General equation for the speed of sound?

[fhqwgads2005]

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?

 

[IsaacOh]

c = \sqrt{P/\rho}

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

 

[IsaacOh]

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

 

[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?

 

[Sunfire]

This gives probably enough of an explanation

http://www.grc.nasa.gov/WWW/K-12/airplane/snddrv.html