Sound absorbers - Building Acoustics - page 171

2

ρω 1 ⎛ 8 ωρ a ⎞

⎟ ()

0 0

Z ≈ + ⋅ j ,

(5.37)

⎝

a

radiation

2 π c S 3 π

⎠ 0

where S and a are the piston radius and area, respectively. The imaginary part of the
impedance is cast into this form to make a comparison with Equation (5.22) . It is easily
seen that it gives an added mass equivalent to an increase Δ d = 8 a /3π ≈ 0.85⋅ a in the
length of the neck. It is commonly assumed that the same correction may be applied to
both ends of the neck, thereby setting the effective length of the neck to
be . This so-called end correction will certainly depend on the actual cross
sectional shape being different for noncircular openings, for slits etc. Data for these other
shapes are listed in the literature. An important case in practice is long and narrow slits
and we shall therefore include this case (see below).

dd ' =+ ⋅ 1.7 a

How good is a single Helmholtz resonator when it comes to absorption? To arrive
at the maximum absorption we have to adjust the system to make the two resistive terms,
given by the Equations (5.33) and (5.37) , equal at the resonance frequency. Doing this,
the effective absorption area of the resonator opening will be given by

2
0

λ
π

A = ,

(5.38)

max

2

where λ 0 is the wavelength at resonance. The effective absorption area is therefore much
larger than the physical size of the opening. However, the resonator will have a small
bandwidth, e.g. the relative bandwidth Δ f / f 0 could be as small as 0.01, which implies a Q
factor as high as 100. In practice, one normally designs for a more broadband absorber
by adding some resistance in the opening, which may be in the form of a porous material,
a metal grid etc. To retain the absorption area one has to increase the volume of the
resonator, which again means that the opening has to be adjusted to maintain the chosen
resonance frequency.

5.4.1.5 Distributed Helmholtz resonators
Single Helmholtz resonators are used in many practical cases where the task is to remove
single frequencies. More commonly used are the types that we may name distributed
Helmholtz resonators, which we referred to in the introduction (see section 5.2.3) . These
are absorbers using perforated panels, perhaps in the form of slats, mounted at a certain
distance A from a hard wall or ceiling. To each opening (see Figure 5.11 a)), or to each
slit, (see Figure 5.11 b)), we then allocate a part of the cavity volume that is used when
calculating the resonance frequency by Equation (5.23) . We then get

c ε S

0

f = with ε = ,

(5.39)

0

2 π A ( dd +Δ ) S

0

where ε is the perforation or the “porosity” of the panel. The distance A is here assumed
to be much less than the wavelength. It should also be noted that for the assumption of a
locally reacting absorber to be valid, the cavity volume has to be subdivided to minimize
lateral wave propagation.
In the same way as for a single resonator one must introduce some resistance in
addition to the natural viscous losses to obtain an absorber for practical use. One will
certainly also have some additional viscous losses due to the air movements on the panel
surfaces around the holes or slits but this is normally not enough. The exceptions to this