This chapter solves the deep water wave problem, a nonlinear dispersive equation that models surface gravity waves in water with infinite depth. This is a classic problem in fluid dynamics and demonstrates advanced numerical techniques:
The deep water equation is physically relevant for ocean waves, water tanks, and fundamental research in nonlinear wave dynamics.
The deep water wave system can be written as:
\[\frac{\partial \eta}{\partial t} = -\frac{\partial \phi}{\partial x} \bigg|_{z=0}\]
\[\frac{\partial \phi}{\partial t} = -g\eta - \frac{1}{2}|\nabla \phi|^2 \bigg|_{z=0}\]
where: - \(\eta(x,t)\) is the surface elevation (the wave height) - \(\phi(x,t)\) is the velocity potential - \(g\) is gravitational acceleration
Using the Matsuno formulation (a clever reformulation of the equations), we can write:
\[\frac{\partial \eta}{\partial t} = -\frac{\partial \phi}{\partial x}\]
\[\frac{\partial \phi}{\partial t} = -g\eta - \frac{1}{2}\left(\frac{\partial \phi}{\partial x}\right)^2 - \frac{1}{2}\mathcal{H}\left[\frac{\partial \phi}{\partial x}\right]^2\]
where \(\mathcal{H}\) is the Hilbert transform and \(g = 1\) (normalized).