What determines the speed of waves in water?

Question

While I was walking my dog this morning, I passed over a canal filled with boats, barges, and kayaks all of different masses and moving at different speeds. I noticed that all of these vessels left behind wakes, and the waves of these wakes moved through the water at different rates, though my ability to make good observations was limited. This made me wonder: what determines the velocity of these waves? I know that sound travels through water at 1,480 m/s or so, but surface waves travel at different rates (tsunamis travel faster than the ripples I make in my bowl of soup when I blow on it). In particular, how is the velocity of wake waves determined?

Answer 1

The speed of water waves depends on a few factors; some of them are inherent to the planet, including gravity and surface tension, whereas others are dependent on the depth of the water, and the wavelength of the waves (which is in turn determined by things like depth and wind speed.)

A deep-water wave moves at about g/2pi or 1.56 times the wave period in seconds. (The units are m/s.) So, long waves move faster.

In a shallow-water situation where the depth of the water is small compared to the wavelength, the velocity is the square root of (gravity times water depth), with units again in meters / second. So, as water gets shallower, the waves travel more slowly. This is what cases the waves to break as they approach the shore. Waves break when the ratio of wave height to water depth is about 3:4.

Note that tsunamis are a special case of a very long wavelength wave, up to 200 km, so the shallow-water calculation applies even though they are in the open ocean. If you run that calculation on water than is 4000 meters deep, you get a speed of over 200 m/s, which is like 450 mph. That's pretty fast!