WKB phase correction for a slowly varying oscillator (${\sigma }_1=3(K^{\prime }{)}^2/(8K^3)-K^{\prime \prime }/(4K^2)$)

For smooth $K=|k|>0$ and fixed amplitude flux, use $\sigma =K+{ϵ}^2{\sigma }_1+\cdots$ and $A_0\propto K^{-1/2}$. The exact amplitude-phase equation gives ${\sigma }_1=A_0^{\prime \prime }/(2K{A}_0)$, which is the displayed formula. The first phase correction is order ${ϵ}^2$ in the local wavenumber, hence order $ϵ$ in its integrated exponential phase.