Why are M2 and S2 both semidiurnal but not identical?

S2 is tied to the solar day and has a 12 h period. M2 follows the Moon relative to Earth rotation, so its semidiurnal period is about 12.42 h. Their slow phase drift creates the spring-neap beat.

What makes spring tides large?

When M2 and S2 are nearly in phase, their semidiurnal phasors add and the daily tidal range grows. About a week later they oppose each other more strongly, producing neap tides.

What is diurnal inequality?

Diurnal constituents such as K1 and O1 make one high tide differ from the next high tide, and one low differ from the next low, especially in mixed-tide regions.

Is the amphidromic map realistic?

Only qualitatively. Real amphidromic systems depend on basin geometry, Coriolis force, shelf bathymetry, friction, and resonance. The canvas shows the phase-rotation idea without solving shallow-water equations.

Tide Constituents & Beats

M2/S2/K1/O1 harmonic superposition, spring-neap beat cycle, diurnal inequality, and an amphidromic phase cartoon.

Live graphs

About this model

Ocean tides at a site are commonly represented as a harmonic sum η(t) = Σ A_i cos(2πt/T_i − φ_i). This simulator keeps four major constituents: M2, the principal lunar semidiurnal tide with period 12.4206 h; S2, the solar semidiurnal tide with period 12 h; K1, a lunisolar diurnal constituent; and O1, a lunar diurnal constituent. Because M2 and S2 have close but unequal frequencies, their vector sum grows and shrinks with a beat period of about 14.77 days, producing spring and neap modulation. K1 and O1 add diurnal inequality, so successive highs and lows need not be equal. The map is an amphidromic cartoon: M2 phase rotates around a point of near-zero amplitude while co-tidal phase lines radiate outward. It is a harmonic teaching model, not a local tide prediction service.

Who it's for: Oceanography, coastal engineering, astronomy, and geophysics students learning tidal constituents, harmonic analysis, and spring-neap modulation.

Key terms

Tidal constituent
M2 tide
S2 tide
K1 tide
O1 tide
Spring-neap cycle
Diurnal inequality
Amphidromic point
Harmonic analysis

How it works

Tide harmonic constituents simulator: M2/S2/K1/O1 superposition, spring-neap beats, diurnal inequality, and an amphidromic phase cartoon.

Key equations

η(t) = Σ A_i cos(2πt/T_i − φ_i), i = M2,S2,K1,O1

T_spring-neap = 1 / |1/T_S2 − 1/T_M2| ≈ 14.77 days

Frequently asked questions

Why are M2 and S2 both semidiurnal but not identical?: S2 is tied to the solar day and has a 12 h period. M2 follows the Moon relative to Earth rotation, so its semidiurnal period is about 12.42 h. Their slow phase drift creates the spring-neap beat.
What makes spring tides large?: When M2 and S2 are nearly in phase, their semidiurnal phasors add and the daily tidal range grows. About a week later they oppose each other more strongly, producing neap tides.
What is diurnal inequality?: Diurnal constituents such as K1 and O1 make one high tide differ from the next high tide, and one low differ from the next low, especially in mixed-tide regions.
Is the amphidromic map realistic?: Only qualitatively. Real amphidromic systems depend on basin geometry, Coriolis force, shelf bathymetry, friction, and resonance. The canvas shows the phase-rotation idea without solving shallow-water equations.

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