Building Acoustics

Waves in fluid and solid media 99

If we wish to calculate the number of modes inside a given frequency interval Δω
(or Δf), we just count the number of points ΔN inside an area defined by two quarter
circular arcs having wave numbers kB(ω) and kB(ω+Δω), these being wave numbers
corresponding to the lower and upper cut-off frequencies for the given band pass filter.
We then get

(^) BB^22 ()() 2.
4
ab
Nk k
π
ωω ω
π

⎡⎤⋅

Δ= ⋅ +Δ −⎣⎦⋅ (3.118)

In the case of kB not being too small, an approximate expression for the modal density is

(^22) BB()() ()B 2
()
444
NSkkS k S m
n
B
ωω ω
ω
ωπ ω πω π

Δ∂⎡⎤+Δ −

==⋅⎢⎥≈⋅ =⋅

ΔΔ ∂⎢⎥⎣⎦

(3.119)

or

() ,
2

NS m

nf

fB

Δ

==⋅

Δ

(3.120)

where S is the plate area. As we can see, the modal density of thin plates is frequency
independent. This is again not the case for other types and shapes of structure; see e.g.
Blevins (1979).
Example The bandwidth of a third-octave-band filter is approximately Δf ≈ 0.23⋅f 0 ,
where f 0 is the centre frequency of the band. Choosing f 0 = 1000 Hz and taking the
concrete slab used in Figure 3.22 as an example we get, even at this relatively high
frequency, ΔN ≈ 14. In contrast to this, a room above this concrete slab (taken as the 24
m^2 floor of the room) having a ceiling height 2.5 m will have approximately 4 400 modes
inside the same bandwidth! The latter number is calculated using the expression

(^) room 3 2
0
4
,
V
N ff
c
π
Δ≈ ⋅⋅Δ
where V is the room volume. This expression is derived using an analogous procedure to
the one used above taking into account the equation for the natural frequencies of a
three-dimensional air-filled space (see section 4.4.1).

3.7.3.6 Internal energy losses in materials. Loss factor for bending waves

In a previous chapter, when dealing with oscillations in simple mass-spring systems, we
introduced the loss factor by way of a complex stiffness. In a similar way we shall, for
bending waves in a given structure, define a complex bending stiffness B′ :

BB'1j,= ( +⋅η) (3.121)

where η is the loss factor. By formal definition, as found in the literature, it is given by
the ratio of the mechanical energy Ed dissipated in a period of vibration to the reversible
mechanical energy Em: