= Solution
For $k>0$, the upper <Langmuir wave> starts at $\omega\approx+\omega_{\mathrm{pe}}$ and approaches the line $\omega=kV$ from above. The lower <Langmuir wave> starts at $\omega\approx-\omega_{\mathrm{pe}}$ and approaches $\omega=-kv_0$ from below. The <electron acoustic wave> starts at the origin with slope $v_1+\delta v_1^2/v_0^2$ and approaches $\omega=kv_1$ from above; it stays inside $kv_1<\omega<kv_2$. These are the only completely undamped branches, as shown by the <cubic equation> in <2/b>.
\Image[/past-exam-of-the-kinetic-theory-course-of-the-university-of-oxford/2025-undamped-dispersion.png]
{title=Three undamped branches for a narrow electron velocity plateau}
{description=Exact roots of the cubic form of the dielectric response, with $v_0=1$, $v_1=0.5$ and $v_2=0.56$. Dashed lines show the three short-wavelength limits. The shaded band contains the electron acoustic branch. The exact outer cutoffs retain the small density-normalization correction explained in <2/d>.}
The small asymmetry between the outer branches reflects the extra positive-<velocity> population introduced by the <velocity plateau>. The negative-$k$ sketch follows by rotating the displayed branches through the origin, since $\epsilon(-\omega,-k)=\epsilon(\omega,k)$ on these undamped solutions. One should not impose symmetry about the $\omega=0$ axis at fixed positive $k$.
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