### Strutt Help

Room Modal Density
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**Strutt|Building Acoustics|Room Modal Density** allows the user to calculate the number of room modes present in a room up to a given frequency cut-off (*number of modes*) as well as the *modal density* (the number of modes per frequency band) using the Bolt and Morse formula.

The number of modes, `N` from `0` Hz up to `f` Hz is given by:

`N=(4 pi f^3 V)/(3 c^3) + (pi f^2 S)/(4 c^2) + (fP)/(8c)`

where:

`f` is the frequency (Hz)

`V` is the room volume (m³)

`S` is the room surface area (m²)

`P` is the total room perimeter (m)

The above terms describe the number of oblique (`prop f^3`), lateral (`prop f^2`) and normal (`prop f`) modes in the room. Especially for high frequencies, the number of oblique modes is generally dominant.

The modal density is given by:

`(dN)/(df)=(4 pi f^2 V)/( c^3) + (pi f S)/(2 c^2) + (P)/(8c)`

Although derived for rectangular rooms, the above expressions give good estimates of the modal behaviour for rooms of arbitrary shape, provided that the aspect ratio of the room is not too extreme.

Reference: Bies and Hansen *Engineering Noise Control*, Third Edition, Section 7.3.1

Comments or suggestions to
strutt@arup.com