Building Acoustics

112 Building acoustics

4.4.2 Sound pressure in a room using a monopole source

We shall proceed by calculating the sound field in a room of rectangular shape where we
have placed a sound source in a given position. This is again a generalization of the one-
dimensional case of a tube with a sound source (see section 3.6). We shall assume that
the source is a monopole, pulsating harmonically in time. The task is then to solve the
Helmholtz equation (4.9) but now modified with a source term on the right side of the
equation. We shall characterize the monopole source by its volume velocity or source
strength Q having unit m^3 /s, i.e. not by the mass q as in Equation (4.1). The pressure
root-mean-square-value in a given point (x,y,z) caused by the source in a position
(x 0 ,y 0 ,z 0 ) may be written

()

2 000

(^0022)
000

(, ,) ( , , )

(, ,) xyz xyz.

xyz xyz xyz

nnn nnn

nnn nnn nnn

xyz x y z

pxyz cQ

V

ω

ρ

ωω

∞∞∞

===

⋅Ψ ⋅Ψ

=

−

∑∑∑

  (4.15)

The quantity ω is the angular frequency of the source, and ωnnnxyz are the

eigenfrequencies according to Equation (4.12). The Ψ-functions are the corresponding

eigenfunctions:

(^) nnnxyz( , , ) cos x cos y cos z.
xy
nx ny nz
xyz
LLL
π π
z
π
Ψ=⋅⋅ (4.16)
Vnnnxyzis a normalizing factor, depending on the modal numbers, given by
,where 1for 0
1
and for 1.
2
nnnxyz nx ny nz n
n
VV n
n
εεε ε
ε

=⋅ ⋅ ⋅ = =

= ≥

(4.17)

The equations are derived assuming no energy losses in the room. However, as shown
earlier in section 3.7.3.6, we may introduce small losses by complex eigenfunctions. We
shall write

4.4

xyz xyz1j xyz1j ,

xyz

nnn nnn nnn

nnnT

π

ωω ηω

ω

⋅

=+⋅=+⋅

⋅

(4.18)

where η is the loss factor and T the corresponding reverberation time. As an example of
the use of Equation (4.15), we shall calculate the pressure at a given position in the same
room as used in the example in section 4.4.1. We shall make the reverberation time 1.0
seconds independent of frequency.
The pressure response is shown in Figure 4.3 represented by the transfer function
p/(Q⋅ω) on a logarithmic scale for a frequency range up to 1000 Hz. This implies that we
have related the pressure to the volume acceleration of the monopole source, both given
by their root-mean-square-values. Also shown in the diagram are the lowest 10
eigenfrequencies. It will appear that only the very low frequency resonances may be
identified. In the higher frequency range we find that the response is made up by
contributions from many modes.