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The spatial Whitham equation

Published online by Cambridge University Press:  03 October 2024

John D. Carter Open the ORCID record for John D. Carter [Opens in a new window] ,
Diane Henderson  and
Panayotis Panayotaros
John D. Carter*
Affiliation:
Mathematics Department, Seattle University, 901 12th Ave, Seattle, WA 98122, USA
Diane Henderson
Affiliation:
Department of Mathematics, Penn State University, 218 McAllister Building, University Park, PA 16802, USA
Panayotis Panayotaros
Affiliation:
Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Apdo. Postal 20-716, 01000 Cd. México, México
*
†Email address for correspondence: [email protected]
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Abstract

The Whitham equation is a non-local, nonlinear partial differential equation that models the temporal evolution of spatial profiles of surface displacement of water waves. However, many laboratory and field measurements record time series at fixed spatial locations. To directly model data of this type, it is desirable to have equations that model the spatial evolution of time series. The spatial Whitham (sWhitham) equation, proposed as the spatial generalization of the Whitham equation, fills this need. In this paper, we study this equation and apply it to water-wave experiments on shallow and deep water. We compute periodic travelling-wave solutions to the sWhitham equation and examine their properties, including their stability. Results for small-amplitude solutions align with known results for the Whitham equation. This suggests that the systems are consistent in the weakly nonlinear regime. At larger amplitudes, there are some discrepancies. Notably, the sWhitham equation does not appear to admit cusped solutions of maximal wave height. In the second part, we compare predictions from the temporal and spatial Korteweg–de Vries and Whitham equations with measurements from laboratory experiments. We show that the sWhitham equation accurately models measurements of tsunami-like waves of depression, waves of elevation, and solitary waves on shallow water. Its predictions also compare favourably with experimental measurements of waves of depression and elevation on deep water. Accuracy is increased by adding a phenomenological damping term. Finally, we show that neither the sWhitham nor the temporal Whitham equation accurately models the evolution of wave packets on deep water.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Mathematical Foundations: General fluid mechanics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 996 , 10 October 2024 , A42
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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