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Retain $K=k^2{\lambda }_{\mathrm{De}}^2$ while making the same narrow-velocity plateau expansion as in d. With $u=v_1+\delta a$, the factor that varies rapidly across the plateau is $(v_2-u)/(u-v_1)=(1-a)/a$. The other factors vary slowly and may be evaluated at $u=v_1$, givingThereforeAt $K\ll 1$, the denominator tends to $v_0^2/v_1^2$, recovering d. At $K\gg 1$,recovering the exponentially close approach to the plateau edge in c. The phase velocity remains strictly inside the velocity plateau for every finite $k$.

As an asymptotic expansion, the formula takes $\delta \to 0$ at fixed $0<v_1/v_0<1$. If the plateau also approaches the outer edge, so that $v_0-v_1$ is comparable to $\delta$, the factors treated as slowly varying need not be slowly varying. In that additional limiting regime, use the exact dispersion relation in b instead.

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 $\omega =-k{v}_0$ from below. The electron acoustic wave starts at the origin with slope $v_1+\delta v_1^2/v_0^2$ and approaches $\omega =k{v}_1$ from above; it stays inside $k{v}_1<\omega <k{v}_2$. These are the only completely undamped branches, as shown by the cubic equation in b.

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 $ϵ(-\omega ,-k)=ϵ(\omega ,k)$ on these undamped solutions. One should not impose symmetry about the $\omega =0$ axis at fixed positive $k$.