Sound transmission. Characterization and properties of single walls and floors - Building Acoustics - page 227

2

⎧ ⎫

⎪ cos cos ⎪

α β

⎡⎤ ⎡⎤
⋅

ππ

⎪ ⎪

⎢⎥ ⎢⎥

22

2

64 kab ⎪ sin 2 sin 2 ⎪

⎣⎦ ⎣⎦

∫∫

σ (, ) nn = ⋅ indd. ϕϕθ (6.46)

⎨ ⎬ xz

622

⎡

2

⎤ ⎡

2

π nn ⎤

⎪ ⎪

⎛ ⎞ ⎛ ⎞

α β

xz

00

⎢ ⎥ ⎢ ⎥

−⋅ 1 − 1

⎪ ⎜ ⎟ ⎜ ⎟ ⎪

⎢ n π ⎥ ⎢ n π

⎥

⎪ ⎝ ⎠ ⎝ ⎠ ⎪

x z

⎣ ⎦ ⎣ ⎦

⎩ ⎭

The quantities α and β are given by

cos

α = ka sin ϕ θβ cos , = kb sin ϕ θ cos , and (6.47)

sin

is to be understood in the following way: cosine should be used when n x or n z 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 ( n x ,n z ) 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 > f c 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 ).

0

-10

(1,1)

-20

(3,1)

(3,3)

-30

(2,1)
(2,2)

-40

-50

0.001 0.01 0.1 1 10

f / f c

Figure 6.14 Radiation index of a square plate as a function of frequency relative to the critical frequency f c . The
mode number ( n x ,n z ) is indicated on the curves.