Next Article in Journal
Balancing Efficiency and Economics in Organic Rankine Cycles with Multistage Turbines for Sustainable Waste Heat Utilization
Previous Article in Journal
Energy-Efficient, Multi-Agent Deep Reinforcement Learning Approach for Adaptive Beacon Selection in AUV-Based Underwater Localization
 
 
Search for Articles:
Title / Keyword
Author / Affiliation / Email
Journal
Article Type
 
 
Advanced Search
 
Section
Special Issue
Volume
Issue
Number
Page
 
Journals
JMSE
Volume 14
Issue 3
10.3390/jmse14030263
jmse-logo
Submit to this Journal Review for this Journal Propose a Special Issue
► Article Menu

Article Menu

share Share announcement Help format_quote Cite question_answer Discuss in SciProfiles
first_page
settings
Order Article Reprints
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

Traveling Wave Solutions of the Extended Displacement Shallow-Water Equation

by
Zheng Yao
1,* and
Feng Wu
2
1
College of Transportation Engineering, Dalian Maritime University, Dalian 116026, China
2
Department of Engineering Mechanics, Dalian University of Technology, Dalian 116023, China
*
Author to whom correspondence should be addressed.
J. Mar. Sci. Eng. 2026, 14(3), 263; https://doi.org/10.3390/jmse14030263
Submission received: 8 January 2026 / Revised: 21 January 2026 / Accepted: 25 January 2026 / Published: 27 January 2026
(This article belongs to the Section Coastal Engineering)
Versions Notes

Abstract

Traditionally, the shallow-water equations have been formulated and developed within the Eulerian framework for studying shallow-water wave problems. In this paper, we present a Lagrangian-based approach based on Hamilton’s variational principle to derive an extended displacement shallow-water equation (EDSWE). Using elliptic functions, we obtain exact traveling wave solutions of the resulting EDSWE. The conditions for the formation of various wave types—including cnoidal waves, looped waves, and peaked waves—are systematically analyzed and summarized. The proposed displacement method, grounded in the Lagrangian description, provides an analytical framework for hydrodynamic problems and can be applied to symplectic formulations in fluid mechanics.
Keywords: Lagrangian system; variational principle; shallow-water waves; elliptic function; traveling wave solution; wave generation Lagrangian system; variational principle; shallow-water waves; elliptic function; traveling wave solution; wave generation

Share and Cite

MDPI and ACS Style

Yao, Z.; Wu, F. Traveling Wave Solutions of the Extended Displacement Shallow-Water Equation. J. Mar. Sci. Eng. 2026, 14, 263. https://doi.org/10.3390/jmse14030263

AMA Style

Yao Z, Wu F. Traveling Wave Solutions of the Extended Displacement Shallow-Water Equation. Journal of Marine Science and Engineering. 2026; 14(3):263. https://doi.org/10.3390/jmse14030263

Chicago/Turabian Style

Yao, Zheng, and Feng Wu. 2026. "Traveling Wave Solutions of the Extended Displacement Shallow-Water Equation" Journal of Marine Science and Engineering 14, no. 3: 263. https://doi.org/10.3390/jmse14030263

APA Style

Yao, Z., & Wu, F. (2026). Traveling Wave Solutions of the Extended Displacement Shallow-Water Equation. Journal of Marine Science and Engineering, 14(3), 263. https://doi.org/10.3390/jmse14030263

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Article metric data becomes available approximately 24 hours after publication online.
Back to TopTop