Tide Harmonics

Chapter 3 of The Nature of Code, Oscillation, covers simple harmonic motion and the sum of sinusoids. Ocean tides are a physical sum of sinusoidal harmonic constituents.

Ocean tides are that sum of sinusoids in the wild, so I plot NOAA water levels at San Francisco against a two-constituent M2+S2 model.

The page plots three days of measured water level from NOAA station 9414290 (San Francisco) and overlays a two-term demo model: the principal lunar semidiurnal constituent M2 and the principal solar semidiurnal constituent S2, with fixed amplitudes and a phase that nudges toward the series. Two sinusoids recover the main beat of the measured curve, which is Chapter 3’s point about how far a small sum of sines extends.

Each constituent is simple harmonic motion \(x(t) = A\sin(\omega t + \phi)\) with \(\omega = 2\pi/T\), and the overlay is \(h(t) = Z_0 + A_1\sin(\omega_{\mathrm{M2}} t + \phi_1) + A_2\sin(\omega_{\mathrm{S2}} t + \phi_2)\) (sine phase convention, equivalent to a cosine form up to a phase shift). The periods are M2 \(T = 12.42\) h and S2 \(T = 12.00\) h. Their beat frequency \(f_{S2} - f_{M2}\) has a period near 14.8 days and produces the spring-neap cycle.

Data source: NOAA CO-OPS (Tides & Currents) API
Endpoint: https://api.tidesandcurrents.noaa.gov/api/prod/datagetter?station=9414290&product=water_level&...
Access: Public, no key. Loaded from an hourly server-side refresh when possible, with a bundled snapshot fallback, and a live browser fetch only when the API allows CORS. Data window: Sep 13, 2026 - Sep 16, 2026 (pulled Sep 16, 2026 UTC)