Formula
A room mode is a standing wave. Between two parallel rigid walls the first resonance is the speed of sound divided by twice the spacing. A rectangular room has three such pairs, and combinations of them. The indices nx, ny, and nz are how many half-wavelengths fit along length, width, and height. They are whole numbers, and they cannot all be zero.
f = (c / 2) √( (nx/L)² + (ny/W)² + (nz/H)² )
c = 343 m/s at 20 °C
c = 331.3 √(1 + T/273.15) otherwise
Axial, tangential, and oblique
One index nonzero is axial. Two are tangential. Three are oblique. Axials are usually the ones you hear, because only one pair of walls has to be reflective and parallel. Tangentials need two pairs. Obliques need all three and tend to be weaker and denser. The table is sorted by frequency. The same frequency can belong to two different index trios. In the 5 × 4 × 2.5 m room, (0,0,1) and (2,0,0) are both 68.6 Hz. That is a degenerate pair, not a bug.
Bonello density
The check counts modes in 1/3-octave bands from 20 Hz through 200 Hz. Band edges are the centre divided by 2^(1/6) and the centre times 2^(1/6). A mode exactly on a shared edge is counted in the higher band. Bonello’s criterion is that each band should contain at least as many modes as the band below it. A drop is flagged. The counts come from the same list as the table, cut off at the frequency you set, so a low cutoff makes the upper bands look empty. This is a room-acoustics planning heuristic from Bonello’s 1981 paper. It is not a building code, and a pass does not mean the room is finished.
Ratios, volume, and what this page leaves out
Volume is length times width times height. Surface area is twice the sum of the three face pairs. The ratio is printed with height as 1 and, in the note when it differs, with the shortest side as 1. Popular starting ratios include 1 : 1.14 : 1.39, 1 : 1.28 : 1.54, and 1 : 1.6 : 2.33. They are starting points. Spacing of the modes you actually have matters more than wearing a named ratio. Sorting writes the longest number into Length and the shortest into Height. It does not rename the fields.
Carpet and curtains do little for the first axials. Placement of the seat and the bass source does more, then dedicated low-frequency absorption, then equalization. An irregular room does not obey this formula. Measure it, or use a model that is not a rectangular cavity. Schroeder’s frequency needs a reverberation time. This page does not invent one, so it does not print a transition frequency.
Worked examples
5.00 × 4.00 × 2.50 m at 20 °C. Length axial 34.3 Hz, width 42.9 Hz, height 68.6 Hz. The first tangential (1,1,0) is 54.9 Hz. (2,0,0) is 68.6 Hz, degenerate with the height axial. The first oblique (1,1,1) is 87.9 Hz. Volume is 50 m³.
16 × 12 × 8 ft, same 343 m/s. The axials are 35.2 Hz, 46.9 Hz, and 70.3 Hz.
Bonello on the 5 × 4 × 2.5 m room, modes to 300 Hz. Counts at 31.5, 40, 50, 63, 80, and 100 Hz are 1, 1, 1, 2, 5, and 7. None of those bands drops.
The same 5 m length at 0 °C. c is 331.3 m/s and the length axial is 33.1 Hz, against 34.3 Hz at 20 °C.
Reference
c = 343 m/s unless the row says otherwise. Frequencies to 0.1 Hz.
| Mode | Frequency |
|---|---|
| (1,0,0) 5 m | 34.3 Hz |
| (0,1,0) 4 m | 42.9 Hz |
| (0,0,1) 2.5 m | 68.6 Hz |
| (1,1,0) | 54.9 Hz |
| (2,0,0), same as height | 68.6 Hz |
| (1,1,1) | 87.9 Hz |
| 16 × 12 × 8 ft axials | 35.2 Hz · 46.9 Hz · 70.3 Hz |
| (1,0,0) at 0 °C | 33.1 Hz |
Questions
What are room modes?
They are standing waves between parallel surfaces. A rectangular room with rigid walls has a frequency for every trio of whole-number indices. Axial modes run along one pair of walls and are usually the strongest. Tangential modes involve two pairs. Oblique modes involve all three and are usually weaker. Small rooms pack these frequencies into the bass, which is why one note booms and the next one disappears.
How do I calculate axial modes?
The first axial along a dimension is the speed of sound divided by twice that dimension. At 20 °C, c is 343 m/s. A 5.00 m length is 34.3 Hz, a 4.00 m width is 42.9 Hz, and a 2.50 m height is 68.6 Hz. The next axial on the length, index 2, is 68.6 Hz, the same frequency as the height fundamental in this room.
What is a good room ratio?
A ratio only spreads the modes. It does not remove them. This 5 × 4 × 2.5 m room is 2.00 : 1.60 : 1 with height as 1. Starting points people try include 1 : 1.14 : 1.39, 1 : 1.28 : 1.54, and 1 : 1.6 : 2.33. Mode spacing matters more than matching a named ratio. Sort the dimensions on this page if you want the longest side in the length field. The labels stay Length, Width, and Height.
What is the Bonello criterion?
Oscar Bonello’s 1981 check counts modes in successive 1/3-octave bands and wants the count to stay the same or rise. A band that thins out is flagged. For this 5 × 4 × 2.5 m room, modes through 300 Hz, the counts at 31.5, 40, 50, 63, 80, and 100 Hz are 1, 1, 1, 2, 5, and 7, which does not decrease. It is a planning heuristic for rectangular rooms, not a building code and not a pass for construction.
Why do I hear a boom at one bass note?
That note is sitting on a mode, and your seat is not on a node. Moving the seat or the subwoofer often changes the boom more than a small equalizer move. Two modes can also land on the same frequency, as the 68.6 Hz pair does in the 5 m example. The list shows clustering. It does not say the room is unusable.
Do carpet and curtains fix room modes?
Mostly no, not at low frequency. Carpet, curtains, and furniture absorb mids and highs. The wavelengths of the first axial modes are several metres, and thin soft goods do not absorb them. Bass traps and placement do more. Equalization is a last resort after the seat and the speakers have moved. This page does not size a trap.
Do metric and imperial dimensions agree?
Yes. The formula runs in metres. 1 ft is 0.3048 m, and the speed of sound stays 343 m/s at 20 °C. A 16 × 12 × 8 ft room has axials at 35.2 Hz, 46.9 Hz, and 70.3 Hz. Air temperature is entered in °C either way. At 0 °C the 5 m length axial falls from 34.3 Hz to 33.1 Hz.
Sources
- Philip M. Morse, Vibration and Sound. Rectangular-room modes with rigid boundaries.
- Oscar J. Bonello, “A New Criterion for the Distribution of Normal Room Modes,” Journal of the Audio Engineering Society, 1981. The non-decreasing 1/3-octave count.
- F. Alton Everest and Ken C. Pohlmann, Master Handbook of Acoustics. Axial, tangential, and oblique modes, and why soft furnishings do not fix the bass.