Ion Acoustic Wave - Derivation

Derivation

We derive the ion acoustic wave dispersion relation for a linearized fluid description of a plasma with multiple ion species. A subscript 0 denotes constant equilibrium quantities, and 1 denotes first-order perturbations. We assume the pressure perturbations for each species (electrons and ions) are a Polytropic_process, namely for species s. Using the ion continuity equation, the ion momentum equation becomes

We relate the electric field to the electron density by the electron momentum equation:

We now neglect the left-hand side, which is due to electron inertia. This is valid for waves with frequencies much less than the electron plasma frequency. The resulting electric field is

Since we have already solved for the electric field, we cannot also find it from Poisson's equation. The ion momentum equation now relates for each species to :

We arrive at a dispersion relation via Poisson's equation:

The first bracketed term on the right is zero by assumption (charge-neutral equilibrium). We substitute for the electric field and rearrange to find

.

defines the electron Debye length. The second term on the left arises from the term, and reflects the degree to which the perturbation is not charge-neutral. If is small we may drop this term. This approximation is sometimes called the plasma approximation.

We now work in Fourier space, and find

is the wave phase velocity. Substituting this into Poisson's equation gives us an expression where each term is proportional to . To find the dispersion relation for natural modes, we look for solutions for nonzero.

where, and . In general it is not possible to further simplify this expression. If is small (the plasma approximation), we can neglect the second term on the term, and the wave is dispersionless with independent of k.