Consider the sound wave propagating through a pipe with cross-section area A. In time interval dt it moves through a tube of length dz=vdt. In the steady state, the mass flow rate dm/dt=ρvA must be the same at the ends of the tube, therefore the mass flux j=ρv=const.→vdρ=−ρdv. The pressure-gradient force provides the acceleration and apply the Newton's second law,
ρdvdt=−dPdz
→dP=−ρdvdtdz=−ρdzdtdv=(−ρdvdt)v=v2dρ.
And therefore,
v2=dPdρ,
where, P is the gas pressure, ρ is the density.
Reference:
https://en.wikipedia.org/wiki/Speed_of_sound#:~:text=The%20speed%20of%20sound%20is,a%20mile%20in%204.7%20s.
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