Building Acoustics

98 Building acoustics

3.7.3.5 Modal density for bending waves on plates

Detailed frequency information is normally not required for quantities used in building
acoustics. In general, one is interested in average values taken over relatively broad
frequency bands such as one-third-octave bands or octave bands. This is true for sound
pressure levels as well as for levels of vibration, e.g. one wish to determine an average
velocity level for a wall or floor in a building. To ensure that the measurement result has
the desired accuracy one has to estimate the required number of measurement points on
the structure and this is directly linked to the number of modes having their natural
frequencies within the frequency band measured. It is therefore important to estimate the
density of natural frequencies in the structure to be measured, the so-called modal
density.
A suitable procedure for this calculation is to make the modes expressed by their
modal wave numbers ki,n, instead of by their natural frequencies. These wave numbers
will be
1

(^222)
,,
,
BB
in^2 in
in
f in
k
cc a b
ωπ⎡⎛⎞⎛ ⎞ππ⎤
== = +⎢⎜⎟⎜ ⎟⎥
⎢⎣⎝⎠⎝ ⎠⎥⎦

(3.116)

or
22
222
in, ix nz,

in

kkk

ab

⎛⎞⎛ ⎞ππ

=+ =+⎜⎟⎜ ⎟

⎝⎠⎝ ⎠

(3.117)

where kix and knz are the wave number components in the x- and z-direction, respectively.
All natural frequencies may therefore be plotted in a wave number diagram as shown in
Figure 3.24, where each point represents a mode (eigenmode).

Figure 3.24 Modal wave numbers for bending waves on a rectangular plate of dimensions a and b. The
eigenmodes within a frequency band Δω are indicated.

Plate

x

z a

b

kx

kz

kB(ω+Δω)

π/b

kB(ω)

π/a