Dispersion relation:
1 = ωₚ²/(ω−kv₀)² + ωₚ²/(ω+kv₀)²
Unstable region: |k| < kc where kc = ωₚ√2/v₀. Modes in this range have Im(ω)>0 — exponential growth.
Top: Dispersion curves Re(ω) vs k. Orange shading = unstable region.
Middle: Phase space (x,v) — two electron streams. Watch electron holes form in the unstable regime.
Bottom: Growth rate Im(ω) vs k — maximum growth at k = ωₚ/(v₀√2).
1 = ωₚ²/(ω−kv₀)² + ωₚ²/(ω+kv₀)²
Unstable region: |k| < kc where kc = ωₚ√2/v₀. Modes in this range have Im(ω)>0 — exponential growth.
Top: Dispersion curves Re(ω) vs k. Orange shading = unstable region.
Middle: Phase space (x,v) — two electron streams. Watch electron holes form in the unstable regime.
Bottom: Growth rate Im(ω) vs k — maximum growth at k = ωₚ/(v₀√2).