Thanks Dan. You hit the nail on the head, or rather identified exactly the point that has been holding me back on the article about gas flow resistivity. Specifically the effect of gaps between porous absorbers and reflective surfaces. I have not been able to determine the best manner in which to explain the effect and that gaps greater than the thickness of the porous material provide flat absorption.
There. I wrote the important thing. Now I just thave to expain it in an effective manner.
In the following explanation I will try to work from the basics in a way that other readers can hopefully understand the priniciples involved also.
The classic way that the effect of gaps is explained is by use of graphs of thin absorbent material spaced away from a wall. The graphs show high values of α where the distance from the wall is 1/4 wavelength and 0 when the wavelength is 1/2. In other words the porous material is effective only where the particle velocity is high. These graphs are appropriate for thin material. The porous materials that we consider when discussing the use of gaps are not thin at the wavelengths significant to us. Therefore the graph is not accurate for our use of gapping!
Gapping is used to lower the effective frequencies of the sound absorption. It is usefull to start with the effect of thickness of homogenous porus material against a solid surface. This the mounting of material used in the reporting of the absorption of materials as used traditionally in studios. With typical porous material, using 703 for the example, at 4" thickness, α is 1 down to ~250 Hz and usually considered effective down to ~125 Hz. At 250 Hz the wavelength is 4.52 feet. Dividing the thickness of the material by the wavelength (.333/4.52)gives us a ratio of .0737, or ~7%. So the thickness of a porus absorber has to be at least 7% of the wavelength for flat absorption. If we consider 703 material absorption at 125HZ to be practically 1, this gives a ratio of 3.5%.
Remember, the previous paragraph deals porous material against a solid surface. This is the area where the particle velocity is lowest in the sound wave. The efectiveness of a non thin absorber is not significantly reduced when located in the relatively low region of particle velocity.
Having shown what the depth of a porous absorber has to be in order to be effective, this leaves the question of the required material depth to gap ratio for effective absorption. This is not as clear as the overall depth calculation due the physics involved and some other non-intuitive factors. The usual belief is that the path of sound sound through an absorber is straight through the material. This is also named the normal incidence. However when sound impinges on an absorber at a non normal angle, the path is greater. The significance of this is a reduction, up to a complete removal, of the point on the thin absorber graph where no absorption occurrs.
When sound travels in air, it doing so in an isothermal manner. That is that at the points where the presuure increases, and the temperature (the combined gas law), the additional heat remains that area. That is, there is movement of the energy in the sound wave. In porous material, the material conducts the heat away from the ares of high temperature to areas of lower temperature.
This is called adiabatic. The ratio of the square root of specific heats of air for constant volume vs constant pressure is the same as the ratio of the speeds of sound in air when traveling isothermally vs. adiabatically. The effect is that that porous absorbers behave with effective thicknesses ~120% greater than the physical depth. The practical result is that for acoustic matching to the peak velocity of a sound wave and covering the full 1/4 cycle, the depth of the gap is 1.2 times the depth of the depth of material.
There is of course the variable sound path length from the various angles of incidence also. So the true effective depth for a gapped porous absorber is more than 1.2 times the material thickness. This leads to the question of how much more? In acoustics, we have the ultimate arbiter of test data. Gapped porous absorbers are used in thousands of spaces with gap to depth ratios up to 20:1. These absorber systems are called acoustic ceilings. This sort of mounting is calld E-405. It consists of absorbent material suspended 405 mm (16") away from a solid surface. The acoustic tiles are as thin as ~20 mm(3/4"). Studying test data on such mounted materials does not show any dip at the 1/2 wavelength frequency implied by the thin absorber graph.
An example of a purpose built absorber using a 2:1 gapping ratio is in the Heinieken Music Hall, construction details in fig 5 in Acoustics for Large Scale Indoor Pop Events
. There is an internationally recognized studio designer, who in the process of designing a wolrd class facility did testing on gaps and used gap to material ratios of up 2.2:1. He is a rather quiet person regarding his work and out of respect for him, I am not disclosing his name or the studio.
The point to answer your question, is no, 1:1 is not a limiting ratio for gap to absorbent material ratios. Ratios up to 20:1 are used regularly in non-critical acoustic spaces, and ~2:1 for critical acoustic spaces, including studios.
I hope this helps with the ostensible question. Reaction to this post and thread and this post will help me compose the major piece on soound absorption and porous absorbers. You got me started in getting past the stumbling block Dan.