Water Waves

Chances are you've been to a beach before, and it's quite likely that there were waves whether you were at the ocean, a lake, or some other sizeable body of water. If you've ever watched the waves for a while, you might have noticed that they tend to arrive parallel, or nearly so, to the shoreline. Even if they are at angle when they are far away, they still reach the shoreline nearly parallel. Why? Another thing noticeable to anyone who has watches waves, and is visible in the above photo, is the crashing and "breaking" of waves as them come towards the shoreline. How does that happen?

The first question can be answered by fluid dynamics. In shallow water, that is at a depth which is smaller than the wavelength (measure of distance between wave crests), the speed of the waves depends on the water depth and gravity only. Specifically, the speed is equal to the square root of gravity times depth. This tells us that waves will move slowest in the shallowest water. Let's assume that there is a nice downward slope as you move away from the shoreline. Start by imagining a wave that begins parallel to the shore. As it moves closer, the shallower water will cause it to slow down. Now, what happens if the wave is rotated so that it starts at an angle to the shoreline?

If you think of the wave crest as a straight line, one end will be closer to shore than the other if it's at an angle. The far end is in deeper water and will travel faster than the shallow end. This causes the wave crest to turn until both ends are at the same depth or it hits the shore (whichever comes first). If you're having trouble imagining this scenario, try thinking about an axle with a wheel on each end. If you move both wheels at the same speed, the axle will move in a straight line. What happens if you hold one wheel stationary (more or less) and move the other? The moving wheel will rotate around the other one. Now, allow both wheels to turn but move one a little faster than the other. The axle will move forward while also gradually turning. This is analogous to our wave scenario.

That's one question answered, so on to the next one: what causes waves topple over? It may seem strange but this answer relies on the same principle as the first one. In this case, we can no longer treat the wave crest as just a line; it does have a finite width, after all. Since the back of the wave is in deeper water than the front, the back will travel faster and eventually catch the front. The water particles will accumulate behind the crest until it reaches a point when the added water forces the wave to topple over. If the wave is large enough, you may notice how the top rolls over the bottom. As water piles up behind the wave, the water becomes even deeper than the water in front. Consequently, the water in back moves even faster, particularly that on the surface. This helps the top of the wave crest to roll as the wave topples over. Keep in mind, however, that this is only valid for waves in shallow water. It's not often that you see breaking waves in the middle of a lake, partly for this reason. When you do see them, the wind is generally very strong and helps force them to topple over.

Fortunately, these are easy things to go see for yourself, so get out and enjoy the coming summer. Maybe the next time you find yourself at a beach, you'll pay a little more attention to the waves.


References
Waves in Fluids (video) by National Committee for Fluid Mechanics Films (1960s; more here)