中国物理B ›› 2026, Vol. 35 ›› Issue (2): 24303-024303.doi: 10.1088/1674-1056/ae1b7c

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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. 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
  • 收稿日期:2025-09-20 修回日期:2025-11-01 接受日期:2025-11-05 发布日期:2026-01-31
  • 通讯作者: Li-Jia Gong E-mail:lijia.gong@hrbeu.edu.cn
  • 基金资助:
    Project supported by the National Natural Science Foundation of China (Grant No. 12204128).

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. 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
  • Received:2025-09-20 Revised:2025-11-01 Accepted:2025-11-05 Published:2026-01-31
  • Contact: Li-Jia Gong E-mail:lijia.gong@hrbeu.edu.cn
  • Supported by:
    Project supported by the National Natural Science Foundation of China (Grant No. 12204128).

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摘要： 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.

关键词: very-low-frequency sound propagation, normal modes, parabolic cylinder function

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.

Key words: very-low-frequency sound propagation, normal modes, parabolic cylinder function

中图分类号:  (Underwater sound)

  • 43.30.+m
43.30.-k (Underwater sound)