Outlet: compute a radiation domain

Whether the run computes a radiation domain at this outlet for the insertion loss. The radiation impedance itself (unflanged or flanged pipe end, or this radiation domain) is chosen after the solve, in Results > Insertion loss (Insertion loss: radiation impedance). The transmission loss always uses an anechoic outlet.

Last updated 2026-11-11

Theory

The frequency sweep is solved once with anechoic ports. Its scattering matrix $S$ does not depend on what is attached to the ports, so a reflecting outlet is applied afterwards as a reflection coefficient $\Gamma$ on the outlet's modes: the outgoing waves become equation (1), and the net power through the outlet is the outgoing minus the reflected power, equation (2). The insertion loss is the ratio of the net outlet powers of the reference pipe and of the muffler.

The terminations:

Anechoic: nothing reflects. Because the inlet is also anechoic, an anechoic outlet makes the insertion loss exactly equal to the transmission loss. The insertion loss therefore NEVER uses it: an outlet stored as anechoic (or without a port) is computed as unflanged.
Unflanged: the end of a thin-walled pipe radiating into free space. The exact plane-mode reflection of Levine & Schwinger (Levine and Schwinger, 1948), equation (3), evaluated with the continuous branch of the arctangent; the end correction $l/a$ is taken from the same paper.
Flanged: the pipe end flush with an infinite rigid baffle, using the radiation impedance of a baffled piston (Rayleigh), equation (4).
Radiation domain (computed): above about 0.8 times the outlet's first non-plane cut-on frequency, equation (5), a small finite-element model of the mouth radiating into a half-space behind a rigid baffle, closed by a perfectly matched layer, gives the FULL modal reflection matrix (all propagating modes and their coupling). Below that frequency the baffled piston is used. The option is offered only when the highest frequency of the case reaches the switch frequency.

The analytic terminations describe the plane mode only. Above the outlet pipe's first non-plane cut-on ( $ka > 1.841$ ) the insertion loss is flagged as unreliable, unless the outlet uses the radiation domain.

Equations

\mathbf{B}_{out} = (\mathbf{I} - \mathbf{S}\boldsymbol{\Gamma})^{-1}\mathbf{S}\,\mathbf{A}_{source}

(1)

W_{net} = \sum_m w_m\left(|B_m|^2 - |(\boldsymbol{\Gamma}\mathbf{B})_m|^2\right), \qquad IL = 10\lg\frac{W_{net,pipe}}{\sum_{outlets} W_{net,muffler}}

(2)

|R| = \exp\left\{-\frac{2ka}{\pi}\int_0^{ka}\frac{\arctan\left(-J_1(x)/N_1(x)\right)}{x\sqrt{(ka)^2 - x^2}}\,dx\right\}, \qquad R = -|R|\,e^{-2ikl}

(3)

z = 1 - \frac{J_1(2ka)}{ka} + i\,\frac{H_1(2ka)}{ka}, \qquad R = \frac{z - 1}{z + 1}

(4)

f_{switch} = 0.8\,f_{c,1}

(5)

where	meaning	unit
$\mathbf{S}$	Scattering matrix of the anechoic sweep	-
$\boldsymbol{\Gamma}$	Reflection matrix of the terminations (diagonal on the plane modes, or the full block of a radiation domain)	-
$\mathbf{A}_{source}$	Incident amplitudes of the excited inlets	Pa
$w_m$	Modal power weight	m⁴ s/kg
$R$	Plane-mode pressure reflection coefficient	-
$a$	Equivalent radius of the outlet, $\sqrt{S/\pi}$	m
$l$	End correction	m
$J_1, N_1, H_1$	Bessel, Neumann and Struve functions of order 1	-
$z$	Normalised radiation impedance of a baffled piston	-
$f_{c,1}$	First non-plane cut-on frequency of the outlet	Hz

Default and rule

Default

off (the radiation impedance is chosen in Results)

Rule

Anechoic is stored on every outlet (the transmission-loss state). For the insertion loss an anechoic outlet is replaced by unflanged, because an anechoic outlet makes the insertion loss equal to the transmission loss and adds no information: on case V1 (a 2.0 m multi-chamber muffler with a perforated pipe and three absorptive packings), IL − TL = −0.0007 dB at every frequency with an anechoic outlet.

Source

Unflanged: (Levine and Schwinger, 1948), pp. 383-385. The end-correction integral as printed evaluates to 0.6127, where the paper states 0.6133 (−0.1 %); the difference is negligible.

Radiation domain, validation on a 100 mm pipe:

$|\Gamma_{00}|$ within 0.01 of the baffled piston up to ka 1.8; phase within 0.6 / 2 / 3.6° at ka 0.3 / 0.6 / 1.0.
A straight pipe against its own reference pipe gives |IL| ≤ 0.0006 dB from 1000 to 3100 Hz, across the switch frequency.

Valid range on | off (on is offered when the frequency range reaches 0.8 × the outlet's first non-plane cut-on)

Needs the insertion loss

A non-anechoic termination without "Compute the insertion loss" ticked is refused when the acoustic computation starts.

Anechoic is not used for IL

Choosing anechoic does not give an anechoic insertion loss; the insertion loss is computed with an unflanged outlet.

Plane mode

Unflanged and flanged are exact for the plane mode only. Keep the outlet pipe straight and perpendicular to the port for at least one diameter, since the pipe end is the reference plane of the radiation model.

Radiation domain

Checked on a straight pipe and on a real muffler (2026-11-11: equal to the flanged outlet below the switch, up to about 3 dB apart above the outlet cut-on). It models a FLANGED mouth; the outlet face must be flat. Below the switch frequency the baffled piston is used.

References 1

Levine, H. and Schwinger, J. (1948). On the radiation of sound from an unflanged circular pipe. Physical Review, 73(4), 383-406. DOI 10.1103/PhysRev.73.383

Theory

Equations

Default and rule

Valid range on | off (on is offered when the frequency range reaches 0.8 × the outlet's first non-plane cut-on)

Related 7

References 1