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We present a numerical study of spatially quasi-periodic travelling waves on the surface of an ideal fluid of infinite depth. This is a generalization of the classic Wilton ripple problem to the case when the ratio of wavenumbers satisfying the dispersion relation is irrational. We propose a conformal mapping formulation of the water wave equations that employs a quasi-periodic variant of the Hilbert transform to compute the normal velocity of the fluid from its velocity potential on the free surface. We develop a Fourier pseudo-spectral discretization of the travelling water wave equations in which one-dimensional quasi-periodic functions are represented by two-dimensional periodic functions on the torus. This leads to an overdetermined nonlinear least-squares problem that we solve using a variant of the Levenberg–Marquardt method. We investigate various properties of quasi-periodic travelling waves, including Fourier resonances, time evolution in conformal space on the torus, asymmetric wave crests, capillary wave patterns that change from one gravity wave trough to the next without repeating and the dependence of wave speed and surface tension on the amplitude parameters that describe a two-parameter family of waves.
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PARAMETER ESTIMATION FOR THE TRUNCATED KdV MODEL THROUGH A DIRECT FILTER METHOD
In this work, we develop a computational method to provide real-time detection for water bottom topography based on observations on surface measurements, and we design an inverse problem to achieve this task. The forward model that we use to describe the feature of the water surface is thetruncated Korteweg-de Vries equation, and we formulate the inversion mechanism as an online parameter estimation problem, which is solved by a direct filter method. Numerical experiments are carried out to show that our method can effectively detect abrupt changes of water depth.
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- PAR ID:
- 10476412
- Publisher / Repository:
- 0.1615/JMachLearnModelComput.2023047711
- Date Published:
- Journal Name:
- Journal of Machine Learning for Modeling and Computing
- Volume:
- 4
- Issue:
- 1
- ISSN:
- 2689-3967
- Page Range / eLocation ID:
- 109 to 132
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1615/JMachLearnModelComput.2023047711
