PX154 - G5 - waves in bulk gases - sound waves

speed of sound

we have longitudinal waves
start with our usual plot of displacement, $u (x, t) = A \cos (k x - ω t)$
consider a loud speaker
remember ideal gas law:

p V = n R T ⟹ p = ρ R T

oscillations of the speaker cone produce density oscillations and pressure gradients that provide the restoring force that leads to SHM
we find that the pressure is described by:

p = A k B \sin (k x - ω t)

where, $B =$ bulk modulus of air [YF eqn 16.4]

we will obtain the speed of sound:

v = \sqrt{\frac{B}{ρ}}

- $v_{a i r} = 344 m s^{- 1}$

speed of sound in a ideal gas

[YF 16.2]

for sound waves in an ideal gas, the propagation is fast enough that there is no heat flow. thus, we can consider it to be an adiabatic process
the bulk modulus is given by:

B = - V \frac{\partial p}{\partial V}

for an adiabatic process:

p V^{γ} = c o n s t a n t = k

where, $γ = \frac{c_{P}}{c_{V}}$

\begin{aligned} \frac{\partial p}{\partial V} & = \frac{\partial}{\partial V} (k V^{- γ}) \\ = - γ k V^{- γ - 1} \\ = - \frac{γ}{V} k V^{- γ} \\ = - \frac{γ}{V} p \end{aligned}

∴ B = γ p

for the wave speed:

v = \sqrt{\frac{B}{ρ}} = \sqrt{\frac{γ p}{ρ}}

to make this more usable, $ρ = \frac{m}{V} = \frac{N M}{V}$
then,

v = \sqrt{\frac{γ p V}{N M}}

using $p V = N k_{B} T$ :

v_{i d e a l g a s} = \sqrt{γ \frac{k_{B} T}{M}}

- oxygen at $300 K : v = 340 m s^{- 1}$ ; $γ = \frac{7}{5}$
- helium at $300 K : v = 1370 m s^{- 1}$ ; $γ = \frac{5}{3}$

we can compare with boltzmann distribution:

v_{r m s} = \sqrt{\frac{3 k_{B} T}{M}}

hence, $v = v_{r m s} \sqrt{\frac{γ}{3}}$
- oxygen at $300 K : v_{r m s} = 464 m s^{- 1}$
- helium at $300 K : v_{r m s} = 1370 m s^{- 1}$