  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.
