RE: Low Frequency Effectiveness of 4" Absorption Panels
Beranek, Keele, et. al. show that modeling a surface (umm, let's say a 4" thick panel such as that discussed) as a collection of co-planar, massless plates (upon which sound impinges), each connected to a mass (freestanding) that is, in turn, connected to a damper is a means by which
to model the acoustic energy-dissipating behavior of said panel. Back of the envelope, 1st order approximation kind of stuff.
The high frequency (where resistive absorption occurs) performance of panels, such as that discussed in this thread, is generally well known, intuitively easy to grasp and frequently documented, whereas the low frequency (LF) (reactive) performance of the same panel is often ignored or otherwise buried in the charts & graphs. Conventional thinking seems to focus on the resistive component of a panel's boundary impedance, ignoring the reactive component.
For diverging spherical sound waves, sound sources located at a distance greater than the wavelength (λ) of interest, the acoustic impedance, Z, is largely resistive (real) and for sources located a distance much smaller than λ, Z is complex and largely reactive (i.e. the acoustic pressure & particle velocity are not always in phase). Referring again to the panel model mentioned in the first paragraph, at high frequencies, the mass is essentially immobile and the energy is absorbed by the damper. At low frequencies, the entire mechanism moves, providing a mass reactance to the impinging wave.
Having said that, the greater the match (at any frequency of interest) between boundary mechanical impedance and the specific acoustic impedance of the incoming sound wave, the more effective said boundary is in reducing the reflection coefficient.
So, can you get LF (resistive) absorption out of a 4" panel? No, but you can get mass reactance! How much LF-taming you can get in practice owing to mass reactance depends on a number of factors or variables. In practical terms, the results are typically minimal to vanishingly small (when compared to the high frequency absorption capabilities of the panel in question) and often difficult to predict. Another words, a 4” panel really isn’t a very effective device for the dissipation of low frequency acoustic energy - even factoring in its mass reactance. This is In contrast to, say, a Helmholtz resonator, an old, well-proven technology that behaves in very predictable ways when dealing with LF acoustic energy
To illustrate an approach that fully exploits mechanical HF resistance & LF reactance, I've attached a photo of an anechoic chamber (Bruel & Kjaer) referred to as an "acoustic jungle" that features not the usual wedges, but a huge collection of contrivances that are purpose built to maximize the matching of boundary and acoustic impedance at all frequencies. Effectively exploiting both resistive and reactive behavior, there is no where to be seen the usual monster-size foam wedges. Dr. techn. Per V. Brüel written quite a bit about the effectiveness of the "acoustic jungle" approach, something that's more popular in Europe (particularly in Germany) than here.
If you're interested in reading more, try these resources:
J. Wang and B. Cai, "Calculation of Free-Field Deviation in an Anechoic Chamber," J.
Acoust. Soc. Am., vol. 85(3), pp. 1206-1202 (1989 Mar.).
D. B. Keele, Jr., "Low-Frequency Loudspeaker Assessment by Nearfield Sound-
Pressure Measurement," J. Audio Eng. Soc., vol. 22, pp. 154 - 162 (1974 Apr.).
Industrial Acoustics Company, "Anechoic Rooms," Bulletin AN-33RI, (Bronx, NY,
USA, 1962).
L. E. Kinsler and A. R. Frey, Fundamentals of Acoustics (Wiley, New York, 1962).
W. Koidan and G. R. Hruska, "Acoustical Properties of the National Bureau of
Standards Anechoic Chamber, "J. Acoust. Soc. Am., vol. 64(2), pp. 508-516 (1978 Aug.).
E. Skudrzyk, The Foundations of Acoustics (Springer-Verlag, New York, 1971).
P. M. Morse and K. U. Ingard, Theoretical Acoustics (Princeton University Press,
New Jersey, 1968).
L. L. Beranek, Acoustics (American Institute of Physics, New York, 1986).
D. B. Keele, Jr., "Anechoic Chamber Walls: Should They be Resistive or Reactive at Low Frequencies?" Audio Eng. Soc., Preprint 3572 (G2-2) (March 1993) Berlin
Germany
G. Rasmussen, "Anechoic Sound Chambers," Bruel & Kjaer design document (1972
Sept.).
C. Fog, "Anechoic Chamber at Bruel & Kjaer," Proceedings of the Nordic Acoustical
Meeting at Aelborg, Denmark, pp. 285-288 (1986 Aug.).