A new approximate solution is given for the general linear second-order differential equation which is especially appropriate in treating the reflection of waves by an inhomogeneous medium. The well-known approximations to a fundamental pair of solutions made by Liouville, Rayleigh, and Jeffreys, which suffer from singularities at the zeros of a particular function, are replaced by another pair of simple approximations ${u}_{1}$, ${u}_{2}$, which in general agree well with the first pair but remain finite at the zeros. Then a corresponding approximation ${\ensuremath{\rho}}_{1}$ is obtained for $\ensuremath{\rho}$, the coefficient of reflection of plane waves by a specified inhomogeneous medium. Also iterative processes are given which from ${\ensuremath{\rho}}_{1}$ (or any other approximation) derive a sequence of approximations ${\ensuremath{\rho}}_{2}, {\ensuremath{\rho}}_{3}, \ensuremath{\cdots}$, which rapidly converge on $\ensuremath{\rho}$. Lastly it is shown that the approximation ${u}_{1}$ for a particular equation leads to a good, simple approximation to the Hankel function ${{H}_{n}}^{(2)}(\mathrm{nz})$ which agrees well with the approximations of Hankel, Debye, and Carlini but has a wider range of validity.