How does SpectraSim calculate muffler transmission loss?

It solves the inhomogeneous Helmholtz equation with second-order finite elements at each frequency of the sweep. Every inlet and outlet is a port that selects the propagating duct modes automatically, and the transmission loss is ten times the logarithm of incident power over transmitted modal power.

What is the difference between transmission loss and insertion loss?

Transmission loss compares incident and transmitted power with anechoic ports: a property of the muffler alone. Insertion loss compares the power radiated with and without the muffler, so it includes the outlet termination. SpectraSim computes insertion loss by the power method against a reference pipe it solves itself.

Does exhaust temperature change transmission loss?

Yes. The speed of sound scales with the square root of absolute temperature, so a muffler carrying gas at 600 °C has its TL domes about 1.73 times higher in frequency than at 20 °C. Mapping the CFD temperature field also captures the cooling of the gas along the muffler.

Does SpectraSim include mean flow in the acoustics?

Perforate impedances use grazing and through-hole Mach numbers from CFD, typed, or estimated as M = Q/(A·c) from the entered flow rate. Convection of sound by the mean flow is an option. Insertion loss and radiated SPL with mean-flow convection are Planned and switched off until a termination model with flow is validated against measurement.

Can I solve acoustics without running CFD?

Yes. Acoustics runs alone with a uniform medium from the Material step and needs no flow solution. Flow and acoustic perforate settings are separate and are linked only when both physics are solved, so an acoustics-only model stays solvable.

Which frequency outputs are available?

Narrow-band transmission loss per inlet–outlet pair, per inlet and for all inlets together, 1/1 and 1/3 octave band TL, return loss and port powers, insertion loss, and radiated SPL with the A-weighted overall level.

Acoustics · finite elements

Finite-element acoustics, built for silencers.

Transmission loss, insertion loss and radiated sound in the frequency domain — in a medium that knows how hot your exhaust is.

SpectraSim computes muffler transmission loss, insertion loss and radiated sound by solving the inhomogeneous Helmholtz equation with the finite-element method in the frequency domain. The medium is either uniform or mapped from the CFD temperature and density field, so hot-exhaust resonances appear at the right frequency.

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Method: Frequency-domain FEM, second-order tetrahedra
Outputs: TL, IL, radiated SPL (A-weighted), 1/1 and 1/3 octave bands
Medium: Uniform, or T and ρ mapped from CFD
Internals: Perforates, absorbing packing, internal walls
Accuracy: 0.12 dB TL rms against a commercial finite-element code

Physics model

How is transmission loss calculated?

SpectraSim solves the Helmholtz equation for the complex acoustic pressure amplitude at every frequency of the sweep, with density and speed of sound allowed to vary in space. The geometry is meshed with second-order tetrahedra by Gmsh, sized from the shortest wavelength at the highest frequency (ten elements per wavelength on auto); a mesh-resolution check judges the mesh as built.

\nabla\cdot\left(\frac{1}{\rho_0}\,\nabla p\right)+\frac{\omega^2}{\rho_0\,c_0^2}\,p=0

(1)

c_0(\mathbf{x})=\sqrt{\gamma R_s\,T(\mathbf{x})},\qquad \rho_0(\mathbf{x})=\frac{p(\mathbf{x})}{R_s\,T(\mathbf{x})}

(2)

Inhomogeneous Helmholtz equation for the acoustic pressure p at angular frequency ω, with the local speed of sound c₀ and density ρ₀ taken from the gas temperature T(x). In (2), p is the absolute static pressure of the gas and R_s its specific gas constant.

From port powers to transmission loss

Each inlet is excited by an incident plane wave of known amplitude or power; all other ports absorb every wave that reaches them. The program solves for all ports and modes as excitations with one factorisation per frequency, then forms the transmission loss from the incident power and the power leaving each outlet in all its propagating modes. With several inlets the per-inlet TL sums the outlets exactly, and the inlets driven together give a combined TL at their entered amplitudes and phases.

W_{\mathrm{in}}=\frac{|A|^2\,S}{2\rho c},\qquad W_m=\frac{|A_m|^2\,S}{2\rho c}\,\frac{\mathrm{Re}(k_{z,m})}{k}

(1)

TL_{ij}=10\log_{10}\frac{W_{\mathrm{in},i}}{W_{\mathrm{out},j}},\qquad TL_i=-10\log_{10}\sum_j 10^{-TL_{ij}/10}

(2)

Incident and modal power at a port of area S (k_z,m axial wavenumber of mode m), pair transmission loss from inlet i to outlet j, and per-inlet TL summed over the outlets.

Medium

How does exhaust temperature change the TL curve?

Because the speed of sound scales with √T, every TL dome of a muffler carrying gas at 600 °C sits about 1.73 times higher in frequency than at 20 °C: $\sqrt{873.15\ \mathrm{K}/293.15\ \mathrm{K}}\approx 1.73$ . The acoustic medium decides how that temperature reaches the model.

Medium option A

Uniform

One density and one speed of sound from the Material step: the gas, its temperature and the ambient pressure. No CFD needed. Perforate Mach numbers are typed, or estimated as M = Q/(A·c) from the entered flow rate when fluid flow is part of the case.

Medium option B

From CFD fields

The solved temperature and absolute pressure of a steady compressible flow run are mapped onto every node of the acoustic mesh; each port uses the mean ρ and c of its own face. Grazing and through-hole Mach numbers at the perforates come from the flow solution.

Hot gas

At 600 °C the TL domes move up by $\sqrt{T/T_{\mathrm{ref}}}$ ≈ ×1.73. With the medium from CFD fields the gas temperature is evaluated where it actually is: wall heat loss cools the gas along the muffler, and the resonances follow. The program's specification estimates that shift at about 3–8 % for a 100–150 K temperature drop at a 600 °C inlet. Wall heat loss and the temperature field

Ports and modes

The two-source method, in software.

Every inlet and outlet is a modal port. Before the sweep the program solves a two-dimensional eigenproblem on each port face, so circular, annular and irregular cross-sections are handled alike, and includes every mode that propagates up to the highest frequency plus one evanescent mode by default. The port condition is non-reflecting for every included mode.

Port powers come out per mode: incident, reflected and transmitted. Because all port–mode excitations are solved together, the full scattering matrix is available — the acoustic equivalent of a laboratory's two-source method, without a second measurement.

f_{c,m}=\frac{c\,k_{t,m}}{2\pi},\qquad k_{z,m}=\sqrt{k^2-k_{t,m}^2}

(0,0)
(1,0)
(2,0)
(0,1)

Illustrative pressure patterns of the first duct modes in a circular cross-section (red and blue: opposite phase). Real mode plots from the program: [MISSING M13 — mode plots from a real run]

Results

Which acoustic results does a run deliver?

Transmission loss

Per inlet–outlet pair, per inlet, and with all excited inlets driven together at their amplitudes and phases.

Band TL

1/1 and 1/3 octave bars; each band is solved at its lower edge, centre and upper edge.

Insertion loss

Power method against a reference pipe that the program builds and solves, with the outlet termination applied to both.

Radiated SPL

At a microphone position from an entered source spectrum, unweighted and A-weighted, with overall levels.

Port powers and return loss

Incident, reflected and transmitted modal power at every port; the power-normalised scattering matrix on request.

Impedance against frequency

Perforate transfer impedance and packing properties, plotted in the report.

Surface acoustic pressure

At any frequency, re-solved on demand, with phase animation: see where each resonance lives.

TL chart — SpectraSim against a reference [MISSING M13 — real plot]

Transmission loss in dB against frequency; solid line SpectraSim, dashed line reference.

1/3-octave TL bars [MISSING M13 — real plot]

Transmission loss per 1/3-octave band, in dB.

Surface |p| colour map on a muffler [MISSING M13 — real plot]

Acoustic pressure magnitude on the muffler shell at one frequency.

Insertion loss and radiated sound

How are insertion loss and radiated SPL computed?

The frequency sweep is solved once with anechoic ports. Its scattering matrix does not depend on what is attached to the ports, so the outlet termination is applied afterwards as a reflection on the outgoing modes. The insertion loss is the net power through the outlet of a reference pipe, solved by the program at the same frequencies with the same inlet excitation, over the net power through all muffler outlets.

Outlets radiate into the outside air, not into their own gas: hot exhaust radiating into cold air has a radiation resistance $(T_p/T_o)^{3/2}$ times the same-gas value at low frequency (Hirschberg and Hoeijmakers, 2014).

The radiated sound pressure level at a microphone follows from an entered source spectrum — a sound power level, or a sound pressure level measured with the reference pipe — minus the insertion loss, with hemispherical spreading over a reflecting ground and air absorption. Results are A-weighted (IEC 61672-1, 2013) and summed to overall levels.

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

(1)

L_p=L_W-IL+10\lg Q-\left(20\lg\frac{d}{1\ \mathrm{m}}+11\right)-\alpha d

(2)

Insertion loss by the power method, and the radiated level at distance d for a sound-power source (Q = 2, α the atmospheric attenuation coefficient).

Outlet terminations for insertion loss
Termination	What it models	Status
Unflanged pipe	Exact plane-mode reflection of a thin-walled pipe end (Levine and Schwinger, 1948). Also used in place of an anechoic outlet, which would make IL equal TL.	Available
Flanged pipe	Pipe end flush with a rigid baffle: radiation impedance of a baffled piston, plane mode.	Available
Radiation domain	A meshed half-space closed by a perfectly matched layer gives the full modal reflection matrix above about 0.8 times the outlet's first non-plane cut-on.	Available

Above an outlet's first non-plane cut-on, the plane-mode terminations are incomplete and those frequencies are flagged in the insertion loss and the radiated SPL, unless the outlet uses the radiation domain.

Perforates and absorbing materials

How are perforated pipes and absorbing packing modelled?

Flow and acoustic perforate settings are separate and linked only when both physics are solved — acoustics stays solvable without flow.

Perforated plates, baffles and pipes

A perforated wall is a cut through the acoustic mesh: the normal particle velocity is continuous through it and the pressure jumps by the transfer impedance. Each perforated wall has its own impedance model, hole diameter, thickness and open-area ratio; the wall impedance is the impedance of one hole divided by the open-area ratio. Grazing-flow and through-hole (bias-flow) terms are included where the model provides them, with Mach numbers from CFD or entered.

p_1-p_2=Z_p\,v_n,\qquad Z_p=\rho c\,\frac{\zeta_h}{\sigma}

Perforate impedance models
Model	Use it for	Source
Elnady (default)	No flow, grazing flow and through-hole flow; discharge coefficient and hole interaction included.	Elnady et al., 2010
Kirby and Cummings	Grazing flow with packing behind the perforate: dissipative straight-through silencers.	Kirby and Cummings, 1998
Lee and Ih	Circular orifices in grazing flow, from measured data.	Lee and Ih, 2003
Cross-flow	Perpendicular baffles and cross-flow mufflers, all flow forced through the holes.	Classical cross-flow form
Thin-plate baseline	Viscous impedance of a short tube with an end correction; an independent no-flow cross-check.	Physics baseline

Absorbing packing

A rigid-frame fibrous packing is an equivalent fluid with a complex, frequency-dependent density and bulk modulus, coupled to the gas directly or through a perforated wall. The empirical models need only the flow resistivity — typed, or estimated from the bulk density of a glass or rock wool with the measured data of Komatsu (2008). The resistivity is entered at room temperature and scaled with the viscosity of the local gas (Sutherland, 1893), in the acoustics and in the flow solution alike.

Frequencies outside an empirical model's published validity range are listed as a warning by the acoustic quality check, never silently extrapolated.

Z_c=\rho_0 c_0\left[1+C_5X^{-C_6}-i\,C_7X^{-C_8}\right],\qquad k_c=\frac{\omega}{c_0}\left[1+C_1X^{-C_2}-i\,C_3X^{-C_4}\right]

(1)

X=\frac{\rho_0 f}{R},\qquad R(T)=R_0\,\frac{\mu(T)}{\mu(293.15\ \mathrm{K})}

(2)

Empirical packing models: characteristic impedance Z_c and propagation constant k_c as power laws in X, with eight published constants per model; the resistivity follows the gas viscosity μ(T).

Absorbing-material models
Model	Inputs	Status
Miki (default)	Flow resistivity (Miki, 1990)	Available
Delany–Bazley	Flow resistivity (Delany and Bazley, 1970)	Available
Komatsu	Flow resistivity (Komatsu, 2008)	Available
JCA	Porosity, tortuosity, viscous and thermal lengths, resistivity (Johnson et al., 1987; Champoux and Allard, 1991)	Available
JCAL	JCA inputs plus static thermal permeability (Lafarge et al., 1997)	Available
User defined	Eight constants of your own fit	Available

Accuracy

How accurate is the acoustic solver?

Acceptance criterion for acoustics against measurement: within ±3 dB over the frequency range of the measurement; flow results are compared case by case as stated.

Full acoustic validation

Acoustics — SpectraSim against a commercial finite-element code and measured transmission loss
Case	Reference	Result
Muffler with an extended-tube chamber, 20 °C, 20–1000 Hz	Code-to-code the same model in a commercial finite-element code	transmission loss rms 0.12 dB, largest 1.3 dB; the 815 Hz notch at the same frequency
Large exhaust muffler with three absorbing-material regions, 20–500 Hz	Code-to-code commercial finite-element code, its finest mesh	rms 0.16 dB, median 0.04 dB, largest 0.49 dB
Concentric-tube resonators (Sullivan and Crocker, 1978)	Measured measured transmission loss	main peak 7–10 % lower in frequency than measured, peak levels within about 2 dB; humps within 1 dB

Code-to-code rows compare two simulations of the same model; only the measured row compares with the real muffler. Measured data: Sullivan and Crocker (1978).

Limits stated plainly

What is not included yet?

Planned

Insertion loss and radiated SPL with mean-flow convection are switched off until a termination model with flow is built and checked against measurement. At a pipe end with flow, part of the sound turns into vortex shedding instead of radiating (Peters et al., 1993), which the available no-flow terminations cannot represent. Transmission loss is computed with convection as usual.

Verified 2026-11-11

Mean-flow convection in the acoustic solve: run on a real muffler with the mapped CFD medium (power factor after Morfey, 1971).
Insertion loss with a radiation domain at the outlet: run on a real muffler.
JCA and JCAL absorbing-material models: run in a full acoustic computation; every quality check passes.

Planned · not in this version

Flow-generated noise is not part of this version.

Not modelled

The amplitude (non-linear) dependence of perforate resistance; limp or elastic packing frames; shell vibration and break-out noise; refraction of sound by a jet shear layer (the convection option is exact for uniform flow).

Quick answers

Duct acoustics, answered

How does SpectraSim calculate muffler transmission loss?: It solves the inhomogeneous Helmholtz equation with second-order finite elements at each frequency of the sweep. Every inlet and outlet is a port that selects the propagating duct modes automatically, and the transmission loss is ten times the logarithm of incident power over transmitted modal power.
What is the difference between transmission loss and insertion loss?: Transmission loss compares incident and transmitted power with anechoic ports: a property of the muffler alone. Insertion loss compares the power radiated with and without the muffler, so it includes the outlet termination. SpectraSim computes insertion loss by the power method against a reference pipe it solves itself.
Does exhaust temperature change transmission loss?: Yes. The speed of sound scales with the square root of absolute temperature, so a muffler carrying gas at 600 °C has its TL domes about 1.73 times higher in frequency than at 20 °C. Mapping the CFD temperature field also captures the cooling of the gas along the muffler.
Does SpectraSim include mean flow in the acoustics?: Perforate impedances use grazing and through-hole Mach numbers from CFD, typed, or estimated as M = Q/(A·c) from the entered flow rate. Convection of sound by the mean flow is an option. Insertion loss and radiated SPL with mean-flow convection are Planned and switched off until a termination model with flow is validated against measurement.
Can I solve acoustics without running CFD?: Yes. Acoustics runs alone with a uniform medium from the Material step and needs no flow solution. Flow and acoustic perforate settings are separate and are linked only when both physics are solved, so an acoustics-only model stays solvable.
Which frequency outputs are available?: Narrow-band transmission loss per inlet–outlet pair, per inlet and for all inlets together, 1/1 and 1/3 octave band TL, return loss and port powers, insertion loss, and radiated SPL with the A-weighted overall level.

References

Champoux, Y. and Allard, J.-F. (1991). Dynamic tortuosity and bulk modulus in air-saturated porous media.Journal of Applied Physics, 70(4), 1975-1979. doi:10.1063/1.349482
Delany, M. E. and Bazley, E. N. (1970). Acoustical properties of fibrous absorbent materials.Applied Acoustics, 3(2), 105-116. doi:10.1016/0003-682X(70)90031-9
Elnady, T., Åbom, M. and Allam, S. (2010). Modeling perforates in mufflers using two-ports.Journal of Vibration and Acoustics, 132(6), 061010. doi:10.1115/1.4001510
Geuzaine, C. and Remacle, J.-F. (2025). Gmsh reference manual, version 4.15.gmsh.info. gmsh.info/doc/texinfo/gmsh.html
Hirschberg, A. and Hoeijmakers, M. (2014). Comments on the low frequency radiation impedance of a duct exhausting a hot gas.The Journal of the Acoustical Society of America, 136(2), EL84-EL89. doi:10.1121/1.4885540
International Electrotechnical Commission (2013). Electroacoustics - Sound level meters - Part 1: Specifications.IEC 61672-1:2013. webstore.iec.ch/en/publication/5708
Johnson, D. L., Koplik, J. and Dashen, R. (1987). Theory of dynamic permeability and tortuosity in fluid-saturated porous media.Journal of Fluid Mechanics, 176, 379-402. doi:10.1017/S0022112087000727
Kirby, R. and Cummings, A. (1998). The impedance of perforated plates subjected to grazing gas flow and backed by porous media.Journal of Sound and Vibration, 217(4), 619-636. doi:10.1006/jsvi.1998.1811
Komatsu, T. (2008). Improvement of the Delany-Bazley and Miki models for fibrous sound-absorbing materials.Acoustical Science and Technology, 29(2), 121-129. doi:10.1250/ast.29.121
Lafarge, D., Lemarinier, P., Allard, J. F. and Tarnow, V. (1997). Dynamic compressibility of air in porous structures at audible frequencies.The Journal of the Acoustical Society of America, 102(4), 1995-2006. doi:10.1121/1.419690
Lee, S.-H. and Ih, J.-G. (2003). Empirical model of the acoustic impedance of a circular orifice in grazing mean flow.The Journal of the Acoustical Society of America, 114(1), 98-113. doi:10.1121/1.1581280
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
Miki, Y. (1990). Acoustical properties of porous materials: modifications of Delany-Bazley models.Journal of the Acoustical Society of Japan (E), 11(1), 19-24. doi:10.1250/ast.11.19
Morfey, C. L. (1971). Acoustic energy in non-uniform flows.Journal of Sound and Vibration, 14(2), 159-170. doi:10.1016/0022-460X(71)90381-6
Peters, M. C. A. M., Hirschberg, A., Reijnen, A. J. and Wijnands, A. P. J. (1993). Damping and reflection coefficient measurements for an open pipe at low Mach and low Helmholtz numbers.Journal of Fluid Mechanics, 256, 499-534. doi:10.1017/S0022112093002861
Sullivan, J. W. and Crocker, M. J. (1978). Analysis of concentric-tube resonators having unpartitioned cavities.The Journal of the Acoustical Society of America, 64(1), 207-215. doi:10.1121/1.381963
Sutherland, W. (1893). The viscosity of gases and molecular force.The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 36(223), 507-531. doi:10.1080/14786449308620508

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