Continued fraction solution of the Riccati equation: Application to acoustic horns and layered inhomogeneous media, with equivalent electrical circuits

Abstract

We present a solution of the general Riccati equation in the form of an ordinary continued fraction expansion and discuss the solution of the second-order wave equation, from which it is obtained. We apply this solution to acoustic horns without losses and obtain an equivalent electrical circuit for each shape of horn. We also extend the method to include acoustic horns with dissipative termination and viscothermal effect. Finally, we apply the solution to the Schrödinger equation and to layered inhomogeneous media equations, where we consider some practical applications of this solution.