Harbour Entrance Wave Diffraction

What is modelled

Long waves on water of constant depth h obey the linear shallow-water equations. A wave maker on the left edge sends in a sinusoid of height H and period T; a breakwater with a gap of width b blocks the rest of the wave; the other edges absorb the waves so that the numerical sea has no walls. Behind the gap the energy spreads by diffraction, and the narrower the gap compared with the wavelength the more nearly circular the spreading.

Equations

• ∂η/∂t + h (∂u/∂x + ∂v/∂y) = 0, ∂u/∂t = −g ∂η/∂x − γ u (same for v).
• Wave speed c = √(g h), wavelength λ = c T.
• Single-slit far field |sin(π b sinθ / λ)| / (π b sinθ / λ), first null at sinθ = λ / b for b > λ.
• Quarter-wave basin period T = 4 L / √(g h).

Checks run in the page

• Phase speed from two probes of a 1D channel against √(g h).
• Volume and energy conservation in a closed 2D basin.
• Quarter-wave resonance period of a channel closed at one end.
• Reflection of the radiation boundary for a pulse.

Honest simplifications

The equations are linear and non-dispersive, so they are only accurate when kh is below about 0.3; storm presets have larger kh and real waves there travel slower. There is flat bottom, no breaking, no wave-structure transmission and no reflection from a real harbour wall. The far-field panel is the idealised single-slit result.

• Choose Narrow gap and watch the waves spread almost circularly behind the opening.
• Choose Wide opening and see a sharp shadow zone behind each end of the breakwater.
• Increase the depth and the wavelength grows with the square root of depth.