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.

 

[paranormal]

I would recommend looking into the field of acoustics and specifically studying conical volumes. There are many texts and resources available that can help you understand the physics and mathematics behind sound amplification, distortion, and dissipation in a conical chamber.

Some potential resources to consider include:

1. "Fundamentals of Acoustics" by Lawrence E. Kinsler, Austin R. Frey, Alan B. Coppens, and James V. Sanders: This is a comprehensive textbook that covers the basic principles of acoustics, including sound propagation, reflection, and absorption. It also includes a section on sound amplification in different types of enclosures, including conical volumes.

2. "Acoustics: An Introduction" by Heinrich Kuttruff: This book provides a more theoretical approach to acoustics, covering topics such as the wave equation, sound fields, and sound sources. It also includes a chapter on sound amplification in different types of enclosures.

3. "The Physics of Musical Instruments" by Neville H. Fletcher and Thomas D. Rossing: While this book focuses on the acoustics of musical instruments, it also includes a chapter on the acoustics of conical volumes and how they affect the sound produced by instruments such as horns and trumpets.

In addition to these texts, there are also many online resources and research papers available that discuss the acoustics of conical volumes. I would recommend searching for specific keywords such as "conical volume acoustics" or "sound amplification in conical chambers" to find relevant articles and studies.

As for the idea of sound amplification being related to the ratio of entry and exit cross sections, this is a simplified explanation and there are many other factors at play, including the shape and material of the chamber, the frequency of the sound, and the position of the listener. Studying the mathematical formulations and principles of acoustics will help you gain a deeper understanding of this phenomenon.