Harbour Tide and Tidal Flat

What is modelled

The water level is the sum of two waves: the main lunar tide with a period of 12.42 hours and the main solar tide with a period of exactly 12 hours. They drift in and out of step, so the daily range grows to spring tides and shrinks to neap tides every 14.77 days. On a gently sloping flat, a small change of level moves the waterline a long way, so large areas dry out. The water at the quay presses on the wall with the hydrostatic pressure, and the lamp of the lighthouse stands at a fixed height above the mean level, so the tide changes how far away it can be seen.

Equations

• Level η(t) = A [sin(2 π t / 12.42) + r sin(2 π t / 12)], spring-neap period 1 / (1/12 − 1/12.42).
• Waterline position x = −N η on a flat of slope 1 : N; dry area W N (ztop − η).
• Pressure p = ρ g h, force on a wall F = ½ ρ g h² per metre.
• Horizon d = R acos(R / (R + h)) ≈ 3.57 √h km.

Checks run in the page

• High-water timing and the spring-neap period.
• Integrated pressure against the wall formula.
• Dry flat area from a step-by-step sum against the formula.
• Horizon distance from two methods against the sailor rule.

Honest simplifications

Only two tidal components, no tidal currents, wind set-up or surge, a perfectly planar flat and a geometric horizon with no atmospheric refraction. Real harbours have dozens of constituents and the 3D scene's numbers are illustrative, so these are not tide-table predictions.

• Press Pure lunar tide, then Harbour as in the 3D scene, and see how the solar tide makes the daily range change.
• Press Very flat mudflat and watch the dry area swing by hundreds of hectares.
• Press Tall lighthouse, small tide and see how little the tide changes the light range.