Solitary waves were first discovered by John Scott Russell whilst observing a canal boat. He observed that a wave formed when the canal boat stopped and it travelled down the river keeping it’s shape regardless of how the river turned and twisted. Later some theory was developed by Diederik Korteweg and Gustav de Vries to try and understand what they came up with and the resulting equation is now called the KdV (or Korteweg de Vries) equation and it is the model equation for the “great wave of translation” that was observed by Scott Russell.
The types of wave that I am studying, generalizes the above wave as I have a moving pressure distribution on the interface of fluid and air or possible some topography on the bottom of the fluid and a vertical electric field. I use use same approximations as Korteweg and de Vries by taking shallow water and small amplitude approximation to obtain a fifth order partial differential equation for the channel flow. I was able to extend my results in 1D physics to 2D physics using a “quasi 1D” approach by examining “thin” waves and obtaining a generalisation of the Kadomtsev–Petviashvili(KP) equation.
I have also been working on “hanging” flows. There are flows which defy the Rayleigh-Taylor instability by using a horizontal electric field. I derived a basic set of equations which describe a periodic wave of a given wavelength. There are a certain set of wavelengths where it isn’t possible to get a set of waves. The paper discussing this work is in the thesis or you can go to the publications page and see the paper that was accepted for publication.
A real fluid however has some viscosity however small and this may change the equations which we need to model fluids with viscosity. This work is joint work with Denys Dutykh. We have currently derived the equations for the infinite depth case and there is a picture of the free surface profile in the blog. My current topic to to try and come up with a generalisation of the equation I derived including a weakly viscous fluid which should be a better description of free surface flows in electrohydrodynamics.