Waves and Tides

Describes surface-wave propagation and the ocean's tidal response. Theory, formulas, worked examples and interactive graphs.

Conceptual context

The ocean is never truly still. Fast disturbances such as wind waves travel across its surface, while slow oscillations such as tides raise and lower broad regions of water. Both involve $g$, but they operate on very different temporal and spatial scales. The central question is not merely how high the water rises. It is how physical information propagates: what controls $c$, how $h$, $T$, and $\lambda$ are connected, and why $\eta$ changes with $\phi$. Keeping these ideas separate allows a student to interpret buoy records, coastal waves, tsunamis, and tide gauges without confusing local water motion with the advance of a disturbance.

🟢 Essential level

Before calculating, identify whether the observation describes rapid propagation, temporal repetition, or a slow tidal oscillation.

A surface wave transports energy and phase, but the water itself does not move forward at the same speed as the crest.

$c$ describes motion of the wave pattern. In shallow water, the seabed influences the water column, so $h$ becomes decisive. $T$ is the time between crests, while $f$ counts cycles per second. $\lambda$ links temporal and spatial descriptions. Tides are slower oscillations: level varies around a reference, phase locates the cycle, and amplitude must be distinguished from the range between extremes. Confusing these two families of phenomena — fast wave propagation and slow tidal oscillation — is a common mistake among students new to oceanography, because both share the same restoring force, gravity, even though they operate on radically different time and space scales. It is therefore worth asking which kind of phenomenon is actually being observed before reaching for any formula.

🔵 Formal level

Basic formulas

Every relation must preserve units, state its depth regime, and distinguish a propagating phase from local water oscillation.

In the shallow-water regime, $c$ depends on $g$ and $h$:

Shallow-water wave speed

This relation assumes that $h$ is much smaller than $\lambda$. The result is a positive phase-speed magnitude; propagation direction must be specified separately. $T$ and $f$ are inverse quantities:

Period-frequency relation

The link between temporal and spatial scales is:

Wavelength relation

Therefore, a wave with a long $T$ may have an enormous $\lambda$ even when its visible height is not remarkable. If the source remains steady, $T$ is often approximately conserved while $c$ and $\lambda$ change as $h$ varies.

In shallow water the $c$ depends on the $h$, so the $\lambda$ of a wave of given $T$ is set by the bottom and the $g$:

Wavelength in shallow water

Real student questions

Does a wave carry water all the way to shore? It mainly carries energy and phase. Water parcels move locally, although currents, drift, and breaking can create net transport.

Why does a wave slow near a beach? Because $h$ decreases and the seabed constrains the whole water column.

Does $T$ change near the coast? In a stationary approximation it remains nearly constant while $c$ and $\lambda$ change.

Are a tide and a tsunami the same thing? No. Both can behave as long waves, but their causes and scales differ.

Why is high tide not always at the same clock time? Because $\phi$ evolves through several astronomical constituents and local effects.

Final synthesis

$h$ controls $c$ for long waves; $T$ and $f$ set the time scale, and $\lambda$ connects time with space. In tides, $\eta$ is interpreted through $\eta 0$, $A$, and $\phi$. Every formula remains useful only inside its physical regime and reference system.