On the Evolution of Shallow-Water Waves
Abstract
This project studies the evolution of shallow-water waves for an initial-value problem using experiments, modeling, and analysis. To model the behavior of a wavetrain at a water surface, we compute solutions to the linearized boundary value problem for water waves. We solve for the dispersion relation between frequency and wavenumber and consider various limits of the solution. For shallow-water waves, we determine a soliton solution of the full KdV equation and solve the linearized equation with given initial conditions, relevant to the experiments. In the WGP Fluid Mechanics Lab, a system fabricated with a submerged plate abruptly moves horizontally to generate a soliton or vertically to generate an evolving wavetrain. We obtain measurements of the surface displacement as a function of distance from the plate using two capacitance-type wave gages. We compare the experimental results to predictions from our mathematical models. Analytic solutions of the KdV equation agree reasonably well with the measurements of the surface displacement obtained from the experiments on solitons. Analytic solutions of the linearized KdV equation provide qualitative insight into the observed evolution of the evolving wavetrains.
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