When water depth is less than one wavelength, waves transition to shallow-water (long-wave) behavior where wave speed depends only on water depth (c = √gh), not frequency. Tidal waves, storm surge, and tsunamis are shallow-water waves that travel across ocean basins with minimal attenuation and amplify dramatically in shallow bays and harbors.
Compare wave speeds in deep ocean and near shore; examine how tsunami waves slow and steepen as they approach land.
Students often think tsunami waves are large everywhere; they are actually small-amplitude in deep ocean and only become dangerous near shore.
From your introduction to gravity waves on the ocean surface, you know that waves involve orbital motion of water particles and that wave behavior depends on the relationship between wavelength and water depth. The critical transition happens when water depth becomes less than about half the wavelength — at that point, the circular orbits of water particles flatten against the bottom, and the wave "feels" the seafloor. Shallow-water waves (also called long waves) are the extreme case: their wavelength is so much greater than the water depth that the entire water column participates in the wave motion, from surface to bottom.
The most important result in shallow-water wave theory is elegantly simple: wave speed depends only on water depth, given by c = √(gh), where g is gravitational acceleration and h is water depth. Frequency and wavelength drop out entirely. This has profound consequences. In the deep ocean where h ≈ 4,000 meters, a shallow-water wave travels at about 200 m/s (roughly 700 km/h — the speed of a commercial jet). In coastal water where h = 10 meters, the same wave slows to about 10 m/s. The wave does not lose energy as it slows; instead, its energy compresses into a shorter wavelength and taller amplitude. This is why tsunamis and storm surges, which are barely detectable in the open ocean, become devastating walls of water at the coast.
A tsunami is the most dramatic shallow-water wave. Generated by seafloor displacement — earthquakes, submarine landslides, or volcanic eruptions — a tsunami can have a wavelength of 200 kilometers or more, making even the deepest ocean "shallow" relative to its wavelength. In the open Pacific, a tsunami might be only 30–50 centimeters tall, spread over a wavelength so long that a ship would rise and fall imperceptibly over several minutes. But as it approaches shore and water depth decreases, the c = √(gh) relationship forces the wave to slow dramatically. The trailing portion of the wave, still in deeper water, continues at high speed, compressing the wave energy into an ever-shorter, ever-taller form. This process — called shoaling — can amplify a half-meter open-ocean wave into a 10-meter-plus coastal surge.
Tidal waves are also shallow-water waves, though driven by gravitational forcing from the Moon and Sun rather than by sudden seafloor displacement. The tidal "wave" has a wavelength equal to half the Earth's circumference — there is no ocean deep enough for this to behave as a deep-water wave. Storm surge operates on the same physics: a broad dome of water pushed by sustained winds and low atmospheric pressure behaves as a long wave whose amplification on approach to shore follows the same depth-dependent speed relationship. Understanding c = √(gh) is the single key that unlocks the behavior of all these phenomena — from the arrival time of a tsunami across the Pacific to the amplification of storm surge in a narrowing bay.
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