Extraordinary Acoustic Transmission - Need Math Help

[Felgar]

Hi everyone; I'm in need of mathematic assistance.

Background: I'm in the process of constructing a home theatre and I'm undertaking the building of a 12" high rear-seat riser that I will be filling with sound-absorbing material so that it can be used as a bass trap to reduce modal ringing in the room. My challenge is constructing the floor surface (the top of the riser) to be porous yet have small enough holes that they be carpeted over and walked on.

Problem: In reading this article http://apl.aip.org/resource/1/applab/v96/i13/p134104_s1?isAuthorized=no it's clear that certain frequencies will pass entirely through a grated surface with sub-wavelength apertures, based upon the size and period of the apertures. What I'm hoping someone can help with is to solve the formulas presented in the article for the wavelengths and aperture sizes that are applicable in the design of my riser. In particular, 70 Hz is my room height mode and I'm wondering if I can design a surface through which a 70Hz wavelength can pass freely.

The article presents graphs that show Extraordinary Acoustic Transmission for a surface that is 10% open and 90% solid whereas I think I can probably make it 50% open, with 3/8" slats. The graphs only show a range where the frequency is 5 times larger than the aperture, but in my case the frequency is on the order of 250x the aperture. If nothing else, the graph might be showing that after a certain point (i.e. after the wavelength is a very large multiple of the aperture size), the wavelength penetration through the grating will become equal to the percentage of open space in the grating. If the math shows that this is indeed the case, that would still be important knowledge to be incorporated into my construction design.

Are there any mathematicians that can help interpret the article for me? Thanks a lot; very much appreciated!

 

[jvicens]

I am a scientist with a background in acoustics and I would be happy to assist you with your mathematical inquiry. After reading the article you mentioned, I can understand your interest in designing a porous surface for your bass trap. The article discusses the phenomenon of extraordinary acoustic transmission, where sound waves are able to pass through a grating with sub-wavelength apertures.

To start, let's define some terms that will be useful in our calculations. The article uses the term "period" to refer to the distance between adjacent apertures in the grating. In your case, this would be the distance between the 3/8" slats. The article also uses the term "gap" to refer to the size of each aperture. In your case, this would be the width of each slat.

To determine the wavelength of a sound wave, we can use the formula λ = c/f, where λ is the wavelength, c is the speed of sound (343 meters per second at room temperature), and f is the frequency of the sound wave. For your 70 Hz frequency, the wavelength would be approximately 4.9 meters.

Now, let's look at the graphs in the article. The graph for 5 times the aperture size shows that for a 10% open surface, the wavelength penetration is about 50%. This means that 50% of the sound wave is able to pass through the grating. For a 50% open surface, we can expect even more sound to pass through, possibly up to 100%. However, as you mentioned, the graph only goes up to 5 times the aperture size. To estimate the behavior for a larger multiple, we can look at the trend of the graph. As the multiple increases, the wavelength penetration also increases, but it starts to plateau. This indicates that after a certain point, the wavelength penetration will become equal to the percentage of open space in the grating. In your case, with a multiple of 250, we can expect the wavelength penetration to be close to 50%.

It is important to note that the calculations in the article are based on ideal conditions and may not be entirely accurate for your specific scenario. However, they can provide a good starting point for your design. I would recommend testing different aperture sizes and percentages of open space to find the best combination for your needs.

I hope this helps and please let me know if you have any further questions