Figure 16

Figure 16: The picture shows a supersonic dispersion relation as a relation ω − vk = ±c |k|Γ k. In particular, we plot a dispersion of the type ∘ ----------- Γ k = 1 + k2 ∕K2. For supersonic velocities the dispersion relation has two real roots. For supersonic velocities and ω less than a critical frequency ωc, the dispersion relation has four real roots.