Source: https://jmrplens.github.io/phonometry/guides/noise-control/ # Industrial noise control: silencers, HVAC and enclosures Three passive measures dominate applied noise control, and the `noise_control` domain covers all three with the engineering theory of Bies, Hansen & Howard, *Engineering Noise Control* (5th ed., CRC Press 2017): **reactive silencers** in a duct (the four-pole transmission-matrix method), the passive attenuations and regenerated noise of an **HVAC** run, and the insertion loss of a **machine enclosure**. The radiating piston of the [electroacoustics](https://jmrplens.github.io/phonometry/guides/electroacoustics/) domain is the companion radiator model. ## 1. Reactive silencers (four-pole method) A reactive silencer attenuates by *reflecting* sound with impedance discontinuities. Each acoustic element is a 2×2 **transfer (four-pole) matrix** relating the sound pressure `p` and the volume velocity `Su` at its two ends (Bies Eq. (8.133); Munjal, *Acoustics of Ducts and Mufflers*), and a compound silencer is the ordered matrix product of its elements. A straight duct of length `L` and area `S` is (Bies Eq. (8.143), no flow) ```text [ cos(kL) j (rho c / S) sin(kL) ] [ j (S / rho c) sin(kL) cos(kL) ] , k = omega / c, ``` and a side branch of acoustic impedance `Z_b` is the shunt `[[1, 0], [1/Z_b, 1]]` (Eq. (8.144)). The **transmission loss** follows from the compound matrix `T` with the port impedances `Z1 = rho c / S_in` and `Zn = rho c / S_out` (Munjal Eq. (3.27); Bies Eq. (8.141) prints the `T11`/`T22` impedance weights of this formula inverted and fails the sudden-expansion limit, see the [errata registry](https://jmrplens.github.io/phonometry/reference/errata/)) ```text TL = 10 log10[ (Zn/Z1) (1/4) | T11 + T12/Zn + Z1 T21 + (Z1/Zn) T22 |^2 ] , ``` which for equal inlet/outlet areas reduces to (Bies Eq. (8.148)) ```text TL = 20 log10( (1/2) | T11 + T12/Zc + Zc T21 + T22 | ) , Zc = rho c / S, ``` and the **insertion loss** for a source impedance `Z_s` and a radiation impedance `Z_r` is the extra attenuation over a direct (zero-length) connection, so a through connection gives `IL = 0`. ### Expansion chamber A chamber of area `S_exp` and length `L` between pipes of area `S_duct` has the closed-form transmission loss (Bies Eq. (8.111)) with area ratio `m = S_exp / S_duct`: ```text TL = 10 log10[ 1 + (1/4) (m - 1/m)^2 sin^2(kL) ] , ``` peaking at `10 log10[1 + (1/4)(m - 1/m)^2]` at `kL = pi/2, 3 pi/2, ...` (1.94 dB for `m = 2`, 6.55 dB for `m = 4`, 12.18 dB for `m = 8`, 18.10 dB for `m = 16`) and dropping to 0 at `kL = n pi`, where the chamber is a half-wavelength long and transparent. The four-pole product reproduces this exactly. ```python import numpy as np from phonometry import expansion_chamber freqs = np.linspace(20.0, 2000.0, 2000) res = expansion_chamber(freqs, length=0.3, chamber_area=0.04, pipe_area=0.01) print(round(res.transmission_loss.max(), 2)) # 6.55 dB peak (m = 4) res.plot() # TL (and IL) vs frequency ``` ### Side-branch and extended-tube resonators A **Helmholtz resonator** (`helmholtz_resonator`) and a closed **quarter-wave tube** (`quarter_wave_resonator`) each short the duct at their tuning frequency, `f_0 = (c / 2 pi) sqrt(S_neck / (l_e V))` (Bies Eq. (8.46)) and `f = c / 4 l_e` (Eq. (8.44)), giving a sharp transmission-loss spike there. An **extended-tube chamber** (`extended_tube_chamber`) buries quarter-wave side branches in an expansion chamber to fill its troughs; with zero extensions it reduces exactly to the plain chamber. Advanced layouts chain elements directly with `duct_matrix`, `shunt_matrix`, `cascade`, `transmission_loss` and `insertion_loss`. ```python import numpy as np from phonometry import ( helmholtz_resonator, quarter_wave_resonator, extended_tube_chamber, ) f = np.linspace(20.0, 600.0, 4000) hr = helmholtz_resonator(f, duct_area=0.01, neck_area=1e-4, neck_length=0.02, cavity_volume=1e-3) print(round(float(hr.resonances[0]), 1)) # tuning frequency, Hz hr.plot() # TL spike at the tuning frequency (needs matplotlib) qw = quarter_wave_resonator(f, duct_area=0.01, length=1.516, branch_area=2e-3, speed_of_sound=343.24) print(round(float(qw.resonances[0]), 1)) # 56.6 Hz (Bies Example 8.1) # An inlet extension of L/4 fills the first expansion-chamber trough. et = extended_tube_chamber(f, length=0.4, chamber_area=0.04, pipe_area=0.01, inlet_extension=0.1) ``` *Each side branch shorts the duct at its own tuning frequency and is nearly transparent elsewhere: the narrow spike is why resonators are matched to a firing frequency or a fan blade-passing tone rather than used broadband.*

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import helmholtz_resonator, quarter_wave_resonator f = np.linspace(20.0, 600.0, 4000) hr = helmholtz_resonator(f, duct_area=0.01, neck_area=1e-4, neck_length=0.02, cavity_volume=1e-3) qw = quarter_wave_resonator(f, duct_area=0.01, length=0.3, branch_area=2e-3) # One line for one device: TL vs frequency with the resonance marked. hr.plot() plt.show() # By hand: both side branches on the same axes. fig, ax = plt.subplots() ax.plot(f, hr.transmission_loss, label="Helmholtz resonator") ax.plot(f, qw.transmission_loss, "--", label="Quarter-wave tube") for fr in (hr.resonances[0], qw.resonances[0]): ax.axvline(float(fr), ls=":", color="#2ca02c") ax.set_xlabel("Frequency [Hz]"); ax.set_ylabel("Transmission loss [dB]") ax.set_ylim(0.0, 50.0) ax.legend() plt.show() ```

Each device returns a `ReactiveSilencerResult` with `transmission_loss`, `insertion_loss` (when source/radiation impedances are given), the compound `transfer_matrix`, the tuning `resonances` and `.plot()`. ## 2. HVAC duct attenuation and flow noise `noise_control.hvac` gathers the Bies Chapter 8 duct methods: - `end_reflection_loss` — the low-frequency reflection back up an open duct end (ASHRAE Table 8.14, interpolated over diameter and frequency; it passes exactly through the tabulated nodes). - `elbow_insertion_loss` — the insertion loss per bend for square/round, vaned/unvaned and lined/unlined elbows keyed by `W / lambda` (ASHRAE Table 8.11). - `plenum_attenuation` — the plenum-chamber transmission loss by Wells' method (Eq. (8.275)), whose reverberant term uses the plenum [room constant](https://jmrplens.github.io/phonometry/guides/room-image-sources/). - `flow_noise_straight_duct`, `flow_noise_bend` — the flow-generated (self) noise sound power of straight ducts and mitred bends (VDI 2081, Eqs. (8.251), (8.254)). ```python from phonometry.noise_control import hvac bands = [63.0, 125.0, 250.0, 500.0, 1000.0, 2000.0] er = hvac.end_reflection_loss(bands, diameter=0.30, termination="flush") el = hvac.elbow_insertion_loss(bands, width=0.3, bend_type="square", lined=True) er.plot() # the band attenuation (or regenerated Lw) in one line (needs matplotlib) tl = hvac.plenum_attenuation(0.1, 1.0, 20.0, 0.2) # Wells' method, dB fn = hvac.flow_noise_straight_duct(bands, flow_velocity=10.0, area=0.04) ``` *The open end of a duct reflects low-frequency energy back up the run — for free, before any silencer: the smaller the duct against the wavelength, the larger the loss, which is why small diffuser necks tame low-frequency fan rumble and why the correction must not be double-counted when a manufacturer's diffuser data already includes it.*

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```python import matplotlib.pyplot as plt import numpy as np from phonometry.noise_control import hvac bands = [63.0, 125.0, 250.0, 500.0, 1000.0, 2000.0] # One line for one duct: the HvacSpectrumResult of the 300 mm flush end. er = hvac.end_reflection_loss(bands, diameter=0.30, termination="flush") er.plot() plt.show() # By hand: the family over duct diameters of the concept figure. fig, ax = plt.subplots() for diameter in (0.15, 0.30, 0.60): er = hvac.end_reflection_loss(bands, diameter=diameter, termination="flush") ax.semilogx(er.frequencies, er.values, "o-", label=f"D = {int(diameter * 1000)} mm") ax.set_xlabel("Frequency [Hz]"); ax.set_ylabel("End reflection loss [dB]") ax.legend(title="Duct diameter") plt.show() ```

Rectangular ducts use the equivalent diameter `D = sqrt(4 S / pi)`. Bies 5th ed. gives the duct end reflection only as the ASHRAE table (no closed form in that edition); this module reproduces and interpolates it. ## 3. Machine enclosures A sealed enclosure reduces the radiated noise by its panel transmission loss `R`, minus a penalty `C` for the reverberant build-up inside the small, hard cavity (Bies Eqs. (7.103), (7.111)): ```text IL = R - C , C = 10 log10( 0.3 + S_E / R_i ) , ``` with the external area `S_E` and the interior room constant `R_i = S_i alpha_i / (1 - alpha_i)` (the same `room_constant` as the steady-state room field). A hard interior wastes much of the panel `R`; lining it drives `C` toward its floor `10 log10 0.3 = -5.2 dB`. **The panel transmission loss `R` is supplied by the caller** — measured, or predicted by a panel model — as a per-band array or a callable of frequency. This module never predicts `R` itself; it combines a given `R` with the interior absorption. ```python import numpy as np from phonometry import enclosure_insertion_loss bands = np.array([125.0, 250.0, 500.0, 1000.0, 2000.0, 4000.0]) panel_R = np.array([18.0, 24.0, 30.0, 36.0, 42.0, 46.0]) # measured, dB enc = enclosure_insertion_loss(panel_R, external_area=6.0, internal_area=5.0, internal_absorption=0.3, frequencies=bands) print(np.round(enc.insertion_loss, 1)) # net IL = R - C per band enc.plot() ``` *What the enclosure delivers is `R − C`, not the panel `R`: even this lined interior (mean absorption 0.3) costs about 5 dB of the panel's rating in every band, and a hard, unlined interior would cost far more. Budget the lining together with the panels, not as an afterthought.*

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import enclosure_insertion_loss bands = np.array([125.0, 250.0, 500.0, 1000.0, 2000.0, 4000.0]) panel_R = np.array([18.0, 24.0, 30.0, 36.0, 42.0, 46.0]) # measured, dB enc = enclosure_insertion_loss(panel_R, external_area=6.0, internal_area=5.0, internal_absorption=0.3, frequencies=bands) # One line — panel R, interior correction C and the net IL = R - C: enc.plot() plt.show() # By hand, from the per-band fields the result carries: fig, ax = plt.subplots() ax.plot(bands, enc.panel_transmission_loss, "s--", label="Panel R") ax.plot(bands, enc.correction, "^:", label="Interior correction C") ax.plot(bands, enc.insertion_loss, "o-", label="Insertion loss (R - C)") ax.set_xlabel("Frequency [Hz]"); ax.set_ylabel("Level [dB]") ax.set_xscale("log") ax.legend() plt.show() ```

`enclosure_insertion_loss` returns an `EnclosureResult` with the panel `panel_transmission_loss`, the interior `correction`, the net `insertion_loss`, the interior `room_constant` and `.plot()`. ## Cross-check against the FDTD solver The four-pole expansion chamber is cross-checked against the independent 2D [FDTD wave solver](https://jmrplens.github.io/phonometry/guides/fdtd-simulation/): a plane-wave duct that widens into a chamber and narrows back transmits far less at the four-pole TL peak (`kL = pi/2`) than at the transparent trough (`kL = pi`), and the measured amplitude ratio reproduces the closed-form peak transmission loss to a fraction of a decibel (test `tests/noise_control/test_fdtd_crosscheck.py`). ## References - Bies, D. A., Hansen, C. H., & Howard, C. Q. (2017). *Engineering noise control* (5th ed.). CRC Press. [doi:10.1201/9781351228152](https://doi.org/10.1201/9781351228152). The muffler four-pole method and expansion-chamber TL (§8.8–8.9), the HVAC duct methods (§8.11–8.17) and the machine-enclosure noise reduction (§7.4). - Munjal, M. L. (2014). *Acoustics of ducts and mufflers* (2nd ed.). Wiley. [doi:10.1002/9781118443767](https://doi.org/10.1002/9781118443767). The transfer-matrix formulation behind the element matrices and the transmission loss from the compound matrix (Eq. (3.27)). - Vér, I. L., & Beranek, L. L. (2006). *Noise and vibration control engineering* (2nd ed.). Wiley. [doi:10.1002/9780470172568](https://doi.org/10.1002/9780470172568). The companion treatment of mufflers, ducts and enclosures.