The Speed of Sound — Question 3

Question 3

Doubling gas pressure need not change sound speed

An ideal gas obeys $\rho=pM/(RT)$ and has adiabatic bulk modulus $B=\gamma p$. Here $M$ is molar mass and $\gamma$ is constant. Compare two equilibrium states at the same $T$ and composition, with $p_2=2p_1$.

1. Derive the sound speed in terms of $T,M,\gamma,R$ and find $c_2/c_1$.

2. Compare with a hypothetical process in which $B$ doubles while density is held fixed. Explain the difference.

Question 3 - Solution

Doubling gas pressure need not change sound speed

(a) Method and calculation

Substitute both state relations into the elasticity-inertia ratio:$$c^2=\frac{B}{\rho}=\frac{\gamma p}{pM/(RT)}=\frac{\gamma RT}{M}.$$At fixed temperature and composition, $B_2/B_1=2$ and $\rho_2/\rho_1=2$, so $c_2/c_1=1$.

(b) Interpretation and further calculation

If density were instead fixed while $B$ doubled,$$\frac{c_2}{c_1}=\sqrt{\frac{B_2/B_1}{\rho_2/\rho_1}}=\sqrt2.$$That is a different constraint and is not the stated isothermal pressure change. Pressure influences stiffness and density together; changing only one in the formula describes another experiment.

Physical check

The ideal-gas cancellation is conditional on the same $T$, composition and $\gamma$; it is not a universal pressure-independence law for all fluids.