Prufer Transformations for the Normal Modes in Ocean Acoustics

In 1926 Prufer introduced a method of transforming the second order Sturm-Liouville (SL) equation into two nonlinear first order differential equations for the phase oe and ''magnitude'', |oe{sup 2}+oe{sup 2}| for a Poincare phase space representation, (oe,oe). The useful property is the phase equation decouples from the magnitude one which leads to a nonlinear, two point boundary value problem for the eigenvalues, or SL numbers. The transformation has been used both theoretically, e.g. Atkinson, [1960] to prove certain properties of SL equations as well as numerically e.g Bailey [1978]. This paper examines the utility of the Prufer transformation in the context of numerical solutions for modes of the ocean acoustic wave equation. (Its use is certainly not well known in the ocean acoustics community.) Equations for the phase, oe, and natural logarithm of the ''magnitude'', ln(|oe{sup 2}+oe{sup 2}|) lead to same decoupling and a fast and efficient numerical solution with the SL eigenvalues mapping to the horizontal wavenubers. The Prufer transformation has stabilty problems for low order modes at high frequecies, so a numerically stable method of integrating the phase equation is derived. This seems to be the first time the these stability issues have been highlighted to provide a robust algorthim for the modes.