Compute the insertion loss

acoustics.insertion_loss.enabled Step 07 · Run Settings

Computes the insertion loss: how much less sound power leaves the outlets of the muffler than leaves the outlet of a plain reference pipe driven by the same source. The program builds and solves the reference pipe automatically at the same frequencies.

Last updated 2026-11-11

Theory

The switch is named "Compute two-port (enables IL)". The muffler's scattering matrix comes from the same factorisation as the transmission loss; the reference pipe is a second, smaller solve. The source impedance, the radiation impedance and the outside air are chosen afterwards in Results (Insertion loss: source impedance, Insertion loss: radiation impedance), without re-solving, and both scattering matrices can be exported as a plane-mode two-port. The panel estimates the solve times from the DOF counts and the measured factorisation times of this project's last acoustic run.

The transmission loss uses anechoic ports and describes the muffler alone. The insertion loss also depends on what the muffler radiates into, because the outlet termination reflects part of the wave back.

1. The program solves the muffler and a reference pipe (Reference pipe length) with the same inlet ports (excitation, amplitude or power, phase, all excited inlets driven together). 2. It applies the outlet termination to both afterwards: the scattering matrix of the sweep does not depend on what is attached to the ports, so a reflecting termination is a reflection coefficient $\Gamma$ on the outgoing modes, equation (1). 3. The net power through an outlet is what leaves minus what the termination sends back, equation (2). 4. The insertion loss is the ratio of the reference pipe's net outlet power to the SUM over all muffler outlets, equation (3).

The termination is the muffler outlet's own setting (Boundary Conditions step); the reference pipe's outlet copies that of the first outlet.

An outlet set to anechoic is treated as an unflanged pipe end for the insertion loss, with the reflection coefficient of (Levine and Schwinger, 1948): with an anechoic outlet and anechoic inlets nothing would reflect, and the insertion loss would equal the transmission loss.
"Flanged" uses the baffled-piston impedance.
"Radiation domain" computes the reflection with a meshed half-space and is the only termination valid above the outlet's first non-plane cut-on.

The outlets radiate into the outside air (Insertion loss: radiation into the ambient air), not into their own gas. A hot exhaust radiating into cold air has a higher radiation resistance, $(T_p/T_o)^{3/2}$ times the same-gas value at low frequency (Hirschberg and Hoeijmakers, 2014) (3.9 for 450 °C into 20 °C), so the outlet reflects less sound back into the muffler.

Equations

\mathbf{B}_{out} = \left(I - S\Gamma\right)^{-1} S\,\mathbf{A}_{src}

(1)

W_{net} = \sum_m w_m\left(\lvert B_m\rvert^2 - \lvert(\Gamma B)_m\rvert^2\right)

(2)

IL = 10\log_{10}\frac{W_{pipe}}{\sum_{outlets} W_{net}}

(3)

where	meaning	unit
$S$	Scattering matrix of the model (anechoic ports)	-
$\Gamma$	Reflection matrix of the terminations (plane-mode coefficient, or the full matrix of a radiation domain)	-
$\mathbf{A}_{src}$	Incident amplitudes of the inlet ports	Pa
$\mathbf{B}_{out}$	Outgoing amplitudes with the terminations attached	Pa
$w_m$	Power per squared amplitude of mode $m$	W/Pa²
$W_{net}$	Net power through an outlet	W
$W_{pipe}$	Net power through the reference pipe's outlet	W
$IL$	Insertion loss	dB

Default and rule

Default

offMeasured

Rule

Off. Switching on the radiated SPL switches it on.

Source

The reference pipe is solved by finite elements, the outlet powers are summed over all outlets, and an anechoic outlet is never used for the insertion loss. Validation results:

A single-inlet test chamber against its own reference pipe: IL = TL within 0.0014 dB with anechoic outlets.
A straight pipe has an insertion loss below 0.01 dB for any termination.
The finite-element power of the reference pipe matches its analytic value $\lvert A\rvert^2 S(1-\lvert R\rvert^2)/(2\rho c)$ within 3e-7 dB.

Unflanged termination: (Levine and Schwinger, 1948).

Valid range on | off

Cost

The reference pipe is a second complete acoustic solve (a 3 m pipe: about 50k unknowns at 900 Hz).

Above the cut-on

Above an outlet's first non-plane cut-on, the unflanged and flanged terminations act on the plane mode only; those frequencies are flagged unreliable unless the outlet uses the radiation domain.

Not with mean-flow convection

Switched off while Mean-flow convection is on, because none of the termination models in the program gives a trustworthy result with flow. (1) With a mean flow out of a pipe end, the net plane-wave power into the end tends to $\pi a^2 \lvert p_+\rvert^2 4M/(\rho c)$ at low $ka$ , and the edge converts almost all of it into vortex shedding instead of radiating it (Peters et al., 1993) (eqs. 32-33). The outlet terminations of the program (Levine & Schwinger, baffled piston, radiation domain) are no-flow models; combined with the convected port powers $(1 \pm M)^2$ they count that absorbed power as radiated (case V4, a 0.3 m expansion chamber with an extended inlet tube, Mach 0.028: insertion loss up to 37 dB in error in a validation run). (2) Published results with flow cover low frequencies only: the reflection magnitude to second order in $ka$ (Cargill, 1982) and the end correction measured for $ka < 0.06$ (Peters et al., 1993), about $ka \le 0.5$ in all (about 1.1 kHz for a 50 mm tailpipe and 550 Hz for a 100 mm one, cold). The radiated power with flow has no published formula, the mean velocity to use (cross-section mean or centreline, about 20 % apart) is disputed (Hirschberg and Hoeijmakers, 2014), hot jets are not covered, and no measured insertion loss with flow is available to validate a model. (3) The reference pipe would need the same outlet flow. The transmission loss is still computed with convection.

Theory

Equations

Default and rule

Valid range on | off

Related 6

References 4