• Choose a category
• All
• Classic-Fiction

By Root 425 0

Sines and cosines can be treated just like any other function in calculus; only the notation is different. We can still take derivatives and integrals, and those values correspond, respectively, to the slope of the tangent line and the area under the curve. There are some interesting connections between sines and cosines that provide shortcuts when taking derivatives and integrals. For instance, note that the sine wave starts at 0 on the graph, rises, and flattens out at the peak; conversely, the slope of the tangent line of our sine wave starts at 1 and goes to 0. This is exactly how the cosine wave behaves, and we can deduce from this that the derivative of the sine is the cosine.52

Similarly, the cosine wave starts at 1 and goes to 0, while its slope starts at 0 and goes to minus 1 on the graph. So, the derivative of the cosine is minus the sine. Working our way full circle, we see that the same holds true when we’re talking about finding the derivatives of minus the sine and minus the cosine: the derivative of minus the sine is minus the cosine, while the derivative of minus the cosine brings us right back to where we started: the sine.

The integral follows the same circular pattern in reverse, as it undoes the work of the derivative. The integral of the sine is minus the cosine; the integral of minus the cosine is minus the sine; the integral of minus the sine is the cosine; and the integral of the cosine is the sine. The above holds true whether we are talking about sound waves, light waves, gravitational waves, or ocean waves. So we can use calculus to analyze any kind of change and motion in wave phenomena.

BREAKING THE WAVES

Most ocean waves eventually “break” as they move into shallower water, which is what happens when the wave base can no longer support its top, causing it to collapse. On this Kona beach, the waves don’t break all at once, but peel to the right or left when they break. We have the pleasant spilling or rolling version of breaking waves. The plunging variety can break too suddenly, dumping surfers and pushing them to the bottom with a lot more force than one might think. There’s a lot of energy in those ocean waves: Depending on the size, it can be as much as five to ten tons per square yard. Surging waves might not even break, but their powerful undertows can drag unwary swimmers and surfers into deeper, more dangerous waters.

Breaking waves produce infrasonic signals as well as audible sounds, and Garces’ work exploits this feature to develop a technique he calls real-time surf infrasonic monitoring, or, as he describes it, “the deep sound of one wave plunging.” Garces is specifically studying breaking waves along Oahu’s North Shore, widely deemed to be a surfer’s Mecca.

There are three types of wave breaks that produce infrasound: plunging breaks, cliff breaks, and reef breaks. Garces’ research focuses on the latter. He is attempting to isolate the sound of a single wave in the process of breaking. Essentially, he’s tracking moving wavefronts with sound sensitive pressure sensors strewn along the ocean floor, enhanced with conventional seismography. The idea is to use the collected raw data to determine wave height and other properties, for example, to better identify potential hazards to surfers. It’s trickier than it seems: Such predictions currently rely on the observations of surfers themselves to determine wave heights. True, there are sensor-equipped buoys in the cove designed to collect that information, but the data are insufficient to make accurate predictions.

This might seem surprising, since a similar buoy system works quite well along the coastline of San Diego, where the Scripps Institute deploys a set of buoys and crunches the raw data using clever algorithms to separate the meaningful signals from background noise. This enables them to plot the direction, speed, and curvature of incoming waves to determine the location of the sound source and to make more accurate predictions.

So why wouldn’t it work on Oahu? I asked Geoffrey Edelmann, an acoustician at the Naval Research