How Does Conical Chamber Shape Affect Sound Amplification?

[the_dialogue]

Homework Statement

This is a problem I've been pondering, and I simply need some recommendations for texts to read on the topic.

Suppose you have an enclosed air chamber with an acoustic source on one end (http://beckerexhibits.wustl.edu/did/images1/hatpatent.jpg" ). How can one predict the amplification, distortion, and dissipation of the sound that ends up on the other end of the chamber?

I am especially interested in what I think are likely basic mathematics that explain the amplification of the sound if the chamber is conical. This is akin to http://www.thehistoryblog.com/wp-content/uploads/2008/05/horn.jpg" used for many years. Why does the sound get amplified? What are the relevant acoustic formulations?

2. General questions and Attempt

What text should I look into to help me understand this topic and answer these kinds of questions?

The only thing that I've heard, in a very unscientific way, is that sound is amplified by a factor equal to the Entry Cross section / Exit Cross section (so in a cone where the sound enters the larger end, the sound is amplified by some factor 'x'). But this alone is insufficient for my cause (i.e. I want some sort of mathematical basis for this kind of statement).


Thank you!

 

[the_dialogue]

Any ideas?

In short, I just need to know where to look to understand how sound travels in a cone.

Thank you!

 

[hikaru1221]

I don't understand why that reasoning is unscientific. When the wave front (it's usually plane wave in practice) comes to one end, its shape is distorted into some sort of shape. As it travels deeper inside the chamber, e.g. from bigger end to smaller end, the surface area of the wave front becomes smaller, and since the energy of the wave front isn't changed, the intensity becomes greater. That's how sound is amplified: it's not the total energy, but the energy per unit area which is amplified.

Making a complete mathematical analysis with for this matter is out of my ability. If you want to see how to find the shape of the wave front inside the chamber, see http://en.wikipedia.org/wiki/Huygens–Fresnel_principle.

 

[the_dialogue]

I don't understand why that reasoning is unscientific. When the wave front (it's usually plane wave in practice) comes to one end, its shape is distorted into some sort of shape. As it travels deeper inside the chamber, e.g. from bigger end to smaller end, the surface area of the wave front becomes smaller, and since the energy of the wave front isn't changed, the intensity becomes greater. That's how sound is amplified: it's not the total energy, but the energy per unit area which is amplified.

Making a complete mathematical analysis with for this matter is out of my ability. If you want to see how to find the shape of the wave front inside the chamber, see http://en.wikipedia.org/wiki/Huygens–Fresnel_principle.

Thanks for the response Hikaru.

As you may know, often in research we need formulations to prove our intuition.

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*Taking a step further, does anyone know where I can find formulations (such as Huygens-Fresnel) that demonstrate the change of a sound wave propagating through a conical (or for that matter, a cylindrical) chamber?

Thank you again.