Sound Speed

Source lecture(s): pc368_lec09_wave1, pc368_lec10_wave2, pc368_lec11_streaming_ins

Intuition

Sound speed in a plasma is the characteristic speed at which pressure perturbations propagate through the ion fluid. It sets the scale for ion acoustic waves and appears in Mach-number criteria like the Bohm criterion.

Formal Definition

\[c_s = \\sqrt{\\frac{\\gamma_e k_B T_e + \\gamma_i k_B T_i}{m_i}}\]

For cold ions and isothermal electrons (\(T_i \\ll T_e\), \(\\gamma_e = 1\)):

\[c_s \\approx \\sqrt{\\frac{k_B T_e}{m_i}}\]

Mathematical Formulation

From the two-fluid ion momentum equation with electron pressure gradient:

\[m_i \\frac{\\partial \\mathbf{u}_i}{\\partial t} = -\\frac{\\nabla p_e}{n_i}\]

Linearizing \(p_e = n_e k_B T_e\) and using Boltzmann response yields the acoustic wave equation with speed \(c_s\).

Derivation

Start with ion continuity and momentum.
Use Boltzmann electrons: \(n_e = n_0 \\exp(e\\phi/k_B T_e)\).
For small \(\\phi\): \(n_{e1} = (e\\phi/k_B T_e) n_0\).
Insert into Poisson and eliminate \(\\phi\) to get wave equation.
The natural frequency is \(\\omega = k c_s\).

Worked Example

For \(T_e = 10\) eV in hydrogen: - \(c_s \\approx \\sqrt{1.6\\times 10^{-18}\\,\\text{J} / 1.67\\times 10^{-27}\\,\\text{kg}} \\approx 3.1\\times 10^4\\,\\text{m/s}\).

Common Mistakes

Using \(T_i\) instead of \(T_e\). Electron pressure dominates the restoring force.
Forgetting the \(\\gamma\) factors in the general definition.

Quiz Questions

How does \(c_s\) change if \(T_e\) doubles?
Why is \(c_s\) called “sound speed” in plasma?