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

[tex] c^2 = \frac{\partial p}{\partial \rho} [/tex]

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 = [itex]\sqrt{P/\rho}[/itex]

Where P = coefficient of "stiffness"
and [itex]\rho[/itex] = 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 = ([itex]\frac{∂p}{∂\rho}[/itex][itex])_{s}[/itex]d[itex]\rho[/itex] + ([itex]\frac{∂p}{∂s}[/itex])[itex]_{\rho}[/itex] ds

I am guessing that because ([itex]\frac{∂p}{∂\rho}[/itex][itex])_{s}[/itex] 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