Parameter · 05 Boundary Conditions

Anechoic termination

acoustic_port.anechoic_mode Step 05 · Boundary Conditions

How an acoustic port absorbs the waves that leave the model in the transmission-loss solve: every computed mode of the port face (multimodal, the default), only the plane wave (plane-wave anechoic), or every mode, evanescent ones included, in an absorbing stub added beyond the face (PML).

Last updated 2026-11-11

Theory

A port is the end of a straight duct of the same cross-section. Its pressure field is a sum of duct modes $\psi_m$ of the port face, computed by the program for each port. A mode propagates when the frequency is above its cut-on frequency, and decays (evanescent) below it. The multimodal termination prescribes, for every computed mode (all propagating ones and the first evanescent ones), the relation between pressure and normal velocity of a wave travelling away from the model, so no mode reflects. It is exact for a straight uniform port at any frequency, and the transmission loss is the ratio of the incident and transmitted power summed over all propagating modes.

The plane-wave termination keeps only the plane mode in that relation; the face is rigid for the higher modes, which therefore reflect. This is what a one-dimensional model or a plane-wave two-microphone measurement assumes. Below the port's first cut-on the two terminations give the same result, because only the plane wave propagates. Above it the higher modes are trapped between the model and the port and the result depends on where the port is cut, so a warning is shown.

The PML termination extrudes the port face outwards into a straight stub one equal-area diameter long: a short buffer layer of ordinary gas, then a perfectly matched layer (PML) in several sub-layers, closed by a rigid end. In the PML the axial coordinate is stretched by a complex factor, so a wave travelling outwards decays without reflection at every frequency, and evanescent modes decay faster; the stub's sides and end are rigid. Checked by the program on an expansion chamber with straight pipes (equal to the multimodal port within 0.002 dB) and on an outlet that ends 0.1 diameter after a right-angled corner (within 0.06 dB of a long straight extension up to just below the outlet pipe's cut-on, except at one sharp transmission zero; the multimodal port with one evanescent mode was up to 38 dB off there). The port's modes are kept, but only to excite the model and to measure the outgoing amplitudes on the face (which now lies inside the mesh); they no longer form the boundary condition. Unlike the multimodal port, which keeps all propagating modes and the first evanescent ones, the PML also absorbs the evanescent modes it would leave out. That matters when the port face sits close to a bend, a step or a perforate. A wave entering through a PML port is introduced on the buffer elements that touch the face (total field up to the face, scattered field beyond it). The PML port needs the port to be one flat face and cannot be combined with the mean-flow convection; it adds the stub's elements to the mesh and makes the system complex.

The first cut-on shown on the card and in the table is the estimate for a circular port of the CAD area, $f_c = \alpha'_{11} c / (2\pi a)$ with $a$ the equal-area radius; the run computes each port's own modes and reports their cut-on frequencies in the acoustic log.

Equations

f_c = \frac{\alpha'_{11}\,c}{2\pi a}, \qquad a = \sqrt{S/\pi}, \qquad \alpha'_{11} = 1.8412

(1)

where	meaning	unit
$f_c$	First non-plane cut-on frequency of the port	Hz
$c$	Speed of sound at the operating state	m/s
$a$	Equal-area radius of the port	m
$S$	Port area (CAD)	m²
$\alpha'_{11}$	First zero of the derivative of the Bessel function $J_1$	-

\frac{\partial p}{\partial n} = \sum_{m} \left( 2\,i k_{z,m} A_m - i k_{z,m}\,a_m \right) \psi_m, \qquad m \in \{0\}\ \text{(plane wave)} \ \text{or all computed modes (multimodal)}

(2)

where	meaning	unit
$k_{z,m}$	Axial wavenumber of mode $m$	1/m
$A_m$	Incident amplitude of mode $m$ (zero at a passive port)	Pa
$a_m$	Modal amplitude of the pressure on the port face	Pa
$\psi_m$	Mode shape on the port face	-

s = \kappa - j\,\frac{\sigma}{k}, \qquad \int_0^{L_p} \sigma\,\mathrm{d}z = 6, \qquad 1 \le \kappa \le 3

(3)

where	meaning	unit
$s$	Complex stretching factor of the axial coordinate in the PML	-
$\sigma$	Absorption of a PML sub-layer, constant in each, growing quadratically over the sub-layers	1/m
$\kappa$	Real stretching of a sub-layer, growing quadratically from 1 to 3: an evanescent mode decays up to three times faster, so a mode just below its cut-on does not reach the stub's rigid end	-
$k$	Wavenumber, $\omega / c$	1/m
$L_p$	PML thickness (the stub length minus the buffer layer)	m

\int_\Omega \frac{1}{\rho}\left[ s\,(\nabla p\cdot\nabla q - \partial_z p\,\partial_z q) + \frac{1}{s}\,\partial_z p\,\partial_z q \right] \mathrm{d}V - \omega^2 \int_\Omega \frac{s\,p\,q}{\rho c^2}\,\mathrm{d}V

(4)

where	meaning	unit
$p, q$	Pressure and test function	Pa
$z$	Coordinate along the outward normal of the port face	m
$\rho, c$	Density and speed of sound of the gas at the port	kg/m³, m/s

Default and rule

Default: Multimodal (port modes)
Rule: Multimodal (port modes).
Source: The modal port condition is the program's finite-element implementation; on every circular port the computed transverse wavenumbers are checked against the zeros of the Bessel function derivatives.

Valid range Multimodal (port modes) | Plane-wave anechoic | PML (absorbing stub)

Below the cut-on

Both terminations give the same result.

PML

A wave leaving the model decays by a factor of about 400 in amplitude through the layer (one way), at every frequency. The stub is one equal-area diameter long; the port must be one flat face.

Above the cut-on

A plane-wave termination reflects the higher modes; the transmission loss then depends on the position of the port. The program warns and still runs.

Theory

Equations

Default and rule

Valid range Multimodal (port modes) | Plane-wave anechoic | PML (absorbing stub)

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