Please wait a minute...
Chin. Phys. B, 2022, Vol. 31(9): 094301    DOI: 10.1088/1674-1056/ac6940

Wave mode computing method using the step-split Padé parabolic equation

Chuan-Xiu Xu(徐传秀)1,† and Guang-Ying Zheng(郑广赢)1,2
1 Hangzhou Applied Acoustics Research Institute, Hangzhou 310023, China;
2 Science and Technology on Sonar Laboratory, Hangzhou 310023, China
Abstract  Models based on a parabolic equation (PE) can accurately predict sound propagation problems in range-dependent ocean waveguides. Consequently, this method has developed rapidly in recent years. Compared with normal mode theory, PE focuses on numerical calculation, which is difficult to use in the mode domain analysis of sound propagation, such as the calculation of mode phase velocity and group velocity. To broaden the capability of PE models in analyzing the underwater sound field, a wave mode calculation method based on PE is proposed in this study. Step-split Padé PE recursive matrix equations are combined to obtain a propagation matrix. Then, the eigenvalue decomposition technique is applied to the matrix to extract sound mode eigenvalues and eigenfunctions. Numerical experiments on some typical waveguides are performed to test the accuracy and flexibility of the new method. Discussions on different orders of Padé approximant demonstrate angle limitations in PE and the missing root problem is also discussed to prove the advantage of the new method. The PE mode method can be expanded in the future to solve smooth wave modes in ocean waveguides, including fluctuating boundaries and sound speed profiles.
Keywords:  parabolic equation      propagation matrix      eigenvalue decomposition  
Received:  29 January 2022      Revised:  22 March 2022      Accepted manuscript online:  22 April 2022
PACS:  43.30.-k (Underwater sound)  
  43.30.Bp (Normal mode propagation of sound in water)  
Fund: Project supported by Young Elite Scientist Sponsorship Program by CAST (Grant No. YESS20200330).
Corresponding Authors:  Chuan-Xiu Xu     E-mail:

Cite this article: 

Chuan-Xiu Xu(徐传秀) and Guang-Ying Zheng(郑广赢) Wave mode computing method using the step-split Padé parabolic equation 2022 Chin. Phys. B 31 094301

[1] Pekeris C L 1948 Geol. Soc. Am. 27 1
[2] Labiance F M 1973 J. Acoust. Soc. Am. 53 1137
[3] Bartberger C L 1977 J. Acoust. Soc. Am. 61 1643
[4] Ellis D D 1985 J. Acoust. Soc. Am. 78 2087
[5] Evans R B 1992 J. Acoust. Soc. Am. 92 2024
[6] Westwood E K 1996 J. Acoust. Soc. Am. 100 3631
[7] Collins M D 1993 J. Acoust. Soc. Am. 93 1736
[8] Collins M D 1991 J. Acoust. Soc. Am. 89 1068
[9] Petrov P S and Ehrhardt M 2016 J. Comp. Phys. 313 144
[10] Porter M B and Bucker H P 1987 J. Acoust. Soc. Am. 82 1349
[11] DiNapoli F R 1971 Tech. Rpt. 4103 (New London:Naval Underwater Systems Center)
[12] Ewing W M, Jardetzky W S and Press F 1957 Elastic waves in layered media (New York:Mcgraw-Hill Book Company) pp. 328-330
[13] DiNapoli F R and Deavenport R L 1980 J. Acoust. Soc. Am. 67 92
[14] Collins M D 1989 J. Acoust. Soc. Am. 86 1459
[15] Jerzak W and Siegmann W L 2005 J. Acoust. Soc. Am. 117 3497
[16] Outing D A, Siegmann W L and Collins M D 2007 IEEE J. Oceanic. Eng. 32 620
[17] Frank S D, Odom R I and Collis J M 2013 J. Acoust. Soc. Am. 133 1358
[18] Frank S D, Collis J M and Odom R I 2015 J. Acoust. Soc. Am. 137 3534
[19] Collins M D 2015 J. Acoust. Soc. Am. 137 1557
[20] Fredricks A J, Siegmann W L and Collins M D 2000 Wave Motion. 31 139
[21] Metzler A M, Collins M D and Collis J M 2013 J. Acoust. Soc. Am. 134 246
[22] Brooke G H, Thomson D J and Ebbeson G R 2001 J. Comput. Acoust. 9 69
[23] Sturm F and Fawcett J A 2003 J. Acoust. Soc. Am. 113 3134
[24] Sturm F 2005 J. Acoust. Soc. Am. 117 1058
[25] Lin Y T and Duda T F 2012 J. Acoust. Soc. Am. 132 EL61
[26] Lin Y T, Collis J M and Duda T F 2012 J. Acoust. Soc. Am. 132 EL364
[27] Lin Y T 2013 J. Acoust. Soc. Am. 134 EL251
[28] Lin Y T, Duda T F and Newhall A E 2013 J. Comput. Acoust. 21 1250018
[29] Sturm F 2016 J. Acoust. Soc. Am. 139 263
[30] Tang J, Piao S C and Zhang H G 2017 Chin. Phys B. 26 114301
[31] Xu C X and Tang J 2019 J. Acoust. Soc. Am. 146 EL464
[32] Porter M 1991 Saclantcen Memorandum, SM-245 (La Spezia:SACLANT Undersea Research Centre)
[1] Quantum partial least squares regression algorithm for multiple correlation problem
Yan-Yan Hou(侯艳艳), Jian Li(李剑), Xiu-Bo Chen(陈秀波), and Yuan Tian(田源). Chin. Phys. B, 2022, 31(3): 030304.
[2] Estimation of co-channel interference between cities caused by ducting and turbulence
Kai Yang(杨凯), Zhensen Wu(吴振森), Xing Guo(郭兴), Jiaji Wu(吴家骥), Yunhua Cao(曹运华), Tan Qu(屈檀), and Jiyu Xue(薛积禹). Chin. Phys. B, 2022, 31(2): 024102.
[3] Theoretical framework for geoacoustic inversion by adjoint method
Yang Wang(汪洋), Xiao-Feng Zhao(赵小峰). Chin. Phys. B, 2019, 28(10): 104301.
[4] Reliable approach for bistatic scattering of three-dimensional targets from underlying rough surface based on parabolic equation
Dong-Min Zhang(张东民), Cheng Liao(廖成), Liang Zhou(周亮), Xiao-Chuan Deng(邓小川), Ju Feng(冯菊). Chin. Phys. B, 2018, 27(7): 074102.
[5] “Refractivity-from-clutter” based on local empirical refractivity model
Xiaofeng Zhao(赵小峰). Chin. Phys. B, 2018, 27(12): 128401.
[6] Three-dimensional parabolic equation model for seismo-acoustic propagation:Theoretical development and preliminary numerical implementation
Jun Tang(唐骏), Sheng-Chun Piao(朴胜春), Hai-Gang Zhang(张海刚). Chin. Phys. B, 2017, 26(11): 114301.
[7] Developments of parabolic equation method in the period of 2000-2016
Chuan-Xiu Xu(徐传秀), Jun Tang(唐骏), Sheng-Chun Piao(朴胜春), Jia-Qi Liu(刘佳琪), Shi-Zhao Zhang(张士钊). Chin. Phys. B, 2016, 25(12): 124315.
[8] Second-order two-scale analysis and numerical algorithms for the hyperbolic-parabolic equations with rapidly oscillating coefficients
Dong Hao (董灏), Nie Yu-Feng (聂玉峰), Cui Jun-Zhi (崔俊芝), Wu Ya-Tao (武亚涛). Chin. Phys. B, 2015, 24(9): 090204.
[9] Estimation of lower refractivity uncertainty from radar sea clutter using Bayesian-MCMC method
Sheng Zheng (盛峥). Chin. Phys. B, 2013, 22(2): 029302.
No Suggested Reading articles found!