Numerical Simulation of Surface and Internal Wave Excitation due to an Air Pressure Wave

Abstract

We consider the irrotational motion of inviscid and incompressible fluids in two layers, as illustrated in Figure 1.The still water depths of the upper and lower layers are h 1 (x) and h 2 (x), respectively, and h(x) = h 1 (x) + h 2 (x). We assume that the densities of the upper and lower layers, ρ 1 and ρ 2 , respectively, are uniform and constant, and that the fluids do not mix even in motion. The water surface displacement, interface displacement, and seabed position are denoted by ζ(x, t), η(x, t), and b(x), respectively. Friction is ignored everywhere for simplicity. The velocity potentials of the upper and lower layers are ϕ 1 (x, t) and ϕ 2 (x, t), respectively.The nonlinear shallow water equations of velocity potential considering the pressure on the water surface, p 0 (x, t), are Upper Layer ∂η/∂t = ∂ζ/∂t + ∇[(ζ -η) ∇ϕ 1 ],(1) ∂ϕ 1 /∂t = -[gζ + p 0 /ρ 1 + (∇ϕ 1 ) 2 /2],(2)