 Solution

The quarter-rate limit is the strong-elasticity limit $\lambda /\alpha \ll 1$, not the weak-elasticity limit. The original PDF repeats $\alpha /\lambda \ll 1$ here; that is inconsistent with the preceding small-$\alpha$ result. In fact at $\alpha =0$ the maximum is $\lambda /4$, directly contradicting a quarter of that value in the same limit. The converted TeX repairs this particular ratio.

For $\alpha \gg \lambda$ and wavenumbers near the maximum, $S=\alpha s\gg L$. In the quadratic equation, the growing root is $O(\lambda )$, so its square and the $-L\beta$ term are small compared with $S\beta$. The leading balance is $S\beta -LS/4=0$. ConsequentlyThe second root is a rapidly damped surfactant-relaxation mode. Thus the stated quarter-rate result is correct after reversing the printed small-parameter ratio.