# General equation for the speed of sound?

1. Feb 5, 2012

I've seen stated in many a physics book that the general case for the speed of sound (for general equations of state p(ρ) ) is given by

$$c^2 = \frac{\partial p}{\partial \rho}$$

where p is pressure and ρ is density.

but I can't for the life of me figure out how on earth to derive that. I've seen tons of derivations for specific cases--gasses, solids, but not for the general case. According to wikipedia, it can be derived using classical mechanics. Can someone point me in the right direction?

Last edited: Feb 5, 2012
2. Feb 22, 2012

### IsaacOh

c = $\sqrt{P/\rho}$

Where P = coefficient of "stiffness"
and $\rho$ = density

3. Feb 22, 2012

### IsaacOh

oops sorry. Didn't understand your initial question. I just jumped to conclusions.

4. Jan 14, 2013

### Sunfire

the equation of state is p=p(ρ,s) thus

dp = ($\frac{∂p}{∂\rho}$$)_{s}$d$\rho$ + ($\frac{∂p}{∂s}$)$_{\rho}$ ds

I am guessing that because ($\frac{∂p}{∂\rho}$$)_{s}$ has units of "velocity squared", it is looked upon as such;
But why this velocity is the sonic one - beats me...

Anyone?

5. Jan 14, 2013