# 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 Z0=1.47x106.

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 [m2s-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 Cp 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 rv>10, but are reasonable down to a value of rv=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.