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For smooth positive $K$ on an interval with regular Dirichlet boundary conditions, there is no turning-point phase shift. Combining conjugate WKB approximation branches to vanish at the first endpoint makes the total phase at the second an integer multiple of $\pi$. The first correction is the integrated WKB phase correction for a slowly varying oscillator. For $K=x+1$ on $[0,1]$ the resulting eigenvalues are $4{\pi }^2{n}^2/9-3/16+O(n^{-2})$.

For $\nabla \cdot (k_0(X{)}^{-2}\nabla p)+p=0$ in a slowly varying rigid duct, a real local transverse normal mode $F$ with squared norm $N(X)$ has longitudinal wavenumber $k(X)$. The WKB approximation is $p=A(X)F(y,X)\exp [-i{ϵ}^{-1}{\int }^{X}k(\xi )d\xi ]$. The moving-wall Neumann boundary condition and next-order solvability give $2kN{k}_0^{-2}{A}_X+(kN{k}_0^{-2}{)}_{X}A=0$, hence the displayed amplitude. For $F=\cos (n\pi y/R)$, $N=R$ for $n\ge 1$ and $N=2R$ for $n=0$. The factor $k_0$ is essential when the coefficient in the divergence-form equation varies. This approximation requires a mode separated from cutoff and from other modes.

A zero of the local squared wavenumber in $E^{\prime \prime }+k^2(x)E=0$. At a simple zero, $(k^2{)}^{\prime }(x_T)\ne 0$, the WKB approximation fails in a layer with scale $|(k^2{)}^{\prime }(x_T){|}^{-1/3}$ and the Airy ordinary differential equation supplies the local replacement. Two nearby wave turning points require a uniform treatment rather than independent Airy turning-point connection formulas.

For a smooth symmetric confining potential energy with four separated simple classical turning points $-x_2<-x_1<x_1<x_2$, defineThe Airy turning-point connection formula applied to a decaying exterior wavefunction gives a central WKB approximation proportional to ${\chi }^{-1/2}[2\cos \theta e^K+\sin \theta e^{-K}]$, where $\chi =\sqrt{2m(V-E)}$ and $K(x)={\hslash }^{-1}{\int }_x^{x_1}\chi dx$. Since ${\chi }^{\prime }(0)=0$ and $2K(0)=\varphi$, the even Neumann boundary condition gives $\tan \theta =2e^{\varphi }$, while the odd Dirichlet boundary condition gives $\tan \theta =-2e^{\varphi }$. For $\varphi \gg 1$, $\theta =(n+\frac{1}{2})\pi \mp \frac{1}{2}{e}^{-\varphi }+O(e^{-3\varphi })$, with the minus correction for even parity. Since ${\theta }^{\prime }=\mathcal{T}/(2\hslash )$ for the single-well classical oscillation period $\mathcal{T}$, the leading double-well tunneling splitting is $2\hslash e^{-\varphi }/\mathcal{T}$.