The asymptotic theory of dispersion relations containing Bessel functions of imaginary order

This paper presents a method of analysing wave-field dispersion relations in which Bessel functions of imaginary order occur. Such dispersion relations arise in applied studies in oceanography and astronomy, for example. The method involves the asymptotic theory developed by Dunster in 1990, and leads to simple analytical approximations containing only trigonometric and exponential functions. Comparisons with accurate numerical calculations show that the resulting approximations to the dispersion relation are highly accurate. In particular, the approximations are powerful enough to reveal the fine structure in the dispersion relation and so identify different wave regimes corresponding to different balances of physical processes. Details of the method are presented for the fluid-dynamical problem that stimulated this analysis, namely the dynamics of an internal ocean wave in the presence of an aerated surface layer; the method identifies and gives different approximations for the subcritical, supercritical and critical regimes. The method is potentially useful in a wide range of problems in wave theory and stability theory. A mathematical theme of the paper is that of the removable singularity.