The High-Order Perturbation Approximate Solution of the Finite Ultrasonic Wave

Abstract

To the nonlinear acoustic wave equation, the general used second harmonic solution is not accurate enough because all perturbated expansion equations higher than the second order are ignored during the equation solving process. The purpose of this paper is to obtain a more accurate solution, i.e., the high-order perturbation approximate solution. Firstly, the nonlinear acoustic wave equation is expanded into many inhomogeneous partial differential equations. the low-order harmonic solutions are obtained manually, then we formulate the form of the high-order harmonic solutions according to the properties of the low-order harmonic solutions. Using symbol calculation tool, we finally obtained higher up to the 14th order perturbation special solutions. Odd order solutions contain only odd order harmonics, and even order solutions contain only even order harmonics. The high-order perturbation solution of the second harmonic is finally achieved by summing up all of the second harmonic solution parts. The simulation results show that the relative amplitude (A 2/A 1) of the second harmonic increases and then decreases with the propagation distance, which is in agreement with experimental results. The high-order perturbation approximated solution can compensate for the theory deficiency and can be used to measure the nonlinear parameter with a good precision.