Stability of traveling wave solutions to the Whitham equation

The Whitham equation was proposed as an alternate model equation for the simplified description of unidirectional wave motion at the surface of an inviscid fluid. An advantage of the Whitham equation over the KdV equation is that it provides a more faithful description of short waves of small amplit...

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Bibliographic Details
Published in:Physics letters. A 2014-06, Vol.378 (30-31), p.2100-2107
Main Authors: Sanford, Nathan, Kodama, Keri, Carter, John D., Kalisch, Henrik
Format: Article
Language:English
Subjects:
Computational fluid dynamics
Cut-off
Dispersion
Fluid flow
Fluids
Fourier–Floquet–Hill method
KdV
Mathematical analysis
Mathematical models
Modulational instability
Stability
Water waves
Wavelengths
Whitham
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Summary:The Whitham equation was proposed as an alternate model equation for the simplified description of unidirectional wave motion at the surface of an inviscid fluid. An advantage of the Whitham equation over the KdV equation is that it provides a more faithful description of short waves of small amplitude. Recently, Ehrnström and Kalisch [19] established that the Whitham equation admits periodic traveling-wave solutions. The focus of this work is the stability of these solutions. The numerical results presented here suggest that all large-amplitude solutions are unstable, while small-amplitude solutions with large enough wavelength L are stable. Additionally, periodic solutions with wavelength smaller than a certain cut-off period always exhibit modulational instability. The cut-off wavelength is characterized by kh0=1.145, where k=2π/L is the wave number and h0 is the mean fluid depth. •The Whitham equation is used as a model for waves on shallow water.•The Whitham equation admits periodic traveling-wave solutions.•All large-amplitude traveling-wave Whitham solutions are unstable.•Small-amplitude solutions with sufficient period are stable.
ISSN:0375-9601
1873-2429
DOI:10.1016/j.physleta.2014.04.067