Helmholtz resonators

Sound absorption and sound absorbers

6.5 Helmholtz resonators

Sound absorption and sound absorbers 171 range so as to give them the desired absorption characteristics. By a suitable combination of several types of resonance absorbers in a room the acous- tical designer is able to achieve a prescribed frequency dependence of the reverberation time. The most common application of vibrating panels is to effect a low-frequency balance for the strong absorption of the audience at medium and high frequencies, and thus to equalise the reverberation time.

This is the reason for the generally favourable acoustical conditions which are frequently met in halls whose walls are lined with wooden panels or are equipped with similar components, i.e. walls or suspended decoration ceilings made of thin plaster. Thus it is not, as is sometimes believed by lay- men, a sort of ‘amplification’ caused by ‘resonance’ which is responsible for the good acoustics of many concert halls lined with wooden panels. Like- wise, audible decay processes of the wall linings, which are sometimes also believed to be responsible for good acoustics, do not occur in practical sit- uations although they might be possible in principle. If a resonance system with the relative half-width (reciprocal of theQ-factor) ω/ω₀ is excited by an impulsive signal, its amplitude will decay with a damping constant δ = ω/2 according to eqn (2.31); thus the ‘reverberation time’ of the resonator is

T=13.8 ω =2.2

f (6.11)

To be comparable with the reverberation time of a room which is of the order of magnitude of 1 s, the frequency half-widthf must be about 2 Hz.

However, the half-width of a resonating wall lining is larger by several orders of magnitude.

In Fig. 6.5, the absorption coefficients of a wooden wall lining and of a resonance absorber with perforated panels are plotted as functions of the frequency, measured at omnidirectional sound incidence.

Figure 6.5 Absorption coefficient of resonance absorbers at random sound incidence (measured in a reverberation chamber, see Section 8.7): (a) wooden panel, 8 mm thick,M=5 kg/m², 30 mm away from rigid wall, with 20 mm rock wool behind; (b) panels, 9.5 mm thick, perforated at 1.6% (diameter of holes 6 mm), 5 cm distant from rigid wall, air space filled with glass wool.

if it were not present:

A=P_abs

I₀ (6.12)

This definition is similar to that of the scattering cross-section in eqn (2.46).

When calculating the reverberation time of a room, absorption of these types of absorbers is taken into account by adding their absorption areasA_i to the sumα_iS_i(see eqn (5.24a)). The same holds for all other formulae and calculations in which the total absorption or the mean absorption coefficient αof a room appears, as for instance for the calculation of the steady-state energy density as described in Section 5.5.

In this section we are discussing discrete sound absorbers with pronounced resonant behaviour. Their characteristic feature is an air volume which is enclosed in a rigidly walled cavity and which is coupled to the surrounding

Sound absorption and sound absorbers 173 space by an aperture, as shown in Fig. 6.6a. The latter may equally well be a channel or a ‘neck’. The whole structure is assumed to be small compared with the wavelength of sound and thus it has one single resonance in the interesting frequency range. It is brought about by interaction of two ele- ments: the air contained in the neck or in the aperture which acts as a mass load, and the air within the cavity which can be regarded as a spring counter- acting the motion of the air in the neck. Arrangements of this type are called

‘Helmholtz resonators’; examples of these are all kinds of bottles, vases and similar vessels. In ancient times, as ‘Vitruv’s sound vessels’, they played an unknown, possibly only a surmised, acoustical role in antique theatres and other spaces.

Figure 6.6 depicts a Helmholtz resonator along with its schematic presen- tation. The neck has a lengthland a cross-sectional areaS; its opening is flush with the surface of a rigid wall of infinite extension. The basic parameters of the resonator are the massM=ρ₀lsof the air enclosed in the neck, and the volumeV of the attached cavity. The shape of the latter is of no relevance.

Furthermore, there are some internal losses represented by a resistanceR₀,

Figure 6.6 Helmholtz resonator: (a) realisation; (b) schematic.

and the opening is loaded by its radiation impedanceR_rdefined by P_r=R_rv²

where—as usual—the bars indicate time averaging. (P_r=radiated power, v= air velocity in the neck). Since the lateral dimensions of the opening are small compared with the wavelength, P_r can be substituted from eqn (1.31) usingQˆ²=(Sv)²=2S²v². However, we must keep in mind that the re-radiation of sound from the aperture is restricted to the half space only and hence is a factor of 2 higher than in eqn (1.31). Thus the radiation resistance is

R_r=ρ₀ω²S²

2πc =2πρ₀c S

λ 2

(6.13) Suppose the resonator is working at its resonance frequencyω₀. In this case all the reactive parts of the mechanical load will mutually cancel each other.

Hence, the ratio of the total forceF acting on the piston and its velocityv is real:

F

v =R₀+R_r (6.14)

The energy converted to heat per second by the internal friction is P_abs=R₀v²= R₀

(R₀+R_r)²F² (6.15)

For a given radiation resistance,P_abs is maximum ifR₀is equal toR_r. The force exerted externally on the piston arises from the sound field. Ifpdenotes the sound pressure, the force isF=2pS. The factor 2 takes into account the reflection from the rigid wall surrounding the piston. By application of eqn (1.28) we can express the sound pressure and hence the force in terms of the intensityIof the incident sound wave:

F²=4S²p²=4ρ₀cS²I (6.16)

Now we are ready to evaluate eqns (6.12) and (6.15) with R₀=R_r by substituting from eqns (6.13) and (6.16), and we obtain as a final result

P_abs= λ²₀

2πI and A_max= λ²₀

2π (6.17)

whereλ₀is the wavelength corresponding to the resonance frequency.

Sound absorption and sound absorbers 175 For the frequency dependence of the absorption area, we can essentially adopt eqn (2.32):

A(ω)= A_max 1+Q²_A

ω/ω₀−ω₀/ω2 (6.18)

with

Q_A=Mω₀

2R_r (6.18a)

The (angular) resonance frequency of the system is given by the general formula

ω²₀= s M

Here sis the elastic stiffness of the air enclosed in the resonator volume.

To calculate it we suppose the air in the neck to be displaced by δx; the corresponding change in air pressure isp_i. According to the usual definition of the stiffness,

s= force

displacement= −p_iS δx

On the other hand, the pressure change is associated with a changeδρin air density, which in turn is due to the volume change Sδx. These quantities are related to each other by (see eqns (1.4) and (1.6))

Sδx V = −δρ

ρ₀ = − p_i ρ₀c²

Thus we obtain for the stiffness of the air cushion:

s=ρ₀c²S² V

Now we can insertM=s/ω₀²=ρ₀c²S²/ω²₀Vinto eqn (6.18a) and expressR_r by eqn (6.13) with the result

Q_A=π V

c ω₀

₃

=π V

λ₀ 2π

₃

(6.19) We observe that the maximum absorption area of a resonator matched to the sound field is fairly large, according to eqn (6.17). On the other hand,

the Q-factor given by eqn (6.19) is very large too, which means that the relative frequency half-width, which is the reciprocal of theQ-factor, is very small, i.e. large absorption will occur only in a very narrow frequency range.

This is clearly illustrated by the following numerical example: suppose a resonator is tuned to a frequency of 100 Hz corresponding to a wavelength of 3.43 m. This can be achieved conveniently by a resonator volume of 1 litre.

If it is matched, the resonator has a maximum absorption area, which is as large as 1.87 m². ItsQ-factor is—according to eqn (6.19)—about 500; the relative half-width is thus 0.002. This means, it is only in the range from 99.9 to 100.1 Hz that the absorption area of the resonator exceeds half its maximum value. Therefore the very high absorption in the resonance is paid for by the exceedingly narrow frequency bandwidth. This is why the application of such weakly damped resonators does not seem too useful.

It is more promising to increase the losses and hence the bandwidth at the expense of maximum absorption.

Finally, we investigate the problem of audible decay processes, which we have already touched on in the preceding section. The ‘reverberation time’

of the resonator can again be calculated by the relation (6.11) T=13.8

ω =13.8Q_A ω₀

In many cases this time cannot be ignored when considering the reverberation time of a room. What about the perceptibility of the decay process?

It is evident from the derivation of the absorption area, eqn (6.17), that the same amount of energy per second which is converted to heat in the interior of the resonator is being reemitted by it, since we have assumed R₀=R_r. Its maximum radiation power is thusI(λ²₀/2π),Ibeing the intensity of the incident sound waves. Thus the intensity of the re-radiated sound at distanceris

I_s= λ₀

2πr ₂

I= c

rω_{0 2}

I (6.20)

Both intensities are equal at a distance

r= c ω₀ =55

f₀ metres (6.21)

The decay process of the resonator is therefore only audible in its immediate vicinity. In the example mentioned above this critical distance would be 0.55 m; at substantially larger distances the decay cannot be heard.

Sound absorption and sound absorbers 177