Application of the split-step Padé approach to nonlinear field predictions

Ultrasonics. 2013 Feb;53(2):432-8. doi: 10.1016/j.ultras.2012.08.008. Epub 2012 Oct 3.


We herein propose a new theoretical approach for analyzing the nonlinear propagation of directive sound beams emitted from a planar piston source with a circular aperture. The proposed approach relies on the split-step Padé approximation, which is an efficient method for obtaining wide-angle one-way wave equations, especially in underwater acoustics. Despite including only two Padé terms in the expansion, the theory was applicable to a beam angle of up to ±40° relative to the main propagation direction, the angle of which is approximately twice that of the Khokhlov-Zabolotskaya-Kuznetsov equation, which is based on parabolic approximation. In order to demonstrate the effectiveness of the newly proposed theoretical approach, we performed an experiment using an airborne ultrasonic emitter with a circular aperture of 7.5cm in radius. We drove the emitter powerfully at a 36-kHz and 40-kHz bi-frequency signal and measured the beam patterns of the primary and secondary waves, such as parametric sounds within wide propagation angles. Excellent agreement between measured data and the corresponding numerical simulations supports the validity of the proposed model equations and the computational methods for their numerical solutions.

Publication types

  • Research Support, Non-U.S. Gov't