Particle Methods for Water Waves

The purpose of this section is to examine the numerical model of water waves, particularly with a method of meshless methods, such as Smoothed Particle Hydrodynamics (SPH). This part of the course will deal with numerical models, numerical interpolation, and meshfree method, with application to SPH for waves. 1 Numerical Modeling The study of water waves has taken the usual scientific paths–experimentation, laboratory studies, theoretical work, and now numerical methods. One of the earliest theoretical models was the linear wave theory due to Airy (1845).1 This model (described completely in Dean and Dalrymple (1991)) assumed that the water was incompressible and the flow irrotational, meaning that the viscosity of the fluid was neglected. These assumptions allowed the use of a velocity potential, φ(x, t), from which wave-induced velocities could be obtained via −∇φ = u (note the use of the arbitrary minus sign.) The equation for the velocity potential is the Laplace equation: ∇2φ(x, t) = 0 (1) Associated with this governing equation for the fluid domain are boundary conditions that exist at the bottom, the free surface, and the lateral sides of the domain. The domain will be taken as a rectangle, wit