Harmonic Tide Predictor

What is modelled

Harmonic tidal prediction treats the sea level as a mean level plus a sum of cosines, each with a speed fixed by the motions of the Earth, Moon and Sun, and an amplitude and phase that depend on the place. Five constituents are used: M2 (principal lunar, 28.984 deg/h), S2 (principal solar, 30 deg/h), N2 (lunar elliptic, 28.440 deg/h), K1 (lunisolar diurnal, 15.041 deg/h) and O1 (lunar diurnal, 13.943 deg/h).

Equations

• Height h(t) = Z0 + Σ Ai cos(ωi t − gi), period Ti = 360° / speedi, so M2 has T = 12.42 h.
• Spring-neap beat of M2 and S2: amplitude |AM2 + AS2 ei(ωS2−ωM2)t| between AM2 − AS2 and AM2 + AS2, period 14.77 days.
• Form factor F = (K1 + O1) / (M2 + S2) classes the tide: below 0.25 semidiurnal, up to 1.5 mixed, above 3 diurnal.
• Harmonic analysis: A ≈ 2 ⟨ (h − Z0) cos(ω t) ⟩ over a long record.

Checks run in the page

• M2 alone: the spacing of numerically found high waters against 360 / 28.984.
• The spring-neap envelope extremes against M2 ± S2.
• Harmonic analysis recovers the set M2 amplitude from the synthetic record.
• A year of samples stays inside Z0 plus the sum of amplitudes and is centred on Z0.

Honest simplifications

Only five of the roughly sixty constituents used by real predictors are included, there are no nodal corrections, no shallow-water terms and no meteorological surge. Amplitudes and phases are free sliders rather than values fitted to a real station, so this is a teaching model, not a tide table.

• Choose Big spring-neap and watch the daily range swell and shrink over two weeks.
• Choose Diurnal gulf: the tide now has one high water a day and the form factor jumps.
• Set the S2 phase lag away from the M2 lag and see the spring tides arrive on a different day.