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Tides are the periodic rise and fall of sea level caused primarily by the differential gravitational pull of the Moon (and secondarily the Sun) on Earth's ocean. The Moon creates two tidal bulges: one facing the Moon and one on the opposite side due to inertia. As Earth rotates, most locations experience two high tides and two low tides per day (semidiurnal), though geographic and basin resonance effects produce diurnal or mixed tidal patterns in many regions. Spring tides (stronger) occur when the Sun, Earth, and Moon align; neap tides (weaker) occur when they form a right angle.
Draw diagrams of tidal forcing geometry for new moon, full moon, and quarter moon configurations. Compare tidal gauge records from different coastal stations to observe how geography influences tidal range and pattern.
From Newton's law of gravitation, you know that every mass attracts every other mass with a force proportional to their masses and inversely proportional to the square of the distance between them. Tides arise not from the Moon's gravitational pull itself, but from the differential force — the difference in gravitational pull across Earth's diameter. The side of Earth facing the Moon is about 12,740 km closer than the far side, so it feels a slightly stronger pull. This difference stretches the ocean into an elongated shape with two bulges: one toward the Moon (where gravity is slightly stronger than average) and one on the opposite side (where gravity is slightly weaker, and inertia from Earth-Moon orbital motion carries the water outward). As Earth rotates through these two bulges roughly once per day, most coastal locations experience two high tides and two low tides in each 24-hour-and-50-minute tidal cycle (the extra 50 minutes accounting for the Moon's orbital advance).
The Sun also exerts a tidal force on Earth's oceans — its enormous mass compensates partly for its much greater distance — but its tidal effect is only about 46% of the Moon's. When the Sun, Earth, and Moon align (at new moon and full moon), their tidal forces add together, producing spring tides with the largest tidal ranges. When the Sun and Moon are at right angles relative to Earth (first and third quarter moon), their forces partially cancel, producing neap tides with the smallest ranges. This fortnightly cycle between spring and neap tides is one of the most predictable rhythms in the ocean, and you can verify it by checking any tide table for two weeks of data.
If Earth were a smooth sphere uniformly covered by deep ocean, the tidal pattern would be simple and symmetric. But continents, ocean basin shapes, and seafloor topography complicate the picture enormously. Each ocean basin responds to tidal forcing as a resonant system — water sloshes back and forth within the basin like water in a bathtub, and the basin's natural resonance period determines how it amplifies or dampens the tidal signal. The result is a pattern of amphidromic points — locations where the tidal range is essentially zero — around which the tidal wave rotates. The Bay of Fundy in Canada has the world's largest tidal range (over 16 meters) not because it faces the Moon most directly, but because its geometry creates a near-perfect resonance with the semidiurnal tidal period.
These geographic effects also explain why some locations experience patterns other than the standard two-highs-two-lows semidiurnal tide. Parts of the Gulf of Mexico have diurnal tides (one high and one low per day), while much of the Pacific coast sees mixed tides (two unequal highs and lows per day). Tidal prediction uses harmonic analysis, decomposing the observed tide into dozens of individual sinusoidal components (called tidal constituents), each corresponding to a specific astronomical forcing frequency. The principal lunar semidiurnal constituent (M₂) is the strongest, but accurate prediction requires summing many constituents. This approach — rooted in the trigonometric ratios you reviewed as a prerequisite — allows tide tables to forecast water levels years in advance with remarkable precision, which is essential for navigation, coastal engineering, and understanding how tidal currents transport sediment and nutrients.