Applied Acoustics - decouple a speaker and a wooden table

[Rick00]

If several sheets made of some kind of rubber are being used as acoustic insulators, which configuration would be more efficient to decouple a speaker and a wooden table, absorbing most vibrations? (Ps: Image not drawn to scale)

 

[berkeman]

If several sheets made of some kind of rubber are being used as acoustic insulators, which configuration would be more efficient to decouple a speaker and a wooden table, absorbing most vibrations? (Ps: Image not drawn to scale)View attachment 219348

Welcome to the PF.

What do you mean "not drawn to scale"? With the scale shown, I think there is an answer, but if things are not to scale, how are we supposed to answer?

Also, is this for schoolwork? Or are you an audiophile wanting to build something?

 

[sophiecentaur]

With the scale shown, I think there is an answer,

I think it's ok to make assumptions here. We want minimal energy to reach the table via longitudinal waves.
I reckon it's all a matter of Impedances. You want a mismatch between the loudspeaker and the support and you want a mismatch between the support and the table so minimal energy reaches the table from the loudspeaker. So the massive / wide end of the support should be against the speaker and the narrow, high compliance end should be against the table.
I think I have got the principle the right way round?

 

[berkeman]

So the massive / wide end of the support should be against the speaker and the narrow, high compliance end should be against the table.

My calculation came out different, but I could be wrong.

However, there are other configurations that would follow the principle that you mention, that the OP did not offer as an option. We should probably wait for the OP to tell us if this is schoolwork before we open the kimono any more...

 

[Rick00]

Hi, this is not a schoolwork :) It's something I might put into practice to solve a real problem (amateur studio), but advanced physics/math (undergrad level) is welcome. About the scale, I didn't draw it with total accuracy, however I'd say the differences can be neglected and everything that is relevant to the problem statement is shown in the image (I exaggerated the thickness and the width difference, so that the sheets are thinner, more similar, and also there are more of them). In practice, these would be the only 2 options, but feel free to discuss other rubber sheet configurations.

 

[berkeman]

What have you found in your reading so far about acoustic isolation? There are lots of great resources for this. Please do some of that reading,and post links to the parts that you are having trouble understanding. Thanks.

 

[sophiecentaur]

Hi, this is not a schoolwork :) It's something I might put into practice to solve a real problem (amateur studio), but advanced physics/math (undergrad level) is welcome. About the scale, I didn't draw it with total accuracy, however I'd say the differences can be neglected and everything that is relevant to the problem statement is shown in the image (I exaggerated the thickness and the width difference, so that the sheets are thinner, more similar, and also there are more of them). In practice, these would be the only 2 options, but feel free to discuss other rubber sheet configurations.

If I were trying to do this, I would just look at as many images on Google as I could and follow up the best looking ones, Thousands (or more) people have worked on this and is there much point in re-inventing the wheel?
I guess the ultimate mounting for a speaker would be on a long rubber bungee, suspended from the ceiling. And that would actually go directly against my original post idea which will provide the most rigid support. Rubber cones would be good as there is no risk of simple resonance due to a non linear elastic characteristic. (The original Austin Mini suspension used rubber cones for suspension and they were quite successful, I believe).

 

[Rick00]

Concerning acoustics, I'm familiar with the production of sound and processes of reflection, interference, etc...at the level of some undergrad textbooks (like Feynman lectures 1), but I have to say I am not familiar with the concept of impedance applied to acoustics. Indeed, I did some research to find out what is the standard solution to the problem and "Isolation Pads" can be found. None of them have this cone/pyramid-like shape, inverted or not. I think I could rephrase the final question as: "What shape would result in the least amount of vibration transferred from the speaker to the table: A cone or an inverted cone?" I'd be glad to be orientated to the resources required to solve the problem, I just want to get the solution right and understand it :)

 

[berkeman]

"What shape would result in the least amount of vibration transferred from the speaker to the table: A cone or an inverted cone?" I'd be glad to be orientated to the resources required to solve the problem, I just want to get the solution right and understand it :)

None of the above. Please post some links to your reading.

isolation in the context of your question means acoustic vibrations on one side of the interface being attenuated well to the other side. Pyramid shapes probably don't enter into the optimum design, IMO. Can you post examples of your reading where pyramid shapes are more important than layering and other considerations?

 

[berkeman]

I apologize if I seem overly aggressive on this, Rick. Sorry. We just prefer to see your reading and research so we can tell where you are coming from.

 

[Rick00]

That's ok, I will. By the way, my phrasing was ambiguous, sorry. I didn't mean that the cone or inverted cone are optimal shapes in general. I meant I'm only interested in these 2 options.

 

[berkeman]

Thanks. I think the more important short-term question has to do with the contact patches on the speaker and the desk. Any thoughts?

 

[sophiecentaur]

Can you post examples of your reading where pyramid shapes are more important than layering and other considerations?

I think it is quite common to stand speakers on conical rubber or metal spikes. I'm only relating what I think I have seen in various HiFi suppliers. The usual caveat applies about the technical validity of most of what you read about HiFi and what is offered in the shops.
There is another issue here and that is the speaker cabinet design'. Many (if not most) speaker cabinets do not have rigid sides. If the bottom of the speaker is designed to flex then it needs to see a low modulus support. Support near the corners would allow flexing and couple less power down to the table.

 

[jrmichler]

result in the least amount of vibration transferred from the speaker to the table: ..." I'd be glad to be orientated to the resources required to solve the problem, I just want to get the solution right and understand it :)

When you state it this way, you have a vibration isolation problem to solve. Vibration isolation problems consist of something vibrating connected to something that is not supposed to vibrate. Your speaker is vibrating, it's setting on a table, the table is not supposed to vibrate. A good Google search term is vibration isolation.

The general principal of vibration isolation is that the natural frequency of the speaker on its mounts must be lower than the lowest frequency of that needs to be isolated. Reread sophiecentaur's posts with that in mind. Keep in mind that a speaker supported by mounts has natural frequencies in both horizontal and vertical directions.

 

[Rick00]

Ok, I did some homework and now I can calculate what's going on with the transmission and reflection coefficients T and R (ratio of transmitted/reflected and incident pressure waves) when sound goes from one medium to another. In specific
$$T = \frac{2z_1}{z_1+z_2}$$
,where z_i are the characteristic impedances, which are dependent on the media.
(The formula comes from applying boundary conditions of pressure and velocity continuity and using the definition for the characteristic impedance of the medium $$z = \frac{\textbf{P}}{\textbf{U}}$$). For minimum T, a big z_2 is desired.

Also, for the case of a wave traveling in a medium, passing through a wall with a certain density \rho, the result is similar, but also exhibits dependence on the density of the material and incident frequency, besides the medium impedance. (Here the boundary conditions take into account the oscillation of the wall)

Am I on the right path? Will I be able to go from these informations to the qualitative understanding of a truncated cone (Several adjacent layers of a material with decreasing/increasing sizes)?