 Neutral mode of the equal-width Hazel model

The zero-phase velocity ansatz $w=(\mathrm{sech}z{)}^k(\tanh z{)}^{1-k}$ solves the Taylor–Goldstein equation away from its critical level of an internal gravity wave exactly when $J=k(1-k)$. Let $y=\tanh z$ and $a=1-k$. Then $w^{\prime }/w=a/y-y$ and $w^{\prime \prime }/w=a(a-1)/y^2-a-1+2y^2$, so the differential-equation residual divided by $w$ is $[J-k(1-k)](y^{-2}-1)$. The curve has maximum $1/4$ at $k=1/2$. The endpoint $k=0$ does not decay at infinity; fractional powers at the critical level require a branch and regularity qualification.