In a very strong magnetic field, we can consider electrons as constrained to move in 1 dimension along the direction of the magnetic field. Consider a beam of relativistic protons propagating with density n_b and speed u_b ∼ c through a cold electron-proton plasma along B. Generalize the dispersion relation (21.74) for modes with k ‖ B.

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pl (w - ku₁)² + w² p2 (w - ku₂)² + . = 1. (21.74)