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Resonant Sound Absorbers

In Part One, we learned how to analyze an acoustical treatment in which perforated metal sheet is mounted over an air- space containing sound absorptive material, in order to make a "Tuned Resonant Absorber" .That is, by the use of a nomogram, we could determine the frequency of resonance where the sound absorption would be especially great; or we could choose the dimensions of the treatment to target a particular frequency range of interest.

Nothing was said there about how much sound absorption would be achieved at the resonance frequency nor about how broad the targeted frequency range would be. We take up these matters here.

Sound Absorption At The Resonance Frequency

In a tuned resonant sound absorber, the sound absorption reaches a maximum value, amax, at the resonance frequency, fR, falling off to lower values at higher and lower frequencies. We can control this maximum value of absorption by the choice of the sound absorptive material with which the airspace is filled. Usually, that material will be a kind of porous blanket or board, made of glass fiber or mineral fiber .

The maximum value of absorption depends only on the flow resistance of that material, and not on any of the physical dimensions of the sound absorptive treatment (such as the depth of airspace, perforation diameter, percent open area, etc.).

Flow Resistance, Flow Resistivity, and Resistance Ratio

The flow resistance of a piece of material tells us how easy it is for air to move through the material. The flow resistance depends upon the density of the fibrous material (Ib/sq ft) and the fiber diameter: generally, the heavier the blanket and the finer the fibers, the higher the flow resistance.

And, naturally, thicker layers have more flow resistance than thin ones. With experience, one can even learn to make a pretty good guess at the flow resistance of a material by seeing how hard it is to blow one's breath through the material. But for our purposes, we will rely on the measured values of flow resistance for some commonly available fibrous materials.

There's good news and bad news here, however. The bad news is that the manufacturers of fibrous materials don't worry much about the flow resistance of their products, so it's not always easy to find accurate information on this parameter .

The good news is that the acoustical behavior of our tuned resonant sound absorbers isn't critically dependent on the exact value of the flow resistance of the filling in the air cavity. We can miss the design goal quite a bit and it won't make much difference.

But first we have to discuss how to characterize the flow resistance of a layer of material. It is usually done by means of a resistance ratio that tells how much harder (or easier) it is for the sound pressure to push air through the layer in question than to push it through the air itself.

That probably sounds peculiar, because it may not have occurred to you that sound actually encounters some resistance in moving through the air. In fact, there is a "characteristic impedance" that relates the pressure in a sound wave to the corresponding particle velocity in the air: it is given by the product of the density of the air, p (gm/cc), and the propagation velocity of sound, c (cm/sec):

Characteristic Impedance = pc = 41 cgs rayls. We always relate the flow resistance, R, of a layer of material to the characteristic impedance of the air, pc, by forming the resistance ratio R/pc. If a layer of material has a flow resistance such that R/pc = 1, then a sound wave will not recognize the existence of that material when it is encountered, because it can't tell the difference between this material and air . If the value of R/pc is either substantially greater or less than unity, then the sound wave will "notice" the layer, and tend to be reflected from it rather than entering and passing through it.

Important distinctions:

Every fibrous material has a property of its own called the flow resistivity , E, which gives the flow resistance per inch of thickness. (We are talking now about the material, itself, not a particular blanket of that material.)

Thus, if a certain type of glass fiber has a flow resistivity E = 60 cgs rayls/inch, then a 2" blanket of the material will have a flow resistance of R = 2 x 60 = 120 cgs rayls. And for this blanket the value of R/pc = 120/41 = 2.93. Remember: the flow resistance E is a property of the material, while the flow resistance R is a property of a blanket of the material with a particular thickness. The resistance ratio R/pc relates the flow resistance of a given blanket to the characteristic impedance of the air.

Now, at last, we are in a position to consider the maximum amount of sound absorption achieved at the resonance frequency of our tuned absorber. As we mentioned above, it depends only on the value of R/pc for the filling in the airspace: 1 IX = max Y2 + V4 (R/pc + pc/R) Table 3 gives values for IXmax (at the resonance frequency) corresponding to different values for R/pc of the cavity filling:

Table 3: Maximum attainable sound absorption (at the resonance frequency), as a function of the flow resistance ratio of the filling material.

~ 0.1 0.2 0.5 0.7 1.0 1.5 2.0 3.0 4.0 5.0

~ 0.33 0.56 0.89 0.97 1.00 0.96 0.89 0.75 0.64 0.56

As we said above, the maximum absorption coefficient at resonance in a tuned absorber is not very sensitive to the filling material: any value of R/pc from 0.5 to 2.0 will yield a value of amax of 0.89 or greater.