The Prufer transform for the calculation of acoustic normal modes

Abstract

The PrüferTransform for the Sturm-Liouvile (SL) equation was introduced by Prüferin 1923. An excellent reference is Numerical Solutions of Sturm-Liouville Problems by Pryce. Porter in Computational Ocean Acoustics introduced it to ocean acoustics. This author stumbled upon working on state variables at CMRE on 1978. The method derives first order equations for the phase and log-magnitude for second order equation in a phase plane. The essential feature is the phase equation is not coupled to the magnitude, so it may be solved as a nonlinear, first order equation and the surface boundary condition leads to a unique solution. New results for normal modes using the Prüfermethod are introduced. i) equations for the phase shifts and log-magnitudes at layer boundaries such as at the seabed; ii) state variable transformations based upon Ricatti equations and requiring evaluating one transcendental quantity, important for numerical methods; iii) issues of numerical stability for coupling the phase solution across the potential barriers of local sound speed maxima and poorly coupled boundaries; iv) an equation for computing the imaginary part leading to modal attenuation’s. The method is fast has survived many students using it, even those writing Matlab versions.