移动载荷激发冰层水弹性波的研究进展

胡刚华; 王展

doi:10.7638/kqdlxxb-2025.0059

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移动载荷激发冰层水弹性波的研究进展

研究专栏——海洋波动与湍流 | 更新时间：2026-05-28

- 移动载荷激发冰层水弹性波的研究进展
- Research progress on hydroelastic waves excited by moving loadson an ice sheet
- 空气动力学学报 2025年43卷第6期页码：46-61
- 作者机构：
  
  1. 中国科学院力学研究所,北京,100190
  2. 中国科学院大学工程科学学院,北京,100049
- 作者简介：
  
  胡刚华（2000—），男，湖北人，博士研究生，研究方向：水动力学. E-mail：huganghua@imech.ac.cn
  王展*，研究员，研究方向：水动力学. E-mail：zwang@imech.ac.cn
- 基金信息：
  
  国家自然科学基金（12325207）
- DOI：10.7638/kqdlxxb-2025.0059
  中图分类号： P731.15
- 纸质出版：2025
- 稿件说明：
移动端阅览
胡刚华, 王展. 移动载荷激发冰层水弹性波的研究进展[J]. 空气动力学学报, 2025, 43(6): 46−61.

HU Ganghua, WANG Zhan. Research progress on hydroelastic waves excited by moving loadson an ice sheet[J]. 2025, 43(6): 46-61.
胡刚华, 王展. 移动载荷激发冰层水弹性波的研究进展[J]. 空气动力学学报, 2025, 43(6): 46−61. DOI： 10.7638/kqdlxxb-2025.0059

HU Ganghua, WANG Zhan. Research progress on hydroelastic waves excited by moving loadson an ice sheet[J]. 2025, 43(6): 46-61. DOI： 10.7638/kqdlxxb-2025.0059.

摘要

在极地海冰覆盖区域，冰水界面波动的回复力除重力外，还需考虑冰层的弹性力，这种波动被称为水弹性波（或挠曲重力波）。与海洋表面波浪不同，水弹性波的传播特性不仅受到流体动力学控制，还显著依赖于冰层的弹性力学特征，其临界共振体系完全不同于纯重力水波，在临界速度附近表现出独特的非线性波动现象。本文系统综述了移动载荷激发水弹性波的研究进展，涵盖理论模型、数值计算与实验观测，重点探讨了势流模型、黏弹性效应、非线性效应以及复杂运动场景（如变速运动、三维效应、渠道边界和水下载荷等）下的冰层动力学特征，发现黏弹性和非线性效应是解释临界速度下有限振幅响应的关键机制，复杂运动场景中的水弹性波传播特性则进一步揭示了冰-水耦合作用的多样性。本文研究可为极地资源开发、船舶航行安全、海洋超大浮体结构设计提供重要的理论支撑和技术参考。

Abstract

In polar ice-covered regions

wave motion at the ice-water interface is governed by a restoring force comprising both gravity and the elastic force of the ice layer. Such waves are known as hydroelastic waves (or flexural-gravity waves). Unlike ocean surface waves

the propagation characteristics of hydroelastic waves are governed not only by hydrodynamics but also significantly depend on the elastic properties of the ice layer. Its critical resonance regime differs entirely from pure gravity waves

exhibiting unique nonlinear wave phenomena near the critical velocity. This paper systematically reviews research progress on hydroelastic waves excited by moving loads

covering theoretical studies

numerical simulations

and experimental observations. Key aspects include potential flow models

viscoelastic effects

nonlinear dynamics

and ice sheet dynamic characteristics under complex motion scenarios (such as accelerated motion

3D effects

channel boundaries

and submerged loads). The study reveals that viscoelastic and nonlinear effects are key mechanisms explaining the finite-amplitude response at critical velocities

while the propagation characteristics of hydroelastic waves in complex motion scenarios further demonstrate the diversity of ice-water coupling interactions. This paper provides critical theoretical and technical insights for polar resource exploitation

ship navigation safety

and the design of large-scale floating marine structures.

关键词

Keywords

references

BENNETTS L G, BITZ C M, FELTHAM D L, et al. Theory, modelling and observations of marginal ice zone dynamics: Multidisciplinary perspectives and outlooks[J]. Philosophical Transactions Series A, Mathematical, Physical, and Engineering Sciences, 2022, 380(2235): 20210265. doi: 10.1098/rsta.2021.0265

STUROVA I V. Oblique incidence of surface waves on an elastic plate[J]. Journal of Applied Mechanics and Technical Physics, 1999, 40(4): 604−610. doi: 10.1007/BF02468434

SQUIRE V A, DIXON T W. How a region of cracked sea ice affects ice-coupled wave propagation[J]. Annals of Glaciology, 2001, 33: 327−332. doi: 10.3189/172756401781818806

NI B, SEMENOV Y A, KHABAKHPASHEVA T, et al. Nonlinear ice sheet/liquid interaction in a channel with an obstruction[J]. Journal of Fluid Mechanics, 2024, 983: A41. doi: 10.1017/jfm.2024.177

WILSON J T. Coupling between moving loads and flexural waves in floating ice sheets[J]. Cold Regions Science and Technology, 1955.

TAKIZAWA T. Response of a floating sea ice sheet to a steadily moving load[J]. Journal of Geophysical Research: Oceans, 1988, 93(C5): 5100−5112. doi: 10.1029/JC093iC05p05100

SQUIRE V A, ROBINSON W H, LANGHORNE P J, et al. Vehicles and aircraft on floating ice[J]. Nature, 1988, 333(6169): 159−161. doi: 10.1038/333159a0

GREENHILL A G. Wave motion in hydrodynamics[J]. American Journal of Mathematics, 1886, 9(1): 62. doi: 10.2307/2369499

PĂRĂU E, DIAS F. Nonlinear effects in the response of a floating ice plate to a moving load[J]. Journal of Fluid Mechanics, 2002, 460: 281−305. doi: 10.1017/s0022112002008236

KOHOUT A L, MEYLAN M H. An elastic plate model for wave attenuation and ice floe breaking in the marginal ice zone[J]. Journal of Geophysical Research: Oceans, 2008, 113(C9): C09016. doi: 10.1029/2007JC004434

SKENE D M, BENNETTS L G, MEYLAN M H, et al. Modelling water wave overwash of a thin floating plate[J]. Journal of Fluid Mechanics, 2015, 777: R3. doi: 10.1017/jfm.2015.378

MEYLAN M H, BENNETTS L G, MOSIG J E M, et al. Dispersion relations, power laws, and energy loss for waves in the marginal ice zone[J]. Journal of Geophysical Research: Oceans, 2018, 123(5): 3322−3335. doi: 10.1002/2018JC013776

JOHNSEN K, KALISCH H, PĂRĂU E I. Ship wave patterns on floating ice sheets[J]. Scientific Reports, 2022, 12(1): 18931. doi: 10.1038/s41598-022-23401-8

SQUIRE V A, HOSKING R J, KERR A D, et al. Moving loads on ice plates[M]. Dordrecht: Springer Netherlands, 1996. doi: 10.1007/978-94-009-1649-4

GOLD L W. Use of ice covers for transportation[J]. Canadian Geotechnical Journal, 1971, 8(2): 170−181. doi: 10.1139/t71-018

DICKINS D F, COX M, THORLEIFSON J. Arctic patrol hovercraft: An initial feasibility study[C]//SNAME 8th International Conference and Exhibition on Performance of Ships and Structures in Ice, Banff, Alberta, Canada. SNAME, 2008. doi: 10.5957/icetech-2008-115

WINKLER E. Die Lehre von der Elastizität und Festigkeit: Mit besonderer Rücksicht auf ihre Anwendung in der Technik, für polytechnische Schulen, Bauakademien, Ingenieure, Maschinenbauer, Architecten, etc, 1. Teil [M]. German: MV-Technik, 2021.

MITTELSTEDT C. Theory of plates and shells[M]. New York: McGraw-Hill, 1996.

MATHEWS P M. Vibrations of a beam on elastic foundation II[J]. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift Für Angewandte Mathematik und Mechanik, 1959, 39(1-2): 13−19. doi: 10.1002/zamm.19590390105

KHEYSIN D Y. Moving load on an elastic plate which floats on the surface of an ideal fluid[J]. 1963.

KHEYSIN D E. Dynamics of the ice cover[G]. Leningrad: Gidrometeoizdat, 1967.

SCHULKES R M S M, SNEYD A D. Time-dependent response of floating ice to a steadily moving load[J]. Journal of Fluid Mechanics, 1988, 186: 25−46. doi: 10.1017/s0022112088000023

DAVYS J W, HOSKING R J, SNEYD A D. Waves due to a steadily moving source on a floating ice plate[J]. Journal of Fluid Mechanics, 1985, 158: 269−287. doi: 10.1017/s0022112085002646

TABATA T. A measurement of visco-elastic constants of sea ice[J]. Journal of the Oceanographical Society of Japan, 1955, 11(4): 185−189. doi: 10.5928/kaiyou1942.11.185

ZHESTKAYA V D. Numerical solution of the problem of an ice sheet under a moving load[J]. Journal of Applied Mechanics and Technical Physics, 1999, 40(4): 770−775. doi: 10.1007/BF02468457

HOSKING R J, SNEYD A D, WAUGH D W. Viscoelastic response of a floating ice plate to a steadily moving load[J]. Journal of Fluid Mechanics, 1988, 196: 409−430. doi: 10.1017/s0022112088002757

COLEMAN B D, NOLL W. Foundations of linear viscoelasticity[J]. Reviews of Modern Physics, 1961, 33(2): 239−249. doi: 10.1103/revmodphys.33.239

DINVAY E, KALISCH H, PĂRĂU E. Waves generated by moving loads on ice plates: Viscoelastic approximations[J]. Wave Motion, 2022, 114: 103011. doi: 10.1016/j.wavemoti.2022.103011

WANG R X, SHEN H H. Gravity waves propagating into an ice-covered ocean: A viscoelastic model[J]. Journal of Geophysical Research: Oceans, 2010, 115(C6): C06024. doi: 10.1029/2009JC005591

ZHAO X, SHEN H H. Three-layer viscoelastic model with eddy viscosity effect for flexural-gravity wave propagation through ice cover[J]. Ocean Modelling, 2018, 131: 15−23. doi: 10.1016/j.ocemod.2018.08.007

ZHAO X, ZHANG C P. A theoretical model of wind-wave growth over an ice-covered sea[J]. Boundary-Layer Meteorology, 2021, 178(1): 1−19. doi: 10.1007/s10546-020-00552-7

MEYLAN M H, BENNETTS L G, PETER M A. Water-wave scattering and energy dissipation by a floating porous elastic plate in three dimensions[J]. Wave Motion, 2017, 70: 240−250. doi: 10.1016/j.wavemoti.2016.06.014

PĂRĂU E, VANDEN-BROECK J M. Nonlinear two- and three-dimensional free surface flows due to moving disturbances[J]. European Journal of Mechanics - B/Fluids, 2002, 21(6): 643−656. doi: 10.1016/S0997-7546(02)01212-8

FORBES L K. Surface waves of large amplitude beneath an elastic sheet. Part 1. High-order series solution[J]. Journal of Fluid Mechanics, 1986, 169: 409−428. doi: 10.1017/s0022112086000708

FORBES L K. Surface waves of large amplitude beneath an elastic sheet. Part 2. Galerkin solution[J]. Journal of Fluid Mechanics, 1988, 188: 491−508. doi: 10.1017/s0022112088000813

MILEWSKI P A, VANDEN-BROECK J M, WANG Z. Hydroelastic solitary waves in deep water[J]. Journal of Fluid Mechanics, 2011, 679: 628−640. doi: 10.1017/jfm.2011.163

VANDEN-BROECK J M, DIAS F. Gravity-capillary solitary waves in water of infinite depth and related free-surface flows[J]. Journal of Fluid Mechanics, 1992, 240: 549−557. doi: 10.1017/s0022112092000193

WANG Z, VANDEN-BROECK J M, MILEWSKI P A. Two-dimensional flexural-gravity waves of finite amplitude in deep water[J]. IMA Journal of Applied Mathematics, 2013, 78(4): 750−761. doi: 10.1093/imamat/hxt020

AKERS B F, AMBROSE D M, SULON D W. Periodic traveling interfacial hydroelastic waves with or without mass[J]. Zeitschrift Für Angewandte Mathematik und Physik, 2017, 68(6): 141. doi: 10.1007/s00033-017-0884-7

GAO T, WANG Z, VANDEN-BROECK J M. New hydroelastic solitary waves in deep water and their dynamics[J]. Journal of Fluid Mechanics, 2016, 788: 469−491. doi: 10.1017/jfm.2015.695

CHO Y, AKYLAS T R. Forced waves near resonance at a phase-speed minimum[J]. Studies in Applied Mathematics, 2009, 123(1): 1−15. doi: 10.1111/j.1467-9590.2009.00444.x

HENTSCHKE R. Small oscillations[M]// Classical Mechanics. Berlin: Spinger, 2016. doi: 10.1007/978-3-319-48710-6_6

MILEWSKI P A, VANDEN-BROECK J M, WANG Z. Steady dark solitary flexural gravity waves[J]. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2013, 469(2150): 20120485. doi: 10.1098/rspa.2012.0485

WU T Y. Generation of upstream advancing solitons by moving disturbances[J]. Journal of Fluid Mechanics, 1987, 184: 75−99. doi: 10.1017/s0022112087002817

WANG Z. A universal bifurcation mechanism arising from progressive hydroelastic waves[J]. Theoretical and Applied Mechanics Letters, 2022, 12(1): 100315. doi: 10.1016/j.taml.2021.100315

DINVAY E, KALISCH H, PĂRĂU E I. Fully dispersive models for moving loads on ice sheets[J]. Journal of Fluid Mechanics, 2019, 876: 122−149. doi: 10.1017/jfm.2019.530

WHITHAM G B, WHITHAM G B. Linear and nonlinear waves[M]. New York Chichester Weinheim: Wiley, 1999. doi: 10.1002/9781118032954

ABLOWITZ M J, FOKAS A S, MUSSLIMANI Z H. On a new non-local formulation of water waves[J]. Journal of Fluid Mechanics, 2006, 562: 313. doi: 10.1017/s0022112006001091

CRAIG W, SULEM C. Numerical simulation of gravity waves[J]. Journal of Computational Physics, 1993, 108(1): 73−83. doi: 10.1006/jcph.1993.1164

GUYENNE P, PĂRĂU E I. Finite-depth effects on solitary waves in a floating ice sheet[J]. Journal of Fluids and Structures, 2014, 49: 242−262. doi: 10.1016/j.jfluidstructs.2014.04.015

XIA X, SHEN H T. Nonlinear interaction of ice cover with shallow water waves in channels[J]. Journal of Fluid Mechanics, 2002, 467: 259−268. doi: 10.1017/s0022112002001477

GUYENNE P, PǍRǍU E I. Computations of fully nonlinear hydroelastic solitary waves on deep water[J]. Journal of Fluid Mechanics, 2012, 713: 307−329. doi: 10.1017/jfm.2012.458

DYSTHE K B. Note on a modification to the nonlinear Schrödinger equation for application to deep water waves[J]. Proceedings of the Royal Society of London A Mathematical and Physical Sciences, 1979, 369(1736): 105−114. doi: 10.1098/rspa.1979.0154

GUYENNE P, KAIRZHAN A, SULEM C. A Hamiltonian Dysthe equation for hydroelastic waves in a compressed ice sheet[J]. Journal of Fluid Mechanics, 2024, 1002: A24. doi: 10.1017/jfm.2024.1161

MILEWSKI P A, WANG Z. Three dimensional flexural-gravity waves[J]. Studies in Applied Mathematics, 2013, 131(2): 135−148. doi: 10.1111/sapm.12005

PLOTNIKOV P I, TOLAND J F. Modelling nonlinear hydroelastic waves[J]. Philosophical Transactions Series A, Mathematical, Physical, and Engineering Sciences, 2011, 369(1947): 2942−2956. doi: 10.1098/rsta.2011.0104

WANG Z, GUAN X, VANDEN-BROECK J M. Progressive flexural-gravity waves with constant vorticity[J]. Journal of Fluid Mechanics, 2020, 905: A12. doi: 10.1017/jfm.2020.750

SEMENOV Y A. Nonlinear flexural-gravity waves due to a body submerged in the uniform stream[J]. 2021, 33(5): 052115. doi: 10.1063/5.0048887

NI B Y, XIONG H, HAN D F, et al. A review of ice deformation and breaking under flexural-gravity waves induced by moving loads[J]. Journal of Marine Science and Application, 2025, 24(1): 35−52. doi: 10.1007/s11804-024-00408-3

ZAKHAROV V E. Stability of periodic waves of finite amplitude on the surface of a deep fluid[J]. Journal of Applied Mechanics and Technical Physics, 1968, 9(2): 190−194. doi: 10.1007/BF00913182

ALBERELLO A, PĂRĂU E I. A dissipative nonlinear Schrödinger model for wave propagation in the marginal ice zone[J]. Physics of Fluids, 2022, 34(6): 061702. doi: 10.1063/5.0089866

GUYENNE P, PĂRĂU E I. Numerical study of solitary wave attenuation in a fragmented ice sheet[J]. Physical Review Fluids, 2017, 2(3): 034002. doi: 10.1103/physrevfluids.2.034002

MILINAZZO F, SHINBROT M, EVANS N W. A mathematical analysis of the steady response of floating ice to the uniform motion of a rectangular load[J]. Journal of Fluid Mechanics, 1995, 287: 173−197. doi: 10.1017/s0022112095000917

NUGROHO W S, WANG K, HOSKING R J, et al. Time-dependent response of a floating flexible plate to an impulsively started steadily moving load[J]. Journal of Fluid Mechanics, 1999, 381: 337−355. doi: 10.1017/s0022112098003875

WANG K, HOSKING R J, MILINAZZO F. Time-dependent response of a floating viscoelastic plate to an impulsively started moving load[J]. Journal of Fluid Mechanics, 2004, 521: 295−317. doi: 10.1017/s002211200400179x

BABAEI H, VAN DER SANDER J, SHORT N, et al. Lake ice cover deflection induced by moving vehicles: Comparing theoretical results with satellite observations[C]// TAC's 2016 Conference of the Transportation Association of Canada, New Research and Developments in Road Safety Session, Toronto. Ottawa: NRC Publications Archive, 2016.

DU D B, QIN D H, MA Y H, et al. The age of cryopolitics is coming[J]. Bulletin of Chinese Academy of Sciences, 2020, 35(4): 514−522(in Chinese).

SUN B, TANG X Y, XIAO E Z, et al. Ice and snow runway engineering in the Antarctica: Current status and prospect[J]. Strategic Study of CAE, 2021, 23(2): 161−168(in Chinese). doi: 10.15302/J-SSCAE-2021.02.022

WHITE G, MCCALLUM A. Review of ice and snow runway pavements[J]. International Journal of Pavement Research and Technology, 2018, 11(3): 311−320. doi: 10.1016/j.ijprt.2017.11.002

MATIUSHINA A, POGORELOVA A, KOZIN V. Effect of impact load on the ice cover during the landing of an airplane[J]. International Journal of Offshore and Polar Engineering, 2016, 26(1): 6−12. doi: 10.17736/ijope.2016.vs08

MILES J, SNEYD A D. The response of a floating ice sheet to an accelerating line load[J]. Journal of Fluid Mechanics, 2003, 497: 435−439. doi: 10.1017/s002211200300675x

HOSKING R J, MILINAZZO F. Two-dimensional response of a floating ice plate to a line load moving at variable speed[J]. Journal of Fluid Mechanics, 2022, 938: A2. doi: 10.1017/jfm.2022.109

ZENG L D. Study on gravity wave of viscoelastic ice plate bending induced by moving pressure surface[D]. Harbin: Harbin Engineering University, 2022(in Chinse).

NI B Y, XIONG H, ZENG L D, et al. Research on the characteristics of flexural gravity waves propagating in a solid ice channel induced by a moving pressure surface[J]. Chinese Journal of Theoretical and Applied Mechanics, 2024, 56(10): 2838−2853(in Chinese). doi: 10.6052/0459-1879-24-216

HUANG Y, SUN C, TIAN Y F. Experimental study of the ice breaking resistance on an air cushion platform[J]. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(3): 714−727(in Chinese). doi: 10.6052/0459-1879-20-418

KHABAKHPASHEVA T, SHISHMAREV K, KOROBKIN A. Large-time response of ice cover to a load moving along a frozen channel[J]. Applied Ocean Research, 2019, 86: 154−165. doi: 10.1016/j.apor.2019.01.020

REN K, WU G X, LI Z F. Hydroelastic waves propagating in an ice-covered channel[J]. Journal of Fluid Mechanics, 2020, 886: A18. doi: 10.1017/jfm.2019.1042

BROCKLEHURST P. Hydroelastic waves and their interaction with fixed structures[D]. Norwich: University of East Anglia, 2012.

TKACHEVA L A. Behavior of a semi-infinite ice cover under a uniformly moving load[J]. Journal of Applied Mechanics and Technical Physics, 2018, 59(2): 258−272. doi: 10.1134/S0021894418020098

STUROVA I V. Motion of an external load over a semi-infinite ice sheet in the subcritical regime[J]. Fluid Dynamics, 2018, 53(1): 49−58. doi: 10.1134/s0015462818010135

NI B Y, HAN D F, DI S C, et al. On the development of ice-water-structure interaction[J]. Journal of Hydrodynamics, 2020, 32(4): 629−652. doi: 10.1007/s42241-020-0047-8

TKACHEVA L A. Edge waves produced by the motion of a vessel in an ice channel[J]. Journal of Applied Mechanics and Technical Physics, 2019, 60(5): 850−864. doi: 10.1134/s0021894419050080

XUE Y Z, ZENG L D, NI B Y, et al. Hydroelastic response of an ice sheet with a lead to a moving load[J]. 2021, 33(3): 037109. doi: 10.1063/5.0037682

ZENG L D, KOROBKIN A A, NI B Y, et al. Flexural-gravity waves in ice channel with a lead[J]. Journal of Fluid Mechanics, 2021, 921: A10. doi: 10.1017/jfm.2021.335

GAO S, ZHANG J, ZHANG J. Key technology and ice-breaking mechanism of polar air-cushion ice-breaking/transportation platform[J]. Ship Boat, 2018, 29(6): 117−122(in Chinese).

LI H Y. Numerical simulation of underwater revolving body making waves and breaking ice[D]. Harbin: Harbin Engineering University, 2023(in Chinese).

POGORELOVA A V, KOZIN V M. Flexural-gravity waves due to unsteady motion of point source under a floating plate in fluid of finite depth[J]. Journal of Hydrodynamics, Ser B, 2010, 22(5): 71−76. doi: 10.1016/S1001-6058(09)60172-4

STUROVA I V. Unsteady three-dimensional sources in deep water with an elastic cover and their applications[J]. Journal of Fluid Mechanics, 2013, 730: 392−418. doi: 10.1017/jfm.2013.303

POGORELOVA A V. Unsteady motion of a source in a fluid under a floating plate[J]. Journal of Applied Mechanics and Technical Physics, 2011, 52(5): 717−726. doi: 10.1134/S0021894411050051

POGORELOVA A V, KOZIN V M. Motion of a load over a floating sheet in a variable-depth pool[J]. Journal of Applied Mechanics and Technical Physics, 2014, 55(2): 335−344. doi: 10.1134/s0021894414020163

LINTON C M. Radiation and diffraction by a submerged sphere advancing in water waves of finite depth[J]. Proceedings of the Royal Society of London Series A: Mathematical and Physical Sciences, 1995, 448(1932): 29−54. doi: 10.1098/rspa.1995.0002

NEWMAN J N. The theory of ship motions[M]//Advances in Applied Mechanics Volume 18. Amsterdam: Elsevier, 1979: 221–283. doi: 10.1016/s0065-2156(08)70268-0

KOZIN V M, POGORELOVA A V. Effect of the viscosity properties of ice on the deflection of an ice sheet subjected to a moving load[J]. Journal of Applied Mechanics and Technical Physics, 2009, 50(3): 484−492. doi: 10.1007/s10808-009-0065-3

ZEMLYAK V L, KOZIN V M, BAURIN N O, et al. Influence of ice cover on the motion of a submerged body[J]. Journal of Physics: Conference Series, 2019, 1214: 012003. doi: 10.1088/1742-6596/1214/1/012003

ZEMLYAK V L, KOZIN V M, VASILYEV A S. Influence of the shape of a submarine vessel on the ice breaking capacity of flexural-gravity waves[J]. IOP Conference Series: Earth and Environmental Science, 2022, 988(4): 042040. doi: 10.1088/1755-1315/988/4/042040

KELLER J, LI J C. Development and prospect of linear wave propagation theory[J]. Advances in Mechanics, 1980, 10(4): 101−118(in Chinese).

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