13.3: The Geometry of Waves

If you take a snapshot of the ocean surface, what do you observe? Its shape appears almost random with hills and valleys and ridges and depressions of different shapes and sizes. But take several photos in succession, and animate them by scrolling rapidly through them, and you see that the hills and valleys and ridges and depressions move in a deliberate manner and interact with each other in unexpected ways.

Though it might not seem possible, the shape of the sea surface at any one point in time can be fairly well described by combining all the individual waves and their motions. The combination of different waves and their shapes provides a kind of model of the sea surface, which is useful for investigating the impacts of waves on structures and ships.

In mathematical terms, waves may be described by three geometric shapes. The most well-known shape is a sine wave, the shape of a smooth, up-and-down motion. This symmetrical rising and falling shape can be visualized as a time series graph of a point on a slowly spinning wagon wheel. The x-axis represents the progression of time, and the y-axis represents the height of the point at any moment as measured from the center of the wheel. The center of the wheel may be thought of as the resting level of the ocean surface, and the height of the point above or below the center—the resting level—determines the height of the point at any given moment.

At the top of the wheel, the point reaches its greatest height, and so the sine wave reaches its maximum height. The point of maximum height of a wave is called the wave crest. At the bottom of the wheel, the point reaches its lowest height, and so the sine wave reaches its minimum height, the point known as the wave trough. The horizontal distance between successive crests is called the wavelength. The vertical distance from the crest to the trough—the diameter of the wheel—is called the wave height. Waves appear as sine waves when their wave height is small compared to their wavelength. Ocean swell often appears sinusoidal in shape.

The second shape is called a trochoid. This too can be visualized by a wagon wheel, only this time, the point is placed on one of the spokes, and the wheel is rolled underneath a flat surface. Trochoids are similar to sine waves, but they’re less smooth and have a pointed crest. Most ocean waves resemble a trochoid.

The third shape is called a cycloid. To generate this shape, we use the same approach as a trochoid, only the point is placed directly at the perimeter of the wheel. Cycloids resemble trochoids, except that their crests are very pointed. Waves breaking in the ocean resemble this shape.

Three lessons can be drawn from this shape-defining discussion. First, waves progress much like a wagon wheel rolling down a trail. Their motion traces out geometric shapes whose equations are known. Second, the motion of water through which a wave passes is circular. Put another way, as a wave moves through the ocean surface, water particles—a term meant to convey something like a dot’s worth of water molecules—move in circular orbits. The circular motions of water particles are called wave orbitals. (See below.) Third, mathematical descriptions of waves provide great insights into how and where they propagate and allow oceanographers to predict their motions and impacts on shores and structures. Mathematical models of waves also give a heads-up to surfers. Surf prediction—predicting the timing, location, and intensity of shore-breaking waves—is big business for commercial and recreational interests alike.

Of course, all of these forms may be present on the sea surface at any time. What happens as these waves encounter each other? Like the people you hang out with, some waves pick you up and some waves let you down. If the crests (or troughs) of two (or more) waves coincide, they add to each other to create a larger wave as they pass through each other, what is called constructive interference. This is like when your friends help you out; you’re better, stronger, faster. When two waves pass through each other so that their crests and troughs do not coincide, they tend to cancel each other out in what’s called destructive interference. If you have friends with whom you can get nothing done, this would be them. The sea surface at any given time is a jumble of waves moving in different directions that constructively and destructively interfere with each other to create the dynamic seascape of the wavy ocean. Check out this phenomenon next time you visit a wavy lake or beach.