I know sound "is" a propagating pressure wave. I'd like to know if there is a relation between the possibility (signal/noise ratio?) of sound propagating and the pressure or particle density in a medium in which it would propagate. I can't think of a straightforward way to deduce this from any relation I learned when studying physics. Furthermore, how would the speed of sound be influenced by pressure or particle density?
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As derived in the answer to this this Phys.SE question, the speed of sound is given by the formula: $$c=\sqrt{\frac{\partial p}{\partial \rho}}$$ where $p$ is pressure and $\rho$ is density. A subtlety of the above equation is that the derivative should be taken keeping the entropy constant. For a gas, this means that it satisfies an equation of state of the form: $$pV^\gamma=\mathrm{constant}$$ where $\gamma = C_p/C_v$ is the ratio of the heat capacities at constant pressure and volume. In terms of density, this means $$p \rho^{-\gamma}=\mathrm{constant}$$ Say that $p = K\rho^\gamma$ where $K$ may depend on other thermodynamic variables. From this we obtain the expression: $$c=\sqrt{\frac{\partial p}{\partial \rho}}=\sqrt{K\gamma \rho^{\gamma-1}}=\sqrt{\gamma\frac{p}{\rho}}$$ which provides the expression you're looking for. One may then use the ideal gas law to simplify it further, $c=\sqrt{\gamma k T/m}$ where $m$ is the mass of a molecule.
John Donne
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Speed of sound does not depend on pressure. Pressure, equal to the bulk modulus, may be in some expressions for the speed of sound, but always together with density. In an ideal gas, the ratio is constant (temperature etc being equal).