Physics
Wave Speed Variation with Depth
Quick fact
In shallow water, wave speed depends almost entirely on water depth, and a tsunami can travel as fast as 800 km/h in the deep ocean but slows to a crawl in coastal shallows.
Why this is interesting
Have you ever noticed that ocean waves always seem to turn to face the shore, no matter how they start? The secret lies in how wave speed changes with water depth.
Read the full explanation
Understanding Wave Speed Variation with Depth
Imagine a long line of waves moving diagonally toward a beach. The part of the wave that reaches shallower water first begins to slow down, while the part still in deeper water keeps moving quickly. This difference in speed makes the wave crest bend, turning the wave front so it becomes more parallel to the shore. This bending is called refraction. Wave speed is not constant: in deep water, it depends on wavelength, but in shallow water, the governing factor is simply the water depth. The deeper the water, the faster the wave travels. As waves approach a coast, their speed decreases, causing them to bunch up (wavelength shortens) and grow taller—this is why waves always seem to rise and break at the shore.
A deeper explanation
The physical reason lies in the restoring force of gravity and how the water particles move. For waves where the wavelength is much longer than the water depth, the wave 'feels' the bottom. The water particles move in flat elliptical paths, and the speed c is given by the shallow-water approximation: c = √(g × h), where g is the acceleration due to gravity and h is the depth. Thus, speed increases with the square root of depth. In deep water where depth is much larger than wavelength, the bottom has no effect, and the speed depends on wavelength. This depth-dependence is why waves slow down in shallower water, bend, and concentrate energy on headlands. It also explains why tsunamis—which have enormous wavelengths—travel as shallow-water waves across the entire ocean, moving at speeds proportional to the square root of the ocean depth. Understanding this principle is crucial for predicting coastal flooding and designing coastal infrastructure.