An effective Lagrangian approach to the problem of parametric interaction of Gaussian acoustic beams

A new technique for studying the nonlinear interaction of collinear Gaussian acoustic beams based on the use of Lagrangian field-theoretic methods is discussed. The technique is based on treating the standard beam parameters, i.e., spot size and on-axis pressure, of the Gaussians as dynamic variables, i.e., unknown functions of distance down the beam axis. An effective one-dimensional Lagrangian derived for these quantities leads to a set of Euler–Lagrange ordinary differential equations that can be solved rapidly on a computer. The method avoids the use of partial differential equations entirely, and gives results that are in good agreement with previous work both on second-harmonic and parametric generation of sound by Gaussian beams.