Building Acoustics

234 Building acoustics

The numerator in this expression represents the power imparted by the force, thereby
giving us the following important relationship between that power and the resulting mean
square velocity

WSmu Emec^2 mec.
ω

ηω ηω

Δ

== (6.57)

The quantity Emec denotes the mechanical (modal) energy of the plate. It should be noted
the validity of this expression is not limited to point excitation but it gives the general
relationship between mechanical power and the mean square velocity of a structure. It is,
however, presupposed that the structure is neither so large nor so heavily damped that we
cannot reasonably determine a representative mean velocity.
We can at this point find an interesting analogue in the relationship between the
sound power Wac injected by a source into a room and the resulting mean square sound
pressure in the diffuse field. See the derivation in section 4.5.1, where we found that

(^226)
ac
00 00 0
4(ln10 )
,
44
pp V
WA
cccT
ωω
ρρ

=⋅=⋅ ⋅ΔΔ



(6.58)

where A, V and T are the total absorbing area, volume and reverberation time of the
room, respectively. Substituting the latter quantity with the equivalent loss factor of the
room using the relationship

ln10^6 2.2

,

2

T

πfηηf

=≈ (6.59)

we get

(^) ac 2 2 ac
0

,

V

WpwVE

c ω

ηω ηω ηω

ρ Δ

=⋅ =⋅= (6.60)

where w is the acoustical energy density in the room and Eac is the corresponding total
acoustical energy. The last expressions in Equations (6.57) and (6.60) we shall meet
again in Chapter 7 when dealing with statistical energy distribution and energy flow in
multimode systems; i.e. when dealing with methods and prediction models labelled as
statistical energy analysis (SEA).
We shall return to our point excited plate, and we shall address the problem of
estimating the part of the mechanical energy transformed into sound power. Knowing
both the mean velocity and the radiation factor we should be able to calculate the
radiated power directly. However, as shown in the next section, the conditions in the
neighbourhood of the driving point do complicate matters.

6.4.2 Sound radiation by point force excitation

The mechanical power imparted to a structure will be dissipated partly by internal losses,
partly transmitted to connected structures and partly radiated as sound. When the
surrounding medium is air, which is the case we shall be concerned with, the latter part
will be small; normally a maximum of 1–2%. As indicated in the preceding section,
radiation will partly be due to the reverberant wave field set up due to reflections from
the plate boundaries. This part will be determined by the resonant modes giving a wave
field having a mean square velocity given by Equation (6.55) or (6.56). In addition, there