  We propose an extension of the Wentzel-Kramers-Brillouin (WKB) approximation
for solving the Schr\"odinger equation. A set of coupled differential equations
has been obtained by considering an ansatz of wave function with two auxiliary
conditions on gauging the first and the second derivatives of the wave
function, respectivey. It is shown that an alternating perturbation method can
be used to decouple this set of equations, yielding the well known Bremmer
series. We derive an analytical scheme to approximate the wave function which
consists of the usual WKB approximation as the zeroth order solution with extra
correction terms up to the first order perturbation. Therefore, we call this
alternative methodology the alternating WKB (a-WKB) approximation. We apply the
a-WKB method to solve the eigenvalue problem of a harmonic oscillator and the
scattering problem of a repulsive inverse-square (centrifugal) potential and
demonstrate its supremacy over the usual WKB approximation.
