Consider a plasma with cold protons and hot electrons with a 1-dimensional distribution function proportional to 1/(v

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Consider a plasma with cold protons and hot electrons with a 1-dimensional distribution function proportional to 1/(v02 + v2), so the full 1-dimensional distribution function is

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(a) Show that the dispersion relation for Langmuir waves, with phase speeds large compared to v0, is

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(b) Compute the dispersion relation for ion-acoustic waves, assuming that their phase speeds are much less than v0 but are large compared to the cold protons’ thermal velocities. Your result should be

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