Multiple-location matched approximation for Bessel function $J_0$ and its derivatives

TL;DR

This paper introduces a matched approximation method for the Bessel function J_0 and its derivatives, useful in nonlinear wave dynamics, demonstrated through acoustic-gravity wave resonance modeling.

Contribution

It develops a novel multiple-location matching technique for approximating J_0 and its derivatives, improving modeling in nonlinear differential equations.

Findings

Effective approximation of J_0 near the origin and in the far field.

Enhanced modeling of nonlinear acoustic-gravity wave interactions.

Application to three-dimensional wave resonance scenarios.

Abstract

I present an approximation of Bessel function $J_0(r)$ of the first kind for small arguments near the origin. The approximation comprises a simple cosine function that is matched with $J_0(r)$ at $r=\pi/\textrm{e}$. A second matching is then carried out with the standard, but slightly modified, far-field approximation for $J_0(r)$, such that first and second derivatives are also considered. The approximation is practical when nonlinear dynamics come into play, in particular in the case of nonlinear interactions that involve second order differential equations as in acoustic--gravity wave theory. A demonstration of the proposed matching technique applied to three-dimensional acoustic--gravity wave triad resonance in cylindrical coordinates is provided.