How is sound in a duct treated?

At low frequency sound travels along the duct as a plane wave uniform across the section, so 1D transfer matrices suffice. As frequency rises and higher-order modes cut on, the cross-sectional distribution must be considered. Cut-on is c/(2a) for a rectangular duct with longer side a, and 1.84c/(πD) for a round duct.

How much do lined ducts attenuate?

Sabine's approximation ΔL = 1.05α^1.4 P/S (dB/m) is a guide. For a 600×300 mm duct lined on all four sides, it gave 1.9 dB/m at absorption coefficient 0.3, 6.4 at 0.7 and 9.1 at 0.9. But absorption falls at low frequency, and at high frequency sound beams down the duct centre, so check frequency by frequency.

What to watch with side-branch silencers?

A closed quarter-wave branch cuts sound strongly at its target frequency but over a narrow band. In the example, a branch tuned to 125 Hz gave over 10 dB only from 121.7 to 128.3 Hz. Without the branch-entrance end correction (about 45 mm) the resonance shifted to 117 Hz, and attenuation at 125 Hz fell to 3.9 dB.

Duct Acoustic Analysis

Category: Coupled Acoustics | Revised 2026-10-01

Theory: sound travelling in ducts

Overview

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I want to reduce fan noise coming out of an HVAC duct. Where should the analysis start?

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First check how sound propagates in the duct at the frequencies of interest. At low frequency it travels as a plane wave uniform across the section, easily analysed with 1D transfer matrices. At higher frequency, higher-order modes with cross-sectional pressure variation start to propagate and 3D analysis is needed; the boundary is the cut-on frequency. Then choose between absorptive lining that trims a broad band and resonators that cut specific frequencies hard.

Cut-on frequency and lined ducts

$$ f_{c,rect} = \frac{c}{2a}, \qquad f_{c,circ} = \frac{1.841\,c}{\pi D}, \qquad \Delta L_{lined} \approx 1.05\,\alpha^{1.4}\,\frac{P}{S}\ \ \text{dB/m} $$

$c$ is the speed of sound, $a$ the longer rectangular side, $D$ the diameter. In Sabine's lined-duct formula, $\alpha$ is absorption coefficient, $P$ lined perimeter and $S$ cross-sectional area; narrower ducts have larger $P/S$ and attenuate more.

Side branch (quarter-wave tube)

$$ TL = 10\log_{10}\left[1 + \left(\frac{S_b}{2S}\tan kL_b\right)^2\right], \qquad L_b = \frac{c}{4f_0} - \delta_{end} $$

At the frequency where the closed branch is a quarter wavelength long, pressure at the branch entrance vanishes and sound doesn't pass downstream. $S_b$ is the branch area and $\delta_{end}$ the entrance end correction (air moving slightly beyond the opening).

Coffee Break Trivia

Organ pipes and silencers

Resonance in pipes has long been known through organ pipes and wind instruments. At a pipe's end the air vibrates slightly beyond the mouth, so the pipe sounds as if a little longer than it is. Lord Rayleigh derived this 'end correction' theoretically in the 19th century. In the 20th century, as noise reduction became important in car mufflers and HVAC ducts, the same principle was used in reverse: silencers that cancel specific tones with pipe resonance.

Worked examples

Example 1: plane-wave range

Duct	Cut-on frequency
600 × 300 mm	286 Hz
400 × 200 mm	429 Hz
1,000 × 500 mm	172 Hz
150 mm round	1,340 Hz
300 mm round	670 Hz
600 mm round	335 Hz

Example 2: lined duct attenuation (600×300 mm, four sides)

Absorption coefficient	0.3	0.5	0.7	0.9
Attenuation	1.9 dB/m	4.0 dB/m	6.4 dB/m	9.1 dB/m

Example 3: side branch tuned to 125 Hz

A closed 150 mm branch on a 300 mm round duct, with length set to c/(4f) = 0.686 m ignoring end correction. Transmission loss:

Frequency	100 Hz	115 Hz	120 Hz	125 Hz	130 Hz	135 Hz	150 Hz	250 Hz
TL	0.6 dB	3.0 dB	6.9 dB	Infinite (ideal)	6.9 dB	3.0 dB	0.6 dB	0.0 dB

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Infinite at 125 Hz, yet barely effective just off it.

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Over 10 dB only from 121.7 to 128.3 Hz — 6.6 Hz. In reality losses such as air viscosity limit the peak to around 20–30 dB. And the biggest danger is end correction: at the branch entrance, air moves slightly beyond the opening, so the branch resonates as if about 45 mm longer. Forget it and resonance shifts to 117 Hz, leaving just 3.9 dB at 125 Hz; the correct length is 0.641 m. If fan speed varies and the tone moves, combining with absorptive treatment is safer than relying on a narrow resonance.

Analysis workflow

Characterise source frequencies and levels (such as fan blade-passing frequency).
Compare with cut-on to decide between 1D transfer matrices and 3D analysis.
Assemble overall transmission including bends, branches, area changes and outlet radiation.
Add absorptive or resonant treatment and verify including end corrections and flow effects.
With flow, check self-generated flow noise of the silencer itself.

Coffee Break Trivia

“Only the low hum won't go”

A meeting room received complaints of a low hum from the HVAC outlet. The ducts had ample absorptive lining, but the tone was about 80 Hz, where lining barely works — and the duct outlet strongly reflects low frequencies, setting up standing waves inside. Adding a side branch tuned to the fan's blade-passing frequency, with its length corrected for end effects, cut the hum substantially. Resonators for low frequencies, absorption for high: that's the basic split.

Common mistakes

Mistakes and fixes

Mistake	Effect	Fix
Forgetting end correction	Resonance misses the target	Size length with correction
1D analysis above cut-on	Higher modes missed	3D analysis
Using lining for low tones	Little effect	Add resonators
Ignoring outlet reflection	Standing waves missed	Outlet radiation impedance
Ignoring flow effects	Silencer generates noise	Evaluate flow noise

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I'd like to try related calculations.

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Try the duct silencer attenuation calculator, muffler transmission loss tool, Helmholtz resonator calculator and acoustic standing wave simulator. Related analyses include muffler design, vibro-acoustics and aeroacoustics.

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