transfer function the excitation is one inlet alone, or, with several excited inlets, all of them together with the amplitudes and phases entered on the Boundary Conditions step.

The absorption coefficient is the part of the incident power that does not come back out of the same port, counting all its modes. With anechoic ports it splits into the power transmitted to the outlets, $\tau$ ( $10^{-TL/10}$ for one excited inlet), and the power dissipated in the perforates and packings, $\delta$ ; for rigid walls without losses $\delta = 0$ to the solver's accuracy.

The four-pole relates pressure and volume velocity $q = S u$ (positive from inlet to outlet) at the two port planes. It is computed from two independent excitations of the stored matrix and inverts exactly to it. With volume velocity, $AD - BC = 1$ for a reciprocal model in one medium; the transfer matrix of a uniform tube is the textbook one, Munjal (2014) eq. 2.159, written there with mass velocity $\rho S u$ , which divides $B$ by $\rho$ and multiplies $C$ by $\rho$ .

With mean-flow convection in the acoustics, the relation $u = \pm p/(\rho c)$ of a plane wave still holds in a uniform flow, so pressure, velocity and impedance are the acoustic ones; the power the mean flow convects with the wave is not in NR or the impedance.

Equations

p = a + b, \qquad u = \frac{a - b}{\rho c}

(1)

where	meaning	unit
$a, b$	Plane-mode amplitude of the wave arriving at / leaving the model at a port	Pa
$p$	Area-mean pressure at the port plane	Pa
$u$	Area-mean particle velocity into the model	m/s
$\rho, c$	Density and sound speed of the port's gas	kg/m³, m/s

R = \frac{b}{a}, \qquad z_{in} = \frac{Z}{\rho c} = \frac{1 + R}{1 - R}, \qquad \alpha = 1 - \frac{W_{refl}}{W_{inc}}

(2)

where	meaning	unit
$R$	Reflection coefficient, port excited alone	-
$z_{in}$	Normalised specific input impedance of the model at the port	-
$\alpha$	Absorption coefficient seen from the port	-
$W_{refl}, W_{inc}$	Power leaving and arriving through the port, all its modes	W

\tau = \frac{\sum_{outlets} W_{out}}{W_{inc}}, \qquad \delta = 1 - \frac{\sum_{ports} W_{out}}{W_{inc}}

(3)

where	meaning	unit
$\tau$	Power transmission coefficient to the outlets	-
$\delta$	Dissipation coefficient (perforates, packings)	-

NR_{ij} = 20 \log_{10} \left| \frac{p_i}{p_j} \right|, \qquad H_{ji} = \frac{p_j}{p_i}

(4)

where	meaning	unit
$NR_{ij}$	Noise reduction from inlet $i$ to outlet $j$	dB
$H_{ji}$	Pressure transfer function	-

\begin{pmatrix} p_1 \\ q_1 \end{pmatrix} = \begin{pmatrix} A & B \\ C & D \end{pmatrix} \begin{pmatrix} p_2 \\ q_2 \end{pmatrix}, \qquad q = S\,u

(5)

where	meaning	unit
$p_1, q_1$	Pressure and volume velocity at the inlet plane	Pa, m³/s
$p_2, q_2$	The same at the outlet plane, positive leaving the model	Pa, m³/s
$A, B, C, D$	Four-pole parameters: A, D dimensionless, B in Pa s/m³, C in m³/(Pa s)

B_{out} = (I - S\,G)^{-1} S\,A_s, \qquad A_{in} = A_s + G\,B_{out}

(6)

where	meaning	unit
$S$	Stored scattering matrix, all ports and modes	-
$G$	Reflection matrix of the outlet terminations (outlets radiating)	-
$A_s$	Waves sent in at the excited inlets	Pa

Source

The relations are the plane-wave decomposition of the program's own finite-element scattering matrices. The four-pole of a straight duct computed this way reproduces the uniform-tube transfer matrix of Munjal (2014) eq. 2.159 exactly.

Valid range

Plane mode

NR, reflection, impedance, transfer function and four-pole use the plane mode of the ports. Above the first non-plane cut-on of a port they describe only the area-mean part of the field there.

Four-pole

Only for a model with exactly one inlet and one outlet.

References 1

Munjal, M. L. (2014). Acoustics of ducts and mufflers. 2nd ed., Wiley, Chichester, ISBN 978-1-118-44312-5. Web page

Theory

Equations

Source

Valid range

Related 4

References 1