Speed of Sound - General Formula

[Curl]

Can anyone show that the speed of sound, or rather, speed of low-energy mechanical waves follows the relationship:

a² = ( bulk modulus / density )
This holds for sound waves, and is also similar to the waves on strings formula.

Can anyone show how this is derived? I read a book and they said "by apply Newton's laws". But how?

 

[tiny-tim]

Hi Curl!

When sound travels, nothing in the material actually moves anywhere, it only oscillates on the spot …

the equation for this oscillation depends on the stiffness (springy-ness) of the material …

there's some details at http://en.wikipedia.org/wiki/Speed_of_sound#Basic_concept"

 

[nasu]

You can derive the equation for mechanical waves in the medium by using Newton's second law. You look at a small piece of the medium and calculate the net force due to the perturbation (the wave).
The net force will depend on the distribution of elastic forces, so you will have the elastic constant(s) in the equation. The mass of the small piece of medium depends on density.
The actual derivation can be found in many textbooks. The actual form depends on the medium (fluid or solid, isotropic or anisotropic, etc) but the idea is along these lines.

 

[Curl]

Yeah no joke, what I am asking is to see the actual derivation.

Landau and Lifgarbagez use a very different method to derive the speed of sound in an ideal gas. I don't have any books that do it for solid materials, otherwise I wouldn't be asking this.

 

[nasu]

Yeah no joke, what I am asking is to see the actual derivation.

Landau and Lifgarbagez use a very different method to derive the speed of sound in an ideal gas. I don't have any books that do it for solid materials, otherwise I wouldn't be asking this.

The derivation for a cubic crystal can be found for example in Kittel - Introduction to solid state Physics (Chapter 3).
I have a couple of slides that show the main steps. I ca send them if you would like. For a crystal you need to be a little familiar the stress and strain tensors to understand the derivation.

For isotropic solid you can find some sketches of the derivation (in the 1 D case) for example here: http://mysite.du.edu/~jcalvert/waves/mechwave.htm