# Duct Acoustics

The acoustics associated with ducts or interconnecting pipes.

- pressure wave propagating in the duct in the x direction

- assuming rigid walls where there are no losses.
- acoustic volume flow is

- The acoustic impedance of the pipe at any point is

- where S is the cross-sectional area of the duct.

- For a typical air intake duct of internal radius 0.03m the impedance Z
_{0}=1.47x10^{6}.

- The acoustic impedance of air is

- The cut-off frequency of the first 'sloshing' mode is

- where a is the radius of the pipe.

- For intake systems the speed of sound is ~343ms
^{-1}and the internal radius of ducts is ~0.03m. This gives a cut-off of ~3300Hz below which only plane wave propagation needs to be considered. Higher order acoustic modes decay very rapidly with distance.

**Losses at the surface of the wall**- The walls of pipes contribute a viscous drag the relative magnitude of this drag is

- where
*η*= viscosity of the fluid [m^{2}s^{-1}]

- The thermal exchange between the fluid and the walls of the pipe is also a loss and this relative loss is expressed by

- where C
_{p}is the specific heat of air at constant pressure and*K*is its thermal conductivity. - For temperatures around 300K (~27°C)

- where ΔT is the temperature deviation from 300K.

- Rewriting the wave vector
*k*as a complex number

- where v is the phase velocity and a is the attenuation coefficient per unit length.

- and

- these formulae are Benade's versions of Rayleigh's approximations which are valid for r
_{v}>10, but are reasonable down to a value of r_{v}=3. - The attenuation coefficient is inversely proportional to the pipe radius and the square root of frequency. Therefore, losses at the wall become significant at high frequency for narrow ducts.

**See also: **Acoustic Duct End Correction, Acoustic Filter Elements, Acoustic Impedance, Acoustic Inertance, Duct.

**Subjects: ** Architectural Acoustics Audio Mechanical Engineering Noise & Vibration