非线性声波方程的几种解法对比

Comparison of solution methods of nonlinear acoustic equations

在医学诊疗领域及微、介观损伤的无损检测行业中,经常需要对介质的材料非线性系数进行表征,以得到局部区域更加精细的力学性能变化。文章在简述各向同性固体和理想流体介质中的非线性声波方程的基础上,证实了它们具有相同的形式,这表明它们的解也应具有相同的形式和性质。介绍了求解非线性声波方程的五种方法,包括有限差分、有限元、摄动法、伪线性解及传统解,并对这些解进行了比较分析,讨论它们的优劣和适用条件。通过与实验信号的对比发现,二次谐波解的几种理论解均能与实验结果取得较好的一致性,相关方法仿真的时域波形也能反映出非线性时域波形随传播距离的演化(为冲击波的)过程。最后,讨论了相关解的优劣和应用范围,高阶的摄动解也能为测量介质的非线性系数提供更宽泛的实验条件和更好的技术方法,几种解法的对比也提升了非线性声学方面的理论研究。

In the medical diagnosis and treatment and the nondestructive inspection of micro or mesoscopic injury it is often necessary to characterize the material nonlinear coefficients of the medium to obtain more accurate change of mechanical properties in local areas.Based on the brief description of nonlinear acoustic equations in isotropic solid and ideal fluid,it can be found that the three equations have similar forms,and it means that the solutions of the three equations have similar forms and propagation characteristics.Then,five methods,including finite difference method(FDTD),finite element method,perturbation method(PERT),Pseudo linear solution(PSEU),and tradition solution,are employed to solve the one-dimensional nonlinear acoustic equation.Through comparing these solutions with experimental results,it is shown that several theoretical solutions of second harmonic propagation characteristics can meet the experimental results well;and,the time domain waveforms calculated by using some method can display the evolutionary process from sinusoidal to shock wave.Finally,the advantages and disadvantages of these solutions are compared and discussed.The higher order perturbation solution can provide broader experimental conditions and better technical methods for measuring the nonlinear coefficients of medium.The comparison of several solutions of the nonlinear acoustic equation also improves the theoretical study of nonlinear acoustics.