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Boundary effects on modal shape in deep ocean via non-integer order parabolic cylinder functions |
| Jian-Kang Zhan(詹建康)1,2,3, Sheng-Chun Piao(朴胜春)1,2,3, and Li-Jia Gong(龚李佳)1,2,3,† |
1 National Key Laboratory of Underwater Acoustic Technology, Harbin Engineering University, Harbin 150001, China;
2 Key Laboratory of Marine Information Acquisition and Security (Harbin Engineering University), Ministry of Industry and Information Technology, Harbin 150001, China;
3 College of Underwater Acoustic Engineering, Harbin Engineering University, Harbin 150001, China |
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Abstract This study investigates the effects of ocean boundaries on modal shapes in very-low-frequency (VLF, 1-10 Hz) sound propagation through the deep ocean. Utilizing a normal mode solution formulated in terms of parabolic cylinder functions (PCF), we demonstrate that boundary interactions induce a phase change reduction below $-\pi$ at frequencies of several hertz. This reduction, in turn, forces a key transition in the solution, shifting the order of the PCF from integer to non-integer values. Analysis of the characteristic shape of the PCF versus its order reveals that these boundary-influenced modes exhibit an energy shift toward deeper regions and a weakened axial convergence of the underwater sound field.
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Received: 20 September 2025
Revised: 01 November 2025
Accepted manuscript online: 05 November 2025
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PACS:
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43.30.+m
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(Underwater sound)
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43.30.-k
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(Underwater sound)
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| Fund: Project supported by the National Natural Science Foundation of China (Grant No. 12204128). |
Corresponding Authors:
Li-Jia Gong
E-mail: lijia.gong@hrbeu.edu.cn
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Cite this article:
Jian-Kang Zhan(詹建康), Sheng-Chun Piao(朴胜春), and Li-Jia Gong(龚李佳) Boundary effects on modal shape in deep ocean via non-integer order parabolic cylinder functions 2026 Chin. Phys. B 35 024303
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