Reflection of Waves by an Inhomogeneous Medium

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 ${\rho }_1$ is obtained for $\rho$, the coefficient of reflection of plane waves by a specified inhomogeneous medium. Also iterative processes are given which from ${\rho }_1$ (or any other approximation) derive a sequence of approximations ${\rho }_2, {\rho }_3, \cdots$, which rapidly converge on $\rho$. Lastly it is shown that the approximation $u_1$ for a particular equation leads to a good, simple approximation to the Hankel function ${H_n}^{\mathrm{}\left(2\right)\mathrm{}}\mathrm{}\left(\mathrm{nz}\right)\mathrm{}$ which agrees well with the approximations of Hankel, Debye, and Carlini but has a wider range of validity.