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Elliptic and hyperelliptic functions describing the particle motion beneath smallamplitude water waves with constant vorticity
A selection of nonlinear problems in water waves, analysed by perturbationparameter techniques
1.  School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne NE1 7RU, United Kingdom 
The classical problem of water waves is formulated, and then various parameter choices are incorporated. The first problem examines the properties of smallamplitude, periodic waves over constant vorticity and, invoking the ideas of breakdown, scaling and matching, the possibility of stagnation is investigated. In the second case, a CamassaHolm equation is derived; this is found to be relevant only for the horizontal velocity component in the flow at a specific depth. However, this special, integrable equation requires the retention of a number of terms of different asymptotic orderwhich is the least satisfactory way to use these methods. The final example, which shows how some new aspects of a familiar problem can be obtained by these methods, develops an asymptotic description of edge waves. This leads to a single equation that captures all aspects of the runup pattern at a shoreline.
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