The run is a subtraction, mostly
Start with the sound power level on the fan submittal for one octave band and take things off it. The branch split is first and it is free: sound power divides with the air, so a branch carrying a sixth of the flow gets a sixth of the power, which is 7.8 dB. Lined duct takes off a published figure per foot. Lined elbows take off a published figure each. A silencer takes off its insertion loss. At the opening, low frequencies reflect back up the duct instead of radiating out, which is the end reflection term.
Run the defaults: 78 dB at the fan in the 250 Hz band, a 200 CFM branch off 1200, 25 feet of lined duct at 0.35 per foot, three lined elbows at 3 dB, no silencer and 2 dB of end reflection. That is 7.8 plus 8.8 plus 9 plus 2, or 27.5 dB total, and 50.5 dB of sound power arrives at the outlet.
Which term is largest matters more than the total, because the three are not interchangeable. The branch split is already happening and costs nothing. Lined duct is cheap per foot but needs length, and on a short run there is nowhere to put it. A silencer works anywhere, costs money, costs static pressure, and is the only one that can be added to a run that is already in the ceiling.
The outlet is often the whole problem
A register generates its own noise from the air squeezing through it, and that figure comes from the catalogue at the neck size and flow you selected. With the defaults it is 30 dB against 50.5 arriving down the duct, so it barely registers and the combined figure is 50.5.
Undersize the neck, though, and the two swap places. Once the outlet generates more than the duct delivers, nothing upstream matters at all — not a quieter fan, not more liner, not a bigger silencer. The only fixes are a larger neck, more outlets sharing the flow, or less air. This is the single most common reason a carefully attenuated system is still noisy, and it is decided at diffuser selection rather than anywhere in the duct design.
From sound power to a level in the room
Sound power describes what the outlet radiates; what somebody hears depends on where they are and what the room is like. Close in, direct sound dominates and the level falls six decibels for every doubling of distance. Further out, the reverberant field takes over and moving away stops helping, because what arrives is the room feeding energy back.
The defaults put the listener eight feet from a flush ceiling diffuser in a room holding 130 sabins. Direct sound is about 7 percent of what arrives, so this is firmly in the reverberant regime: the level comes out at 46.2 dB, and the absorption in the room is what sets it. Strip that room — take the carpet up, empty it for a refit — and the same ductwork gets several decibels louder with nothing changed in the ceiling.
One band is not the job
Every term on this page behaves differently at the two ends of the range. Duct liner is nearly useless at 63 Hz and very effective at 2000. End reflection does the reverse: it is largest at low frequency and negligible high up. Fan sound power is usually highest at the bottom, which is exactly where the attenuation is weakest. The result is that a run which looks comfortable at 250 Hz can be rumbling at 63, and a single band answer conceals it completely.
Three things are not modelled here at all. Breakout is noise radiating through the duct wall into a space the duct merely passes through, and it is the usual explanation for a rumble in a corridor with no outlet in it. Regenerated noise from dampers, takeoffs and fittings does not appear on the fan submittal and a half closed balancing damper can dominate everything upstream of it. And structure borne transmission — fan vibration through its mounts, through the hangers, into the slab — is untouched by any amount of duct attenuation.
None of this is a measurement. It is a design estimate from figures you supplied, and whether the finished space is acceptable is judged across all bands against a criterion someone chose, on an instrument, in the room as built.
Questions people ask
Why does a branch take noise out for free?
Because sound power divides with the airflow. A branch carrying a sixth of what leaves the fan receives about a sixth of the sound power, which is ten times the log of one sixth, or 7.8 dB. It costs nothing, it is already happening, and on a small branch off a large trunk it is frequently the largest single term on the whole run.
My system is still noisy after adding a silencer. Why?
The likely candidates are the outlet, the band, and regenerated noise. If the register is generating more than the duct delivers then nothing upstream matters and the fix is a bigger neck or more outlets. If the complaint is a rumble, the silencer probably does little in that band while doing plenty at 500 Hz. And a partly closed balancing damper generates noise downstream of everything you installed.
Where do the attenuation figures come from?
The liner manufacturer data for the duct size, liner thickness and band you have; the elbow figures from the same source for the type and size; the silencer figures from its selection data at the face velocity you are running; and the outlet noise from the register catalogue at your flow and neck size. Every one of them varies by band, and mixing bands produces an answer that means nothing.
Does room absorption really change duct noise?
It changes what you hear, which is the number that matters. With the defaults, direct sound from the diffuser is about 7 percent of what reaches a listener eight feet away — the rest is the reverberant field, and that is set entirely by the absorption in the room. The same ductwork in a hard empty room is several decibels louder, which is why a space measured before the furniture arrives is not the space anyone occupies.
Can I use this to show the system meets a noise criterion?
No. It is a single band design estimate built from figures you supplied. A criterion is judged across all the octave bands together, against a curve or a target somebody chose, on a measurement taken with an instrument in the finished room — and anything the code touches is decided by the authority having jurisdiction rather than by any calculation.