General equation for the speed of sound?

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Discussion Overview

The discussion revolves around the general equation for the speed of sound, particularly in relation to different equations of state for pressure and density. Participants explore derivations, definitions, and the underlying principles of the equation.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant states that the general case for the speed of sound is given by the equation c² = ∂p/∂ρ, where p is pressure and ρ is density, and expresses difficulty in deriving this for the general case.
  • Another participant provides a simplified expression for the speed of sound as c = √(P/ρ), where P is described as a coefficient of stiffness.
  • A later reply acknowledges a misunderstanding of the initial question and retracts an earlier assumption.
  • Another participant discusses the equation of state in terms of pressure as a function of density and entropy, suggesting that the partial derivative ∂p/∂ρ has units of velocity squared, but questions why this relates specifically to the speed of sound.
  • A participant shares a link to an external resource that may provide further explanation on the topic.

Areas of Agreement / Disagreement

Participants express varying degrees of understanding and confusion regarding the derivation and implications of the speed of sound equation. Multiple competing views and interpretations remain, with no consensus reached on the derivation or its significance.

Contextual Notes

Participants note the complexity of deriving the general case for the speed of sound and the dependence on specific equations of state, which may not be universally applicable.

fhqwgads2005
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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?
 
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c = \sqrt{P/\rho}

Where P = coefficient of "stiffness"
and \rho = density
 
oops sorry. Didn't understand your initial question. I just jumped to conclusions.
 
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?
 

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