Building Acoustics

Sound transmission 227

2

2 22

(^62222)
00
cos cos
64 sin 22 sin
(, ) sindd.
11
xz
xz
xz
kab
nn
nn
nn
ππ αβ
σ φφθ
π αβ
ππ

⎧⎫

⎪⎪⎡⎤ ⎡⎤

⎪⎪⎢⎥ ⎢⎥⋅

=⋅⎪⎪⎣⎦ ⎣⎦

⎨⎬

⎪⎪⎡⎤⎛⎞ ⎡⎤⎛⎞

⎪⎪⎢⎥⎜⎟−⋅⎢⎥⎜⎟−

⎪⎪⎢⎥⎣⎦⎝⎠ ⎢⎥⎣⎦⎝⎠

⎩⎭

∫∫ (6.46)

The quantities α and β are given by

cos

sin cos , sin cos , and

sin

αφθβφθ==ka kb (6.47)

is to be understood in the following way: cosine should be used when nx or nz is an
uneven number, sine when they are even. The radiation factor, given by the radiation
index 10⋅log σ, is shown in Figure 6.14 as a function of relative frequency, i.e. relative to
the critical frequency. The plate is square (a = b), and the index is calculated for a
number of the lower modes, the mode numbers (nx,nz) are indicated on the curves.
A Gaussian numerical integration is used to evaluate the integral in Equation (6.46)

. The accuracy is relatively low for f > fc and high mode numbers (>8–10). The important
point is, however, to show the behaviour of the radiation factor at low frequencies and, at
the same time, to link the results to the observations above and to compare with
calculated results using the idealized source types. A plate vibrating in the fundamental
mode (1, 1) will represent a monopole, whereas the vibration pattern in the (1, 2) mode
or (2, 1) mode will represent a dipole. (Do compare Figure 6.8 and Figure 6.14).

Figure 6.14 Radiation index of a square plate as a function of frequency relative to the critical frequency fc. The
mode number (nx,nz) is indicated on the curves.

0.001 0.01 0.1 1 10

f/fc

-50

-40

-30

-20

-10

0

10 lg

σ (

dB)

(1,1)

(2,1)

(2,2)

(3,1)

(3,3)