Resonance absorbers

Sound absorption and sound absorbers

6.4 Resonance absorbers

Sound absorption and sound absorbers 167 The correction 2b, known as the “end correction”, accounts for the fact that the streamlines (see Fig. 6.2b) cannot contract or diverge abruptly but only gradually when entering or leaving the hole.

For circular apertures with radius a and with relatively large lateral distances the end correction is given by

b=π

4a (6.7)

Finally, the absorption coefficient of a perforated panel is obtained from eqn (6.3) by substitutingMfrom eqn (6.4).

Very often perforated plates are so light that they will vibrate as a whole when a sound waves strikes them. In this case M in eqn (6.5) must be replaced by the effective mass

M= 1

M+ 1 M₀

(6.8) Mis the equivalent mass per square metre of the panel itself after eqn (6.4) andM₀is the specific mass of the panel material.

If frictional forces within the holes and other loss processes are dis- regarded, the absorption coefficient given by eqn (6.3) represents the fraction of incident sound energy which is transmitted by a wall or a perforated panel. It thus characterises the sound transparency of the wall. Let us illustrate this by an example: to yield a transparency of 90%, ωM/2ρ₀c in eqn (6.3) must have the value 1/3. At 1000 Hz this is the case with a mass layer with an (equivalent) mass per unit area of about 45 g/m². If realised by a perforated panel, this can be achieved, for instance, by a 1-mm thick sheet with 7.5% perforation and with holes having a diameter of 2 mm.

Figure 6.3 Absorption coefficient (calculated) of resonance absorbers as a function of fre- quency at normal sound incidence, forMω=10ρ₀c. Parameter is the ratio r_s/ρ₀c.

Forr_s> ρ₀cthe maximum absorption is less than unity and the curves are broadening. This is qualitatively the same behaviour as that of a porous layer which is arranged at some distance and in front of a rigid wall (compare Fig. 2.7). The frequency-selective absorption characteristics make this device a useful tool for the control of reverberation, mainly in the low- and mid- frequency range. In practical applications the ‘membrane’ consists usually of a panel of wood, chipboard or gypsum (see Fig. 6.4a). The vibrational losses occurring in this system may have several physical reasons. One of them has to do with the fact that any kind of panel must be fixed at certain points or along certain lines to a supporting construction which forces the panel to be bent when it vibrates. Now all elastic deformations of a solid, including those by bending, are associated with internal losses that depend on the material and other circumstances. In metals, for instance, the intrinsic losses of the material are relatively small, but they may be substantial for plates made of wood or of plastic. If desired the losses may be increased by certain surface layers, or by porous materials placed in the space between the panel and the rigid rear wall.

According to the discussion of the preceding section, the mass layer can also be realised as a perforated or slotted sheet (see Fig. 6.4b). This device is often called a Helmholtz resonator, although the original Helmholtz res- onator consists of a single, rigid-walled cavity with a narrow opening, as described in the next section. If the apertures in the sheet are very narrow, the frictional losses occurring in them may be sufficient to ensure the low

Sound absorption and sound absorbers 169

Figure 6.4 Resonance absorbers: (a) with vibrating panel; (b) with perforated panel.

Q-factors that are needed for high efficiency of the absorber. Such devices are known as microperforated absorbers and have found considerable atten- tion during the past two decades,² prompted by the technical progress in drilling numerous thin holes. When applied to transparent materials such as acrylic glass, they offer the possibility of manufacturing transparent reso- nance absorbers, although it may prove difficult in practice to keep apertures with diameters in the sub-millimeter range free of dust particles and other obstructions.

For wider holes it is usually necessary to provide for additional losses.

This can be achieved, for instance, by covering the holes with a porous fabric. Another method of adapting the magnitude ofr_s to a desired value is to fill the air space behind the panel partially or completely with porous material.

For normal sound incidence the (angular) resonance frequency of this absorber is given by eqn (2.28), provided the space behind the membrane or perforated panel is empty. For its practical application, the following form may be more useful:

f₀= 600

√(Md) Hz (6.9)

whereMis in kg/m²anddin cm. If the air space behind the panel is filled with porous material, the changes of state of the air will be rather isothermal than adiabatic. This means that eqn (1.4) is no longer valid and has to be replaced with p/p₀ =δρ/ρ₀; the sound velocity becomes c_iso=√

(p₀/ρ₀).

Accordingly, the numerical value in eqn (6.9) is reduced by a factor√ κ, i.e.

by about 20%, and is 500.

This formula is relatively reliable if the mass layer consists of a perforated panel, as in Fig. 6.4b, or a flexible membrane. Then the way in which the mass layer is fixed has no influence on its acoustical effect, at least as long as the sound waves arrive frontally. Matters are different for non-perforated wall linings made of panels with noticeable bending stiffness. Since such pan- els must be fixed in some way, for instance on battens which are mounted on the wall (Fig. 6.4a), their vibrations are controlled not only by the air cush- ion behind but also by their bending stiffness. Accordingly, the resonance frequency will be higher than that given by eqn (6.9), namely

f₀=√

(f₀²+f₁²) (6.10)

with f₁ denoting the lowest bending resonance frequency of a panel sup- ported (not clamped) at two opposite sides. The typical range of f₁ is 10–30 Hz, whereas f₀ is typically 50–100 Hz. This shows that the influ- ence of the bending stiffness on the resonance frequency can be neglected in most practical cases, and eqn (6.9) may be applied to give at least a clue to the actual resonance frequency.

Another consequence of the strong lateral coupling between adjacent elements of an unperforated panel is that the angle dependence of sound absorption is more complicated than that in eqn (2.17) (with frequency- independent ζ). In fact, a sound wave with oblique incidence excites a forced bending wave in the panel, even if it is of infinite extension. The propagation of this wave is strongly affected by the supporting construc- tion and the air layer behind the panel. Since general statements on the way it influences the absorption coefficient and its dependence on the inci- dence angle are not possible, we shall not discuss this point in any further detail.

For resonators with perforated panels, lateral coupling of surface elements is affected by lateral sound propagation in the air space behind the panels.

It can be hindered by lateral partitions made of rigid material, or by filling the air space with porous and hence sound-absorbent materials like glass or mineral wool. Neighbouring elements of the panel can then be regarded as independent; the wall impedance and similarly the resonance frequency is independent of the direction of sound incidence. In any case it is difficult to assess correctly the losses of a resonance absorber which determine its absorption coefficient. Therefore, the acoustical consultant must rely on his experience or on a good collection of typical absorption data. In cases of doubt it may be advisable to measure the absorption coefficient of a wall lining by putting a sufficiently large sample of it into a reverberation chamber (see Section 8.7).

Resonance absorbers of the described type are typically mid-frequency or low-frequency absorbers. Their practical importance stems from the possi- bility of choosing their significant data (dimensions, materials) from a wide

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.