Ocean tides, the Moon and changing coastal water levels: sources & model

Keep one clock at three real waterfronts. Reveal which water level rises next, build a tide from repeating rhythms, and discover why the Moon does not give every coast the same pattern.

What supports the explanation?

What this model assumes

1. Pinned historical records cover June 1 through July 7, 2024; no live tide or navigation service. Historical-date predictions were retrieved September 9, 2026, not asserted to be forecasts issued before the observations.
2. The pier, ruler and water surface are authored monitoring representations, not surveyed stations, bathymetry or documentary time-lapse imagery.
3. Four fitted rhythms approximate NOAA predictions over a limited window; their amplitudes, phases and fitted mean are not recovered physical constants or permanent mean sea level.
4. The static force lens and dimensionless spring/neap illustration do not solve local ocean dynamics or turn force multipliers into coastal water heights.
5. Hourly or six-hour samples do not identify exact observed extrema; connecting lines are illustrative interpolation.
6. Local MLLW references differ by station. Units and datum changes must preserve physical water position and original source flags.
7. An observation residual is not a weather attribution, current velocity, hazard judgment or long-term trend estimate.
8. The paper activity is an original data-reading exercise, not a tested classroom protocol, tide tank or beach-safety assessment.
9. Subtract a shared acceleration: In an Earth-centered translating, nonrotating frame, let μ be the distant body’s gravitational parameter, D its position from Earth’s center and r a surface marker’s position. The differential acceleration is aₜ(r)=μ[(D−r)/|D−r|³−D/|D|³]. We subtract the distant body’s acceleration at Earth’s center—not Earth’s own gravity. Both original lunar attractions point toward the Moon. The far-side difference points outward relative to Earth’s center because that side is attracted less strongly.
10. An arrow is not a solved ocean: For unit surface normal n=r/|r|, the radial part is (aₜ·n)n and the tangential part is aₜ−(aₜ·n)n. Tangential forcing helps redistribute water. Our static force lens does not solve ocean motion, friction, the Coriolis effect, continents, elasticity or loading. Its exaggerated arrows are acceleration directions and magnitudes, not predicted water heights or literal paths traveled by water.
11. Near and far are not exactly equal: With the source along +x, Earth radius 6,371 km and Moon distance 384,400 km, the exact near-side differential x acceleration is about +1.128×10⁻⁶ m/s²; the far-side value is about −1.073×10⁻⁶ m/s². The unequal magnitudes are a finite-distance result. These distances define an illustrative static geometry, not a dated ephemeris. Lunar μ comes from JPL DE440, with km³/s² converted to m³/s².
12. Why inverse cube appears: For Earth’s radius small compared with the distance D, the leading term is aₜ≈(μ/D³)[3(nD·r)nD−r], where nD points toward the source. A distance of 0.9 times the reference gives leading strength 1/0.9³≈1.3717 times as large. This is a differential-strength approximation; whole point-mass attraction instead scales as inverse square. Exact surface-vector ratios differ slightly. Neither factor multiplies the historical station records.
13. Four frequencies, local fingerprints: The source speeds are M2 28.984104°/hour, S2 30°/hour, K1 15.041069°/hour and O1 13.943035°/hour. Period equals 360° divided by speed: about 12.4206, 12, 23.9345 and 25.8193 hours. M2 is principal lunar semidiurnal; S2 is principal solar semidiurnal; K1 is lunisolar diurnal; O1 is principal lunar diurnal. K1 is not a Moon-only switch. Different fitted amplitudes and phases give the three stations different curves.
14. Exactly what our fit means: Let t be hours after June 1, 2024 00:00 UTC. Our model is h₄(t)=b₀+Σ[cⱼ cos(ωⱼt)+sⱼ sin(ωⱼt)], where ωⱼ is source speed times π/180 radians per hour. Nine coefficients—one mean offset and four cosine/sine pairs—were fitted by least squares to 720 June hourly NOAA predictions at each station. These coefficients are authored short-window fits, not new NOAA harmonic constants, observations or a complete hydrodynamic model.
15. Phase changes timing, not amplitude: Each fitted pair can also be written A cos(ωt−φ), with A=√(c²+s²) and φ=atan2(s,c). Adding positive lag δ gives A cos(ωt−φ−δ); a +90° lag moves the corresponding peak later by one-quarter period. This is a mathematical change to one modeled component. It does not physically move a harbor or determine what an altered coastline would do.
16. Why published phase constants need more context: NOAA’s fuller harmonic equation includes a mean level, amplitudes, time-varying amplitude factors, equilibrium arguments and phase constants: h=H₀+ΣfH cos[at+(V₀+u)−K]. Published station amplitude H and epoch K are not a self-contained dated prediction. A phase labeled GMT must not be treated as the angle at June 1 without the other arguments. A 30-day fit also cannot cleanly separate all close frequencies, such as K1/P1 or S2/K2.
17. Try data the fit did not use: July 1–7 contributes 168 withheld hourly predictions per station. Full baseline four-rhythm RMS differences from NOAA are about 0.185 m for Boston, 0.115 m for San Diego and 0.038 m for Pensacola; maximum absolute differences are about 0.425, 0.299 and 0.094 m. These quantify this approximation against the pinned NOAA prediction product. They are not NOAA operational forecast errors or observational uncertainty. Refitting on July would end its status as unused test data.
18. Spring and neap as interacting rhythms: In a separate dimensionless illustration, q(t)=cos(ωM2t)+0.46 cos(ωS2t). Its amplitude envelope is E(t)=√[1+0.46²+0.92 cos((ωS2−ωM2)t)], with maximum 1.46 and minimum 0.54. Repeated alignment takes about 14.7653 days. Neap does not mean no tide. The 0.46 is an illustrative amplitude, not every station’s ratio; the envelope is not a measured height or exact tidal-cycle range.
19. A lunar phase is not a universal coastal appointment: Alignment near new and full Moon supports spring-tide tendencies; quadrature supports neap tendencies. Actual local timing depends on phases and other components, and need not coincide exactly with the named lunar phase. K1/O1 have a different alignment interval, about 13.6608 days. Do not substitute that interval for the M2/S2 beat or infer a station’s highest water from a phase calendar alone.
20. A datum belongs to a place and an epoch: MLLW is based on the mean of lower low waters over a tidal datum epoch. The selected station metadata use 1983–2001. MSL-minus-MLLW offsets are 1.586 m at Boston, 0.897 m at San Diego and 0.188 m at Pensacola. Thus hMSL=hMLLW−offset. Equal numeric readings at different local MLLW zeros are not necessarily equal elevations above a common worldwide plane. A negative height can be valid below the chosen zero.
21. Quality flags remain attached to values: The NOAA hourly_height flag f lists inferred-value status first and expected-level-limit status second. A 1 means the condition is flagged. Boston and San Diego have 888 unflagged rows each; Pensacola has 884 unflagged rows and four flagged entries in this pack. Inferred readings are shown distinctly and excluded from default observation summaries. Sigma s is the standard deviation of the short-interval samples used to calculate the level, not a 95% uncertainty band or forecast-error bound.
22. A residual is a question, not an attribution: Define the observation residual as observed height minus NOAA’s prediction at the same station, time, unit and datum. The opening Boston value is 2.838−2.699=0.139 m. Wind, atmospheric pressure, river flow and other influences, along with model and measurement issues, can contribute to water-level differences. This scalar residual cannot uniquely identify a cause, diagnose a hazard or establish a sea-level-rise rate.
23. Samples, extrema and range are different: An hourly reading is not necessarily the instant of high or low water; the six-hour paper subset is sparser still. NOAA’s separate predicted event list includes Boston’s low at June 1 04:51 UTC, 0.242 m MLLW. Its 05:00 hourly prediction is 0.247 m. A plotted line between samples is interpolation. The largest minus smallest value over a month is not automatically the range of one adjacent high/low tidal cycle.
24. Classification is useful, with boundaries: Parker discusses several constituent-ratio classifications and identifies Boston as semidiurnal in his example; the selected current API metadata label Boston Mixed. We preserve that source discrepancy. Boston’s chosen records are dominated by the twice-per-lunar-day rhythm rather than forcing a universal category or two identical daily highs. A predominantly diurnal station can have secondary structure and an unexpected event count within one UTC calendar day.
25. Water level is not current speed: Tide describes vertical water-level changes; tidal current describes horizontal water motion. Their phase relationship depends on local dynamics. High or low water does not universally mean slack current, and a station’s tide classification need not describe its currents. This lesson contains water-level records, not measured current-speed or safe-clearance data.

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About the cover illustration

Actual browser render of Brytalearn’s authored monitoring pier, ruler and water cutaway, driven by pinned NOAA CO-OPS records. Geometry is an educational representation, not surveyed station CAD, bathymetry or documentary footage.