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The science
Shallow-water equations
For long waves in a lake of depth h the linearised equations are ∂η/∂t = -∂(h u)/∂x and ∂u/∂t = -g ∂η/∂x + τ/(ρ h) - r u, with surface elevation η, depth-mean velocity u, wind stress τ = ρa CD U² (ρa = 1.2 kg/m³, CD = 1.3e-3) and bottom friction rate r. The waves travel at c = sqrt(g h), and the lake is closed at both ends (u = 0).
Setup and seiche
A steady wind tilts the surface until the pressure gradient balances the stress: ∂η/∂x = τ/(ρ g h), a total level difference of τ L/(ρ g h), i.e. half of that above and below the mean level at the ends. Stop the wind and the tilted surface is a superposition of standing waves with periods given by Merian's formula Tn = 2L / (n sqrt(g h)). The third graph shows kinetic and potential energy swapping every quarter period; friction drains the total.
Checks
(1) A free first-mode hump in a flat, frictionless lake oscillates with the Merian period (and the discrete energy stays constant). (2) A steady 20 m/s wind on a 12 m deep lake gives the analytical end setup after the friction damps the transient. (3) The flux form of the scheme conserves water volume exactly.
Simplifications
One dimension, linear (no wave steepening or wetting), uniform wind stress, no Coriolis force, no stratification (real lakes also have internal seiches that are much slower), the surface exaggeration in the picture is displayed on the canvas. The 3D page is a general scene viewer; this companion models the lake physics itself.
What to try
- Choose the friction-free preset and compare the measured period with the Merian value.
- Halve the depth and see the period grow by sqrt(2).
- Raise the wind to 30 m/s on a shallow lake and compare the setup curve.