= Neutral mode of the equal-width Hazel model
{title2=$J=k(1-k)$}
= Neutral modes of the equal-width Hazel model
{synonym}
The zero-<phase velocity> ansatz $w=(\operatorname{sech}z)^k(\tanh z)^{1-k}$ 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 $w'/w=a/y-y$ and $w''/w=a(a-1)/y^2-a-1+2y^2$, 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.
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