Phase Loss in WKB Waves Due to Reflection by a Potential

We study the WKB method away from the short-wave limit. Incorporating the correct phase loss due to reflection at a classical turning point, in place of the usual choice $\pi /2\mathrm{}$, can greatly improve the accuracy of the WKB wave function in the classically allowed region. For a repulsive ${1/x}^2$ potential this leads to far more accurate WKB wave functions that the usual Langer modification of the potential.