Source: https://jmrplens.github.io/phonometry/devices/noise-control/duct-path/ # Duct-borne noise: fan to room Air-conditioning noise is not predicted, it is *accounted for*. You start from the sound power the fan puts into the duct, walk down the path, and at every element subtract what it attenuates and add back what its own airflow regenerates. What survives to the terminal device is turned into a sound pressure level by the room, the supply and the return paths are added together, and the total is laid against the design criterion. If it fails, the sheet itself tells you which element to change: the row with the small attenuation, or the row whose self-noise is now the floor. That bookkeeping is what `noise_control.duct_path` implements, with the element models of `noise_control.hvac` feeding it and `noise_control.duct_modes` marking the frequency above which the whole one-dimensional picture stops being exact. The reference throughout is Long, *Architectural Acoustics* (2nd ed., Academic Press 2014), Chapters 13 and 14, whose Table 14.9 is the worked sheet this guide is built around, with the ASHRAE *HVAC Applications Handbook* Chapter 49 for the air terminal devices and Bies, Hansen & Howard for the splitter silencers and the plenums. The reactive four-pole silencers of [Silencers](https://jmrplens.github.io/phonometry/devices/noise-control/silencers/) and the rest of the installation methods in [Industrial noise control](https://jmrplens.github.io/phonometry/devices/noise-control/noise-control/) are the companion pages. ## 1. The sheet, and how it adds up A duct-borne calculation is a table: octave bands across the columns (63 Hz to 8 kHz, the range the published procedures use), one block of rows per physical element. Each block prints what the element takes out, the running level after subtracting it, what the element puts back, and the level leaving it. `DuctElement` carries exactly the two spectra an element owns, and `duct_path` walks them: ```python from phonometry import DuctElement, duct_path bands = [63.0, 125.0, 250.0, 500.0, 1000.0, 2000.0, 4000.0, 8000.0] fan = [90.0, 86.0, 82.0, 79.0, 77.0, 75.0, 71.0, 61.0] path = duct_path( bands, fan, [ DuctElement("Elbow, 36 x 24 in, unlined", attenuation=[0, 1, 2, 3, 3, 3, 3, 3], self_noise=[41, 39, 36, 29, 20, 6, 0, 0], code="2"), DuctElement("Silencer, 3 ft, standard pressure drop", attenuation=[7, 12, 16, 28, 35, 35, 28, 17], self_noise=[49, 43, 44, 42, 42, 45, 35, 24], code="3"), ], source_label="Fan, centrifugal FC, 5000 cfm, 2 in w.g.", ) for row in path.table(): print(f"{row['code']:>2} {row['label'][:38]:<38} " f"{[round(float(v)) for v in row['values']]}") ``` Three conventions matter, and they are worth stating once because every published sheet states them differently. **Attenuations are positive.** Every element model in `noise_control.hvac` returns a loss as a positive number of decibels, and the cascade subtracts it. Printed worksheets show the same quantity as a negative level change, so `DuctPathResult.table()` flips the sign back for the `"attenuation"` rows: the table reads like the reference, the arithmetic does not have to. **Regenerated noise adds on a power basis.** The self-noise of an element is a sound power level in its own right, not a correction to the incoming level, so it is combined as $10\log_{10}(10^{L/10} + 10^{L_\text{sn}/10})$ rather than added arithmetically. That is why the `Self-noise` row sits between the `Sum` and the `Combined` row and never touches the attenuation. **There is a self-noise floor.** Long's sheet uses a 0 dB sound power level wherever an element has no regenerated-noise data, and also as a floor under any computed level that would go negative, which is why his received spectrum bottoms out near 0 dB instead of running off to minus infinity. `self_noise_floor` reproduces that (default `0.0`) and `None` switches it off entirely. ## 2. The source: fan sound power The fan is the one element whose spectrum you can build from the operating point alone. `fan_sound_power` implements the ASHRAE scaling law printed as Long Eq. 13.1, $$ L_W = K_\mathrm{F} + 10\log_{10}\frac{Q_\mathrm{F}}{Q_\text{REF}} + 10\log_{10}\frac{P_\mathrm{F}}{P_\text{REF}} + C_\text{EFF} + C_\text{BFI}, $$ with the spectral constant $K_\mathrm{F}$ of Table 13.5 (one row per fan type), the off-peak efficiency correction $C_\text{EFF}$ of Table 13.6 and the blade frequency increment $C_\text{BFI}$ of Table 13.7 dropped into the single octave band that contains the blade passing frequency. In SI the references are $Q_\text{REF} = 0.472$ L/s and $P_\text{REF} = 249$ Pa, so the two logarithmic terms take the same values as the foot-pound form in cfm and inches of water gauge. ```python from phonometry import ( blade_passing_frequency, fan_casing_attenuation, fan_efficiency_correction, fan_sound_power, ) CFM, IN_WG = 0.0004719474432, 249.0 fan = fan_sound_power(volume_flow=5000 * CFM, static_pressure=2 * IN_WG, fan_type="forward_curved", relative_efficiency=80.0) print([round(float(v)) for v in fan.values]) # [99, 99, 89, 84, 82, 77, 72, 67] print(fan_efficiency_correction(80.0)) # 6.0 dB off the peak print(blade_passing_frequency(1200.0, 24)) # 480.0 Hz, in the 500 Hz band print(fan_casing_attenuation().values) # what the housing holds back # [ 0. 0. 5. 10. 15. 20. 22. 25.] fan.plot() # the band spectrum, one line ``` Two habits keep this honest. The law assumes ideal inlet and outlet flow conditions, so a fan boxed into a plant room with a bad inlet is louder than it says; and ASHRAE's own current guidance is that a fan's sound power "is best obtained from manufacturers' test data" to AMCA Standard 300 or ASHRAE Standard 68. Treat Eq. 13.1 as the early-design fallback, not as the answer. The fan radiates the same power from its intake and from its discharge, which is why the supply and return paths of a real sheet start from the *same* row. `fan_efficiency_correction` is a step function, and a brutal one: a fan running at 90 per cent of its peak static efficiency adds nothing, one at 80 per cent adds 6 dB, one below 50 per cent adds 16 dB. Selecting a fan away from its best point is the cheapest way to lose a duct-noise budget before any silencer is priced. `fan_casing_attenuation` (Table 13.8) is the other side of the same source: the power the housing radiates into the plant room instead of into the duct, zero at 63 and 125 Hz because a vibrating casing radiates low frequency as freely as the unhoused fan. ## 3. What the run takes out Everything between the fan and the room removes something, and most of it is free. The models are Long Chapter 14 with the Reynolds (1990) regressions, and they all return an `HvacSpectrumResult` with `.values`, `.plot()` and `.report()`. **Straight ducts.** An unlined rectangular duct loses energy into the induced motion of its own walls, so the loss grows with the perimeter-to-area ratio: a wide shallow duct has floppier side walls than a square one. `unlined_rectangular_duct_attenuation` fits that below 250 Hz and holds a flat rate above it; an external fibreglass blanket (`wrapped=True`) doubles the low-frequency part. A circular duct is far stiffer in its breathing mode, so it hardly responds at all, and `unlined_circular_duct_attenuation` is a bare length rate of 0.03 to 0.07 dB/ft. Lining the duct changes the order of magnitude: `lined_rectangular_duct_attenuation` and `lined_circular_duct_attenuation` evaluate the Reynolds regressions, valid for 25 mm to 52 mm linings and clipped at 40 dB per run because flanking takes over beyond that. ```python from phonometry import ( lined_rectangular_duct_attenuation, unlined_rectangular_duct_attenuation, ) import numpy as np IN, FT = 0.0254, 0.3048 bands = [63.0, 125.0, 250.0, 500.0, 1000.0, 2000.0, 4000.0, 8000.0] bare = unlined_rectangular_duct_attenuation(bands, 36 * IN, 24 * IN, 5 * FT) lined = lined_rectangular_duct_attenuation(bands, 36 * IN, 24 * IN, 5 * FT, 1 * IN, include_unlined=True) print(np.round(bare.values, 1)) # [1.1 0.7 0.5 0.2 0.2 0.2 0.2 0.2] print(np.round(lined.values, 1)) # [ 1.3 1.3 2.5 6.7 12.8 10.6 9.7 9. ] ``` The `include_unlined=True` switch is not cosmetic. The lined-duct regression was fitted to an *insertion loss*, measured by substituting the lined section for an unlined one of the same face size, so the side-wall contribution has been subtracted out of it; Long recommends adding it back for rectangular ducts, and ignoring it for circular ones where it is negligible. **Flexible duct.** The last run of a supply branch is usually flexible duct, and its published insertion loss is startling: 2 to 3 dB per foot in the mid bands. `flexible_duct_insertion_loss` interpolates ASHRAE Table 14.4 over length and log diameter. Part of that number is the duct's own breakout rather than dissipation, which is exactly why a serpentine run of flexible duct in a joist space works as an improvised silencer. **Elbows and splits.** `elbow_insertion_loss` is keyed by $W/\lambda$ and covers square and round bends, vaned and unvaned, lined and unlined; a lined square bend is worth 10 to 11 dB where a round one gives 3. `split_loss` handles a duct division: the power is shared between the branches in proportion to their areas, plus a reflection when the total branch area does not match the feeder, and a 25 per cent branch therefore costs 6 dB. ```python from phonometry import elbow_insertion_loss, split_loss IN = 0.0254 area = 36 * IN * 24 * IN print(round(split_loss(area, [0.25 * area, 0.75 * area], branch=0), 1)) # 6.0 print(elbow_insertion_loss( [63.0, 125.0, 250.0, 500.0, 1000.0, 2000.0, 4000.0, 8000.0], 24 * IN, bend_type="round").values) # [0. 1. 2. 3. 3. 3. 3. 3.] ``` **End reflection.** An open duct end reflects low frequency back up the run, for free, before any silencer. `end_reflection_loss` offers both published methods and neither replaces the other: `method="bies"` (the default) interpolates the ASHRAE table of Bies Table 8.14, and `method="long"` evaluates Reynolds' closed form $$ R = 10\log_{10}\!\left[1 + \left(\frac{a\,c}{\pi f d}\right)^{1.88}\right], $$ with $a = 0.8$ for a flush termination and $a = 1$ for a free one. The two agree within a decibel or so over the bands both cover. Use `equivalent_diameter(area)` for a rectangular duct, and do not apply the correction at all when the duct terminates in a diffuser: the flare smooths the impedance transition, and a manufacturer's diffuser rating already contains whatever is left of it. ```python from phonometry import end_reflection_loss, equivalent_diameter import numpy as np bands = [63.0, 125.0, 250.0, 500.0, 1000.0, 2000.0] print(np.round(end_reflection_loss(bands, 0.30, method="bies").values, 1)) # [12. 7. 3. 1. 0. 0.] print(np.round(end_reflection_loss(bands, 0.30, method="long").values, 1)) # [12.7 7.7 3.7 1.3 0.4 0.1] print(round(equivalent_diameter(0.36 * 0.24), 3)) # 0.332 m ``` **Silencers and plenums.** A parallel-splitter attenuator reduces, in Bies §8.10.5, to a set of lined ducts whose liner thickness is half the splitter thickness, combined by the energy average of Eq. 8.241 so that the leakiest airway dominates. `plenum_attenuation` is Wells' method for a lined plenum chamber (Bies Eq. (8.275)), whose reverberant term uses the plenum [room constant](https://jmrplens.github.io/phonometry/buildings/rooms/room-image-sources/). ```python from phonometry import plenum_attenuation, splitter_silencer_insertion_loss import numpy as np IN, FT = 0.0254, 0.3048 sil = splitter_silencer_insertion_loss( None, height=24 * IN, length=5 * FT, airway_widths=[0.10] * 5, splitter_thickness=0.10, ) print(np.round(sil.values, 1)) # [ 5.8 8.6 14.1 28.2 34.5 33.5 18.4 12.1] print(round(plenum_attenuation(0.36, 2.4, 74.0, 0.5), 1)) # 16.1 dB ``` Read that estimate for what it is. The 5 ft unit Long's return path actually specifies is a low-frequency design worth 16 and 21 dB at 63 and 125 Hz, where this geometry-only model gives 6 and 9. Published dynamic insertion loss from the manufacturer, measured with the design airflow and in the design direction, is what belongs in the sheet; the model is for sizing the airway before there is a manufacturer. ## 4. What the system puts back Attenuation is only half the sheet. Every disturbance of the airflow generates noise of its own, and past a certain velocity the silencer bought to remove the fan becomes the loudest thing in the duct. `silencer_self_noise` is Fry's estimate as Long Eq. 14.31, $$ L_W = 55\log_{10}\frac{V}{V_0} + 10\log_{10} N + 10\log_{10}\frac{H}{H_0} - 45, $$ spread over the octave bands by the corrections of Table 14.8. The exponent is the whole message: the fifth-and-a-half power of the airway velocity means that *doubling the face velocity of a silencer adds about 17 dB*. ```python from phonometry import silencer_self_noise import numpy as np IN = 0.0254 slow = silencer_self_noise(None, airway_velocity=10.0, passages=5, height=24 * IN) fast = silencer_self_noise(None, airway_velocity=20.0, passages=5, height=24 * IN) print(np.round(slow.values, 1)) # [40.8 40.8 38.8 36.8 31.8 26.8 21.8 16.8] print(round(float(fast.values[0] - slow.values[0]), 1)) # 16.6 dB ``` Straight duct and bends regenerate too, through `flow_noise_straight_duct` and `flow_noise_bend` (VDI 2081 as Bies Eqs. (8.251) and (8.254)); the bend model carries the Strouhal-number transition from the sixth-power inner-corner dipole to the eighth-power outer-corner quadrupole. The terminal device is the last one in the path and the one nothing downstream can fix, because there is no ductwork left after it. Its sound power is normally manufacturer data measured to ASHRAE Standard 70, and that is what a real sheet uses. When there is none to hand, `diffuser_sound_power` is Reynolds's estimate as Long Eqs. 13.27 to 13.33: an overall level $L_W = 10\log_{10} S_\mathrm{G} + 30\log_{10}\xi + 60\log_{10} U_\mathrm{G} - 31.3$ from the face area, the approach velocity $U_\mathrm{G} = Q/S_\mathrm{G}$ and the normalised pressure-drop coefficient $\xi$, spread over the octaves by the shape function $C_D = -11.82 - 0.15 A - 1.13 A^2$ ($-5.82$ for a round device) about the peak band $f_\mathrm{P} = 48.8\,U_\mathrm{G}$. ```python from phonometry import diffuser_sound_power import numpy as np IN, CFM, IN_WG = 0.0254, 0.0004719474432, 249.0 # The supply diffuser of Long's worked sheet: 24 x 24 in, 312 cfm, 0.05 in pd. print(np.round(diffuser_sound_power(None, (24 * IN) ** 2, volume_flow=312 * CFM, pressure_drop=0.05 * IN_WG).values, 1)) # [ 33.4 32.4 29.1 23.6 15.9 5.9 -6.4 -21. ] # Long Table 14.9 prints 33/32/29/23/15/4/0/0 for that row. ``` The sixth power of velocity in Eq. 13.27 is the design rule: about 18 dB per doubling of the approach velocity once the pressure drop follows it, and about 15 dB back for every doubling of face area at the same air volume. Two screening rules from ASHRAE Chapter 49 come with it. `air_terminal_velocity_limit` (Table 9) gives the maximum neck velocity for a design RC, and `air_terminal_damper_correction` (Table 10) gives the penalty for throttling a balancing damper, which is where a great many finished installations fail. ```python from phonometry import air_terminal_damper_correction, air_terminal_velocity_limit print(air_terminal_velocity_limit(30, opening="supply")) # 2.2 m/s print(air_terminal_velocity_limit(30, opening="return")) # 2.5 m/s print(air_terminal_damper_correction(3.0, location="diffuser_neck")) # 15.0 dB print(air_terminal_damper_correction(3.0, location="supply_duct")) # 2.0 dB ``` Fifteen decibels in the neck against two decibels 1.5 m back in the duct, for the same pressure ratio, is the entire design rule: throttle far from the outlet, or balance the system by sizing the ductwork instead. ## 5. From sound power to room level The last step converts the sound power arriving at the terminal device into a sound pressure level where somebody is sitting, through the steady-state room relation $L_p = L_W + 10\log_{10}\left[Q/(4\pi r^2) + 4/R\right]$. `room_effect` returns that as a positive attenuation so it drops into the cascade beside every other loss, with $Q = 2$ by default for a diffuser flush in a ceiling. ```python from phonometry import room_constant, room_effect # The 20 x 20 x 8 ft room of Long's worked sheet, drywall and carpet. area = 2 * 6.10 * 6.10 + 4 * 6.10 * 2.44 # 134.0 m2 r_const = room_constant(area, 0.15) # 23.6 m2 print(round(float(room_effect(1.83, r_const, directivity=2.0)), 1)) # 6.6 dB ``` Long's sheet prints 5 to 7 dB for that room across the bands, so a single mean absorption of 0.15 lands in the right place; a per-band absorption gives a per-band room effect, which is what the carpet actually does. Pass `target=` and `criterion=` to `duct_path` and the result rates itself. `criterion_curve` samples the NC or RC curve at the analysis bands, `exceedance` is the band-by-band excess over it, `meets_target` is the band-by-band verdict a design sheet applies, and `rating` is the full `NCResult` or `RCResult` derived by the ANSI/ASA S12.2-2019 procedure, which is a different question and can differ from the tangency verdict. ## 6. The worked example: Long's Table 14.9 Long's Chapter 14 closes with a complete sheet: a 5000 cfm forward-curved fan at 2 in w.g., feeding one room through a supply path (elbow, silencer, lined duct, a 25 per cent branch split, a second lined duct, flexible duct, a rectangular diffuser) and a return path (elbow, low-frequency silencer, lined elbow, plenum, grille), each ending in the room effect of a 20 x 20 x 8 ft carpeted office, combined and checked against NC 30. Every row below is the one Long prints, including the manufacturer data for the silencers and the terminal devices, which is what a real sheet uses. ```python import numpy as np from phonometry import DuctElement, combine_duct_paths, duct_path bands = [63.0, 125.0, 250.0, 500.0, 1000.0, 2000.0, 4000.0, 8000.0] fan = [90.0, 86.0, 82.0, 79.0, 77.0, 75.0, 71.0, 61.0] source = "Fan, centrifugal FC, 5000 cfm, 2 in w.g." supply = duct_path( bands, fan, [ DuctElement("Elbow, 36 x 24 in, unlined", [0, 1, 2, 3, 3, 3, 3, 3], [41, 39, 36, 29, 20, 6, 0, 0], code="2"), DuctElement("Silencer, 3 ft, standard pressure drop", [7, 12, 16, 28, 35, 35, 28, 17], [49, 43, 44, 42, 42, 45, 35, 24], code="3"), DuctElement("Duct, 36 x 24 in, 5 ft, 1 in lining", [2, 2, 3, 7, 15, 12, 11, 9], code="4"), DuctElement("Split, 25 per cent", 6.0, code="5"), DuctElement("Duct, 18 x 12 in, 6 ft, 1 in lining", [3, 3, 5, 11, 25, 22, 16, 13], code="6"), DuctElement("Flexible duct, 12 in, 6 ft", [14, 14, 16, 15, 17, 22, 16, 13], code="7"), DuctElement("Rectangular diffuser, 312 cfm", None, [33, 32, 29, 23, 15, 4, 0, 0], code="8"), ], room_effect=[6, 6, 5, 5, 6, 7, 6, 6], source_label=source, target=30.0, label="Supply", ) ret = duct_path( bands, fan, [ DuctElement("Elbow, 36 x 24 in, unlined", [0, 1, 2, 3, 3, 3, 3, 3], [43, 42, 39, 33, 24, 12, 0, 0], code="2"), DuctElement("Silencer, 5 ft, low-frequency type", [16, 21, 35, 41, 41, 28, 21, 15], [51, 49, 53, 56, 56, 59, 60, 53], code="3"), DuctElement("Elbow, 36 x 24 in, lined, 1 in", [1, 2, 3, 4, 5, 6, 8, 10], [39, 38, 34, 28, 18, 4, 0, 0], code="4"), DuctElement("Plenum, 800 sq ft, 50 per cent lined", [12, 13, 19, 20, 20, 20, 21, 21], code="5"), DuctElement("Rectangular grille, 24 x 24 in, 563 cfm", None, [30, 29, 26, 20, 12, 1, 0, 0], code="6"), ], room_effect=[9, 8, 6, 8, 8, 8, 9, 10], source_label=source, target=30.0, label="Return", ) total = combine_duct_paths([supply, ret], label="Supply + return") print(np.round(supply.received_level, 0)) # Long: 52 42 30 18 9 -2 -2 -1 print(np.round(ret.received_level, 0)) # Long: 52 41 27 25 23 25 22 12 print(np.round(total.received_level, 0)) # Long: 55 45 32 26 23 25 22 12 print(total.meets_target) # True print(round(float(total.rating.rating), 1)) # 26.7, governed at 63 Hz ``` Every printed row comes back within the sheet's own 1 dB rounding: the supply is 1 dB low at 4 kHz, the return 1 dB low at 500 Hz, and the combination 1 dB low at 500 Hz and 1 dB high at 8 kHz, everything else exact. The room lands at NC 27, comfortably inside its NC 30 target, and the 63 Hz band is what governs the rating, which is the usual outcome of a duct-noise design and the reason low-frequency silencer performance is worth paying for. *The two paths and their sum against NC 30. The supply, with its silencer, two lined runs, a branch split and six feet of flexible duct, has nothing left above 1 kHz; the return, with a plenum but a silencer whose own self-noise floors it near 25 dB, is what the room actually hears in the mid and high bands. Adding low-frequency attenuation to the supply would change nothing at all: the return already sets the answer everywhere except at 63 Hz, and that is the row to argue about.*

Show the code for this figure

```python import matplotlib.pyplot as plt from phonometry import combine_duct_paths # `supply` and `ret` are the two DuctPathResult objects built above. # One line for one path: the cascade of the supply run against NC 30. supply.plot() plt.show() # The concept figure: both paths, their energy sum and the criterion. total = combine_duct_paths([supply, ret], label="Supply + return") total.plot() plt.gca().set_ylim(-6.0, 62.0) plt.show() ```

`DuctPathResult` also prints and files itself. `.table()` returns the sheet row by row with the worksheet sign convention, and `.report()` renders a one-page PDF in the layout of the published procedures (AHRI Standard 885 Table 8; Long Table 14.9): the element table, the cascade chart against the criterion curve, the boxed room-criterion rating and the verdict. ```python for row in total.table(): print(f"{row['kind']:<12} {row['label'][:26]:<26} " f"{[round(float(v)) for v in row['values']]}") total.report("duct-path.pdf") # needs phonometry[report] ``` The example fiche, regenerated with `make reports`, is kept rendered in the repository. Click the preview to open the PDF: [](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/reports/duct_path_example.pdf) *Duct-borne noise path fiche (`DuctPathResult.report`): the supply path of the worked sheet element by element, rated NC-22.6 and passing NC 30 with 5 dB to spare at 63 Hz.* ## 7. What reproduces, and what does not Long's Table 14.9 was produced by a commercial computer program, not by hand from the tables printed alongside it, and being honest about that is more useful than pretending otherwise. The *arithmetic* of the sheet is reproduced exactly, as section 6 shows. Several of its **element rows**, however, do not follow from the book's own printed data, and the functions in this library implement the printed equations and tables. Verified band by band: - **The fan row does not come from Eq. 13.1.** The sheet prints 90/86/82/79/77/75/71/61 dB. Eq. 13.1 with the Table 13.5 forward-curved constants at 5000 cfm and 2 in w.g. gives 99/99/89/84/82/77/72/67 dB, and the printed spectrum is not a level shift of the tabulated one, so it comes from other data (a manufacturer's, most likely). - **The flexible-duct row is not Table 14.4.** The sheet prints 14/14/16/15/17/22/16/13 dB for 12 in by 6 ft; `flexible_duct_insertion_loss` reads 3/5/10/15/17/16/9 dB out of the table for that duct. - **The lined rectangular ducts agree in the mid and high bands.** For the 18 x 12 in, 6 ft, 1 in run the library returns 11/25/22/16/13 dB from 500 Hz up, exactly the printed row, and is 1 to 2 dB high below it (5/4/6 against 3/3/5). For the 36 x 24 in, 5 ft run it matches at 250, 500 and 8 kHz and is 1 to 2 dB low elsewhere. - **The split and the unlined elbow reproduce exactly.** `split_loss` gives the 25 per cent branch as 6.0 dB against the printed -6 dB, and `elbow_insertion_loss` gives 0/1/2/3/3/3/3/3 dB against the printed row when the elbow is read as round (Table 14.7) at $w = 24$ in. - **The supply diffuser row reproduces too.** `diffuser_sound_power` on a 24 x 24 in rectangular device at 312 cfm and 0.05 in pd returns 33.4/32.4/29.1/23.6/15.9/5.9 dB against the printed 33/32/29/23/15/4, inside the sheet's own rounding in the five bands that carry the level. The return grille row (30/29/26/20/12/1) does not follow from the same equations at its 563 cfm, so it is manufacturer data. - **The NC 30 row differs by 1 dB at 1 kHz.** Long prints 57/48/41/35/31/29/28/27; the library's `nc_curve(30)` returns 57/48/41/35/**32**/29/28/27, the values of ANSI/ASA S12.2-2019 Table 1. Long is using the original Beranek 1957 curve. The difference does not change the verdict here, but it is worth knowing which NC you are quoting. None of that is a defect of the sheet. It is what a real duct-borne calculation looks like: the elements a manufacturer publishes (fans, silencers, diffusers, grilles) come from test data, and the elements nobody publishes (duct runs, elbows, splits, end reflections, the room) come from the tables. `DuctElement` takes both without caring which is which, which is the point. ## 8. The plane-wave limit Every element model above, and every four-pole silencer in [Silencers](https://jmrplens.github.io/phonometry/devices/noise-control/silencers/), is one-dimensional. It assumes a single sound pressure describes the whole duct cross section, which is true only below the frequency at which the first higher-order acoustic mode cuts on. Above it several modes propagate at once, each with its own axial wavenumber, and a plane-wave prediction quietly stops being right. `noise_control.duct_modes` implements the cut-on analysis of Norton & Karczub, *Fundamentals of Noise and Vibration Analysis for Engineers* (2nd ed.), section 7.3: circular ducts by Eq. 7.6 with the $\pi\alpha_{pq}$ eigenvalues of Table 7.1 that solve $J'_p(\kappa_{pq} a_\mathrm{i}) = 0$, rectangular ducts by Eq. 7.10, and the mean-flow correction of Eqs. 7.8 and 7.9, in which a uniform axial flow of Mach number $M$ lowers every cut-on frequency by $\sqrt{1 - M^2}$ and moves the cut-on itself from $k_x = 0$ to $k_x = -M\kappa_{pq}/\sqrt{1 - M^2}$. ```python import numpy as np from phonometry import plane_wave_limit, rectangular_duct_cut_on # Norton problem 7.2: a 0.65 x 0.4 m air-conditioning duct at 15 m/s. modes = rectangular_duct_cut_on(0.65, 0.40, flow_velocity=15.0, count=6) print(modes.modes[:3]) # ((1, 0), (0, 1), (1, 1)) print(np.round(modes.cut_on[:3], 1)) # [263.6 428.3 502.9] Hz print(np.round(modes.cut_on_no_flow[:3], 1)) # [263.8 428.8 503.4] Hz print(round(modes.plane_wave_limit, 1)) # 263.6 Hz IN = 0.0254 print(round(plane_wave_limit(width=36 * IN, height=24 * IN), 1)) # 187.6 Hz print(round(plane_wave_limit(diameter=12 * IN), 1)) # 659.5 Hz ``` Those ventilation numbers are blunt: in that duct plane waves are the whole story only up to the 250 Hz octave, and a 36 x 24 in supply trunk gives up at 188 Hz. What a number cannot say is that cut-on is a **switch**, not a slope. The clip below simulates the 0.65 m dimension of that same duct as a 2D slice with rigid walls, and drives it twice with one off-axis source that excites the plane mode and the first transverse mode together: at 180 Hz, below the 263.8 Hz just printed, and at 400 Hz above it. Below cut-on the transverse mode is evanescent, decaying as $e^{-\alpha x}$ with $1/\alpha = 0.28$ m, which puts it 20 dB down one duct width from the source, so the lumpy near field dies there and every later wavefront crosses the section flat. Above cut-on it propagates, the two modes run at different axial wavenumbers ($k_x = 5.51$ against $k = 7.33$ rad/m) and beat along the duct every 3.45 m, and the profile across the section never settles. [Watch the high-resolution video (WebM)](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_fdtd_duct_cut_on.webm) The panel beside each strip draws the instantaneous profile across the section at a station 5 m downstream against its own section average, which is the single pressure a one-dimensional model carries. That is why a transfer-matrix silencer model has a **ceiling frequency** rather than an accuracy budget: below cut-on the second mode is not small, it is absent, and above it the model describes one of several waves that are all there. At 15 m/s the flow correction is invisible: $M = 0.044$ gives $\sqrt{1 - M^2} = 0.999$, which moves the first cut-on by 0.2 Hz. It earns its place in high-speed pipework instead. Norton's problem 7.1 is that case, a 254 mm line carrying steam ($c = 405$ m/s) at 200 m/s, $M = 0.494$, and there the two ladders separate by more than a hundred hertz at every rung. ```python import numpy as np from phonometry import circular_duct_cut_on # Norton problem 7.1: a 254 mm circular duct carrying steam at 200 m/s. steam = circular_duct_cut_on(0.254, flow_velocity=200.0, speed_of_sound=405.0, count=6) print(steam.modes) # ((1, 0), (2, 0), (0, 1), (3, 0), (4, 0), (1, 1)) print(np.round(steam.cut_on_no_flow, 1)) # [ 934.5 1550.1 1944.7 2132.3 2698.9 2705.9] print(np.round(steam.cut_on, 1)) # [ 812.6 1347.9 1691.1 1854.1 2346.8 2352.9] print(np.round(steam.axial_wavenumber, 2)) # [ -8.23 -13.66 -17.13 -18.79 -23.78 -23.84] ``` *Norton's problem 7.1, the case where the mean flow is worth drawing: half the speed of sound in the pipe pulls every cut-on down by $\sqrt{1 - M^2} = 0.870$, so the first higher-order mode appears at 813 Hz instead of 935 Hz and the plane-wave band, shaded, is 13 per cent narrower than the still-air calculation would promise. The axial wavenumber at cut-on is negative in every rung: with flow, the mode is already travelling upstream at the frequency at which it appears.*

Show the code for this figure

```python import matplotlib.pyplot as plt from phonometry import circular_duct_cut_on, rectangular_duct_cut_on # One line for one duct: the cut-on ladder with the plane-wave band shaded. steam = circular_duct_cut_on(0.254, flow_velocity=200.0, speed_of_sound=405.0, count=6) steam.plot() plt.show() # The ventilation duct of problem 7.2, where the flow shift is negligible. rectangular_duct_cut_on(0.65, 0.40, flow_velocity=15.0, count=6).plot() plt.show() ```

Two results carry this limit for you. Every `ReactiveSilencerResult` now reports the first cut-on of its widest cross section as `plane_wave_limit`, and `duct_path` accepts a `section=` description of the duct it is walking. Both raise a `PlaneWaveWarning` when the analysis grid runs past that frequency: the numbers are still returned, and above cut-on they describe the plane-wave mode alone, which a measurement will not. ```python import warnings from phonometry import DuctElement, PlaneWaveWarning, duct_path IN = 0.0254 bands = [63.0, 125.0, 250.0, 500.0, 1000.0, 2000.0, 4000.0, 8000.0] with warnings.catch_warnings(record=True) as caught: warnings.simplefilter("always") duct_path(bands, [90.0] * 8, [DuctElement("Straight run", 3.0)], section={"width": 36 * IN, "height": 24 * IN}, flow_velocity=6.0, label="Supply") print(caught[0].category is PlaneWaveWarning) print(str(caught[0].message)) # Supply: 6 of 8 frequencies are above the first duct cut-on frequency # (188 Hz), where higher-order modes propagate and the plane-wave result # describes the plane-wave mode only. ``` Six of the eight octave bands of a standard duct-noise sheet sit above the cut-on of a 36 x 24 in duct. That is not a reason to distrust the sheet: the ASHRAE element models it is built from are *empirical*, fitted to measurements of real ducts in which those modes were present, so they carry the multimode behaviour inside their regression constants. The warning is aimed at the analytical methods, the four-pole silencer algebra above all, where the plane-wave assumption is structural rather than statistical, and where the peaks and troughs of a computed transmission loss simply do not survive past cut-on. ## What this guide covers **Covered.** The fan source (`fan_sound_power`, `fan_efficiency_correction`, `blade_passing_frequency`, `fan_casing_attenuation`; Long Eq. 13.1 with Tables 13.5-13.8); the run attenuations (lined and unlined rectangular and circular ducts, `flexible_duct_insertion_loss`, `elbow_insertion_loss`, `split_loss`, `end_reflection_loss` in both published methods, `splitter_silencer_insertion_loss`, `plenum_attenuation`); the regenerated noise (`silencer_self_noise`, `flow_noise_straight_duct`, `flow_noise_bend`, `diffuser_sound_power`, and the ASHRAE Chapter 49 screening rules `air_terminal_velocity_limit` and `air_terminal_damper_correction`); `room_effect` and the criterion machinery; the cascade itself (`duct_path`, `combine_duct_paths`, `.table()`, `.report()`); and the `duct_modes` cut-on analysis. **Not covered.** Nothing is predicted where the manufacturer publishes it: the sheet expects the published fan and terminal figures. There is no dissipative liner model — the splitter estimate is a lined-duct regression and the plenum is Wells' closed form — and above the first cut-on the element models rest on their empirical fit alone. Four paths of a real installation sit outside the calculation entirely, which is usually why a passing sheet meets a failing room: **duct breakout** from a trunk radiating through its own walls into a space it merely crosses; **break-in and crosstalk** between two rooms served by the same run, whatever the partition between them; **structure-borne transmission** of fan vibration through hangers, plinths and slab, whose dynamic transfer stiffness belongs to [Transfer stiffness (ISO 10846)](https://jmrplens.github.io/phonometry/vibration/structural/transfer-stiffness/); and the **octave resolution itself**, which averages a blade-passing tone into its band and so understates how a pure tone will be judged. ## See also - [Silencers](https://jmrplens.github.io/phonometry/devices/noise-control/silencers/): the reactive four-pole elements (expansion chambers, Helmholtz, quarter-wave and extended-tube resonators) whose validity ends at the same cut-on frequency. - [Industrial noise control](https://jmrplens.github.io/phonometry/devices/noise-control/noise-control/): the rest of the installation, the individual HVAC duct methods and machine-enclosure insertion loss. - [Room-noise criteria (NC / RC Mark II)](https://jmrplens.github.io/phonometry/buildings/rooms/room-noise/): the ANSI/ASA S12.2-2019 families the received spectrum is judged against. - [Steady-state room field](https://jmrplens.github.io/phonometry/buildings/rooms/room-image-sources/): the room constant behind the room effect and the plenum reverberant term. - API reference: [`noise_control.duct_path`](https://jmrplens.github.io/phonometry/reference/api/noise_control/duct-path/), [`noise_control.duct_modes`](https://jmrplens.github.io/phonometry/reference/api/noise_control/duct-modes/), [`noise_control.hvac`](https://jmrplens.github.io/phonometry/reference/api/noise_control/hvac/). ## References - Long, M. (2014). *Architectural acoustics* (2nd ed.). Academic Press. ISBN 978-0-12-398258-2. The fan sound-power model (Ch. 13, Eq. 13.1 and Tables 13.5-13.8) and the diffuser self-noise model (Ch. 13, Eqs. 13.27-13.33), the duct attenuation, flexible duct, split loss, end reflection, silencer self-noise and room effect of Ch. 14, and the worked duct-borne sheet of Table 14.9 this guide is built around. - 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 end-reflection table (§8.13), the elbow insertion loss (§8.11), the splitter-muffler reduction to lined ducts (§8.10.5, Eq. 8.241), the plenum chamber (§8.17) and the flow-generated noise of ducts and bends (§8.15). - Norton, M. P., & Karczub, D. G. (2003). *Fundamentals of noise and vibration analysis for engineers* (2nd ed.). Cambridge University Press. [doi:10.1017/CBO9781139163927](https://doi.org/10.1017/CBO9781139163927). The higher-order duct modes, the cut-on frequencies of circular and rectangular ducts and the mean-flow correction (§7.3, Eqs. 7.6-7.10). - ASHRAE (2019). *ASHRAE handbook: HVAC applications* (SI ed.), Chapter 49, Noise and vibration control. ASHRAE. The air terminal velocity limits (Table 9) and the volume-damper corrections (Table 10), and the guidance that fan sound power is best taken from manufacturer test data. - Air-Conditioning, Heating and Refrigeration Institute. *AHRI Standard 885: Procedure for estimating occupied space sound levels in the application of air terminals and air outlets*. The industry row structure of the duct-borne calculation sheet (Table 8) that `DuctPathResult.table()` and `.report()` follow.