均匀海洋波导本征值问题的多模态分析方法
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摘要
针对传统搜根法求解本征值依赖于初值的设定、精度不高且容易丢根等问题,提出了基于多模态展开方法的本征值、本征函数的快速求解方法。用正弦函数作为深度方向声压的正交函数基,对波动方程进行模态展开,将超越方程的搜根问题转化为正交展开系数矩阵的特征值分解,在求解本征值的同时得到本征函数。运用该方法对一些典型的海洋波导进行了数值计算,得到了单层等声速波导、具有声速剖面的波导,以及双层波导的本征值与本征函数结果,并与标准计算结果进行对比,证明该方法是合理且可行的。
Due to the inefficiency of traditional root-finding method and its dependence on initial value, in this paper, a multimodal expansion method is proposed to solve the problems of eigenvalues and eigenfunctions in the homogeneous ocean waveguide. The sine function is chosen as the orthogonal basis function of the sound pressure in the depth direction. The eigenvalues can be obtained by decomposing the matrix of the expansion coefficients together with the eigenfunctions rather than finding root in transcendental equation. Numerical validation of this method is carried out in the single-layer isovelocity waveguide, the waveguide with sound velocity profile and the double-layered waveguide and proves that this algorithm is reasonable and feasible.
Due to the inefficiency of traditional root-finding method and its dependence on initial value, in this paper, a multimodal expansion method is proposed to solve the problems of eigenvalues and eigenfunctions in the homogeneous ocean waveguide. The sine function is chosen as the orthogonal basis function of the sound pressure in the depth direction. The eigenvalues can be obtained by decomposing the matrix of the expansion coefficients together with the eigenfunctions rather than finding root in transcendental equation. Numerical validation of this method is carried out in the single-layer isovelocity waveguide, the waveguide with sound velocity profile and the double-layered waveguide and proves that this algorithm is reasonable and feasible.
引文
[1]JENSEN F B,et al.Computational ocean acoustics[M].New York:Springer Science&Business Media,2011:337-456
[2]郭杨阳,范军,熊磊.波导中弹性波频散曲线计算的谱方法[J].上海交通大学学报,2016,50(11):1784-1788.GUO Yang yang,FAN Jun,XIONG Lei.Spectral Method for Computing Frequency Dispersion Curves of Elastic Waves in Wave Guide[J].Journal of Shanghai Jiao Tong University,2016,50(11):1784-1788.
[3]任永亮.谱方法在工程波动分析中的应用研究[D].南京:南京工业大学,2008.REN Yongliang.Analysis of spectral method in engineering application[D].Nanjing:Nanjing Tech University,2008.
[4]PAGNEUX V,MAUREL A.Lamb wave propagation in elastic waveguides with variable thickness[J].Proceedings Mathematical Physical&Engineering Sciences,2006,462(2068):1315-1339.
[5]BI W P,PAGNEUX V,LAFARGE D,et al.Efficient Modelling of sound propagation in nonuniform lined intakes[C]//13th AIAA/CEAS Aeroacoustic Conference,2007.
[6]EVANS R B.A coupled mode solution for acoustic propagation in a waveguide with stepwise depth variations of a penetrable bottom[J].J.Acoust.Soc.Am.,1983,74(1):188-195.
[7]PAGNEUX V,AMIR N,KERGOMARD J.A study of wave propagation in varying cross‐section waveguides by modal decomposition.Part I.Theory and validation[J].J.Acoust.Soc.Am.,1996,100(4):2034-2048.
[8]JENSEN F B.Computational ocean acoustics[M].Springer Science&Business Media,2011:345
[9]THOMSON D J,BROOKE G H,DESANTO J.Numerical implementation of a modal solution to a range-dependent benchmark problem[J].J.Acoust.Soc.Am.,1990,87(4):1521-1526.
[2]郭杨阳,范军,熊磊.波导中弹性波频散曲线计算的谱方法[J].上海交通大学学报,2016,50(11):1784-1788.GUO Yang yang,FAN Jun,XIONG Lei.Spectral Method for Computing Frequency Dispersion Curves of Elastic Waves in Wave Guide[J].Journal of Shanghai Jiao Tong University,2016,50(11):1784-1788.
[3]任永亮.谱方法在工程波动分析中的应用研究[D].南京:南京工业大学,2008.REN Yongliang.Analysis of spectral method in engineering application[D].Nanjing:Nanjing Tech University,2008.
[4]PAGNEUX V,MAUREL A.Lamb wave propagation in elastic waveguides with variable thickness[J].Proceedings Mathematical Physical&Engineering Sciences,2006,462(2068):1315-1339.
[5]BI W P,PAGNEUX V,LAFARGE D,et al.Efficient Modelling of sound propagation in nonuniform lined intakes[C]//13th AIAA/CEAS Aeroacoustic Conference,2007.
[6]EVANS R B.A coupled mode solution for acoustic propagation in a waveguide with stepwise depth variations of a penetrable bottom[J].J.Acoust.Soc.Am.,1983,74(1):188-195.
[7]PAGNEUX V,AMIR N,KERGOMARD J.A study of wave propagation in varying cross‐section waveguides by modal decomposition.Part I.Theory and validation[J].J.Acoust.Soc.Am.,1996,100(4):2034-2048.
[8]JENSEN F B.Computational ocean acoustics[M].Springer Science&Business Media,2011:345
[9]THOMSON D J,BROOKE G H,DESANTO J.Numerical implementation of a modal solution to a range-dependent benchmark problem[J].J.Acoust.Soc.Am.,1990,87(4):1521-1526.