The Calculus Diaries: How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse

by Jennifer Ouellette

  The lab is also near at least one prime local surfing haunt; lunchtime surf outings are a common occurrence. So it seems perfectly natural when Garces insists that if I really want to understand waveforms and wave dynamics, I should experience the phenomenon firsthand by hopping on a surfboard and hitting the warm Hawaiian water. I’m a strong swimmer, and I’ve always wanted to try surfing, so I jump at the chance. Everyone piles into various four-wheel-drive vehicles, and we trundle our way over unpaved rocky terrain to surfing paradise.

  That’s how my pasty-white, city-dwelling self ends up on a borrowed surfboard in the bright sunshine, gamely paddling out to meet the incoming waves with the rest of Garces’ acoustical crew, along with his wife (a scientist in her own right) and young daughter. I do not, alas, remain pasty-white. By the end of the afternoon, my entire back is bright red, even the soles of my feet. I look like a haddock that has only been seared on one side.

  Sunburn aside, there is a great deal of fundamental physics involved in the sport of surfing—potential and kinetic energy, surface tension, friction, buoyancy, hydrodynamics—and in the study of waves themselves. Waves are fundamental to nearly every field of physics, from water, sound, and light, to the wave nature of elementary particles and gravitational ripples in the fabric of space-time. Not to harsh anyone’s mellow or anything, but once again, wherever there is physics, there is also calculus.

  BALANCING ACT

  In 1778, Captain James Cook stopped off at Waimea Harbor on Kauai, en route from Tahiti to the northwest coast of North America in search of a fabled passage through that continent connecting the Pacific and Atlantic oceans. They were the first Europeans on record to visit the Polynesian chain, and their reception was warm and inviting, as they arrived smack in the middle of the season of worship for Lono, Polynesian god of peace. Islanders paddled out to where the HMS Discovery and Resolution were anchored to trade wares, so the ships could restock provisions. Cook returned after a year’s fruitless searching for the Northwest Passage to restock and make repairs to his ships. But this time, he ran afoul of the natives when he stopped at the Big Island—possibly because his second landing overlapped with their season of worship for Ku, Polynesian god of war.

  Historical accounts differ about the details, but it seems the conflict arose when some of the natives began pilfering items from the ships. First there was a dispute concerning a stolen pair of tongs, and then one over a stolen boat. Cook’s men attempted to take a chief hostage for the return of the boat—a common leveraging practice in negotiations by British mariners—but were rebuffed. Tensions mounted, the British opened fire, and a chief named Kalimu was killed. The enraged Hawaiians attacked in revenge, and when the British stopped firing to reload their muskets, they were driven to the water’s edge at Kealakekua Bay. Cook was stabbed repeatedly with an iron dagger his crew had traded to the natives, and his body was dragged off and disemboweled, the flesh stripped from the bones. As barbaric as it sounds, it was meant as a great honor. Such were the funerary rites for the remains of a deceased high priest.

  Despite the hostilities, when Lieutenant James King finally recorded the details of that ill-fated voyage in the late Captain Cook’s journals, he included not just an account of the fighting, but also of the more joyous aspects of Hawaiian culture—notably surfing, “a diversion that is common upon the water, where there is a very great sea, and surf breaking on the shore. . . . They seem to feel a great pleasure in the motion which this exercise gives.”

  There is very little record of how surfing came to the Hawaiian islands, but by the time of Cook’s visit, surfing was deeply embedded into the culture, with myths and legends about surfing heroes (and heroines) and an annual celebration called Makahiki in which surfing played a central role in honoring Lono. There were even separate reefs and beaches for royalty and commoners, a stratification that still exists in some form today: There are surf sites that cater to tourists and more-hidden local spots favored by residents.

  King couldn’t help admiring the skill of those eighteenth-century Hawaiians as they rode the waves, and for good reason. La famille Garces makes it look easy, but surfing is one of those activities that is quite straightforward in concept yet difficult to master—as Twain found out over a century ago. Following Garces’ instructions, paddle a decent way out from shore, turn the board around, and wait for a promising wave. At this point, the primary physical mechanisms at work are gravity and buoyancy. (Think Archimedes and his eureka moment.) There is no acceleration and thus no net force. There is just me, on my surfboard, bobbing gently in the ocean, waiting for the perfect wave.

  Whenever he spots a promising wave, Garces calls out and urges me to paddle furiously toward the shore. The trick is to accelerate to match the speed of an incoming wave just as it arrives at my position in order to “catch” it; otherwise it just shoots right past, leaving me bobbing forlornly behind on my surfboard, watching everyone else have all the fun. This happens far more often than I care to admit, due in part to my lack of upper body strength. But every now and then, I succeed and feel that telltale tug as the wave pulls me with it. At least that’s what it feels like; from a physics standpoint, the moving wave pushes my surfboard forward, accelerating me to match its speed. At that point, I must paddle with wild abandon to ensure I end up “riding” the wave.

  The first time this happens, I am so exhilarated that I throw myself off balance and promptly take a nosedive into the salty surf—a common occurrence for first-time surfers. A moving wave is literally a slippery slope, with constantly shifting forces acting on the surfboard—not just gravity and buoyancy at this point, but also hydrodynamic forces (the force exerted by a moving fluid) that push the board forward, along with a certain amount of friction or drag along the bottom of the board. You’ve got to keep shifting your weight back and forth to stay near the board’s center of mass as you ride the wave to keep the proper balance of forces: between the downward force of gravity and the upward buoyant force. When these forces are out of balance, the board torques, or twists. If the nose is too low, you pitch forward; if you shift too far back and the nose is too high, you lose your momentum, the board stops, and you pitch into the water. In this case, the nose dipped too low, just for a second, but that was all it took: I pitched myself forward into the ocean.

  It is easier to maintain that critical balance on a shorter board; the tradeoff is that it’s harder to catch the initial waves. So for a beginner, like me, a longboard is best, and that is what I am using. Garces assures me the board will “catch anything” (or it would, with a better surfer wielding it); but it means it gets a bit trickier when I try to stand up once I’ve caught a wave. Like Twain a century before me, I wipe out on a regular basis and never quite get into a full stand; the best I can manage is a low crouch.

  Hawaiian legend tells of Mamala the Surf-Rider, an Oahu chieftess who skillfully rode the biggest and roughest of waves, far from shore. I am no Mamala. Still, twice I manage to maintain my balance sufficiently to ride a baby wave all the way to the shore, with no fancy turns, but no spills, either. I’m relying on hydrodynamic forces to work their magic as water moves up the front of a wave, collides with my surfboard, and is deflected around it. If I were moving faster, there would have been a telltale spray in my wake. A good surfer—defined as “not me”—is skilled enough to keep just ahead of the break, turning up and down the face of the wave all the way into shore.

  Ultimately, surfers are dancing with the waves, exploiting the same basic principles as roller coasters. They gain kinetic energy by dropping down the face of the wave and exploiting gravity, although they trade off potential energy as they lose altitude. But then they use that accumulated kinetic energy to ride back up the face of the wave to the crest, and the whole process begins all over again. Ideally, at the end of the ride, a good surfer will shift his or her weight to the back of the board, causing it to drop and the nose to rise, effectively applying “brakes.” The wave rolls past, and the surfer is
ready to drop back down onto the board and paddle out to catch another wave. Alternatively, you can try my cunning strategy of wiping out before I reach the shore.

  That is the basic physics of surfing; where is the calculus? One simple example can be found in the knotty problem of catching that initial wave: so simple in concept, so tricky to execute. Recall that I need to reach a specific velocity—the same velocity as the traveling wave—at a specific time and place: the point at which the incoming wave reaches me bobbing in the water on my borrowed surfboard. A baseball outfielder merely has to be in the right place (position) at the right time; a surfer must match velocity as well. From a calculus standpoint, it’s a matter of integrating acceleration over time in order to hit the matching velocity at precisely the moment the wave reaches me. Technically, we have to take two separate integrals—one to determine velocity by integrating over acceleration, and another to determine position by integrating over velocity—to ensure I catch that incoming wave.

  “Really, it’s amazing that anyone can possibly surf at all,” Sean observes as he ponders the mathematical realities of the sport. And yet excellent surfers abound, every last one of them a master at making that intricate calculation within seconds, many without consciously realizing they are doing so. The human brain is capable of performing amazing feats of calculation, although this is as much a learned as an innate ability. When it comes to sports and motor skills—and calculus, for that matter—practice makes perfect.

  WHAT’S YOUR SINE?

  Surfing entered the international mainstream in 1959 when the film Gidget hit the silver screen, coining the term “the Big Kahuna” to describe the best surfer on the beach. Traditionally, a kahuna was a local priest or magician who would intone special chants to christen new surfboards and bring promising surf conditions. In reality, the size and shape of ocean waves depends not on mystic chants, but on three variables: wind speed, the “fetch” (the distance of open water the wind has been blowing over to form the waves), and how long the wind has been blowing over a given area. The best waves, according to experienced surfers, are those produced by intense distant storms that generate heavy winds. Those winds blow continuously for several days, creating lots of waves that slam into each other repeatedly to create a “chop.” Gradually, all the little waves accumulate into a larger swell. By the time they reach the shores of Hawaii, they’ve become a series of powerful, large swells.

  It is not a big-wave day when I have my outing—good news for me, as a beginner, because the waves are smaller and the waters less crowded with hard-core surfers. One of the keys to surfing is choosing the right incoming wave. This is not an easy call to make; ocean wave dynamics are pretty complex. That’s why surf forecasters rely on real-time meteorological data from satellites to locate the biggest waves. Avid surfers get pretty adept at eyeballing the incoming waves to identify the most promising (by size, by when they’re likely to break, and so forth) and also at estimating how fast those waves will be traveling by the time they reach the surfer. But to a novice like me, they all look the same, and it’s tough to predict when they’ll crest and break.

  Calculus comes into play when analyzing the waves themselves. All types of waves have three basic properties: wavelength, frequency, and amplitude. In the case of sound waves, the distance between compressions determines the wavelength. Objects that vibrate very quickly create short wavelengths because there is very little space between the compressions, creating a high-pitched sound. Objects that vibrate very slowly create long wavelengths because the compressions are spaced further apart. Frequency measures how many crests, or compressions, occur within one second; the measurement of this speed of vibration is called a Hertz (Hz), and 1 Hz is equivalent to 1 vibration per second. A sound wave’s amplitude, or range of movement, determines the volume (loudness) of the sound.

  Ocean waves have these properties, too. Generally, waves are measured according to height (measured from trough to crest), wavelength (measured from crest to crest), and period (the interval between the arrival of consecutive crests at a fixed point), corresponding in turn to amplitude, wavelength, and frequency. Mathematically, waves are described as periodic functions: They repeat over regular intervals, forming a series of crests and troughs over time. Graph a periodic function on a Cartesian grid, and you get the signature ideal sine wave, the “face” of a periodic function:

  The above sine wave represents the function sin(x), the mathematical idealization of a wave. The cosine, or cos(x), is the complement of the sine. It looks very much the same, except everything is shifted slightly to the left along the x axis, such that the cosine wave appears to start at its maximum while the sine wave starts at zero on the graph.

  Any wave can be thought of as a sine or cosine, merely shifted by different amounts. Mathematicians simply call such waves sinusoids. We can glean quite a bit of information about a waveform from its “face.” The number of crests and troughs we can count in a given period of time, such as one second, gives us the frequency of the sinusoid. A large number of crests and troughs means it is a high-frequency wave; a low number of crests and troughs means it is a low-frequency wave. If we multiply x by a number, like 2, we increase the frequency of the wave (above), described by the function sin(2x). This means that the periods between crests and troughs will be shorter, giving us a wave with a higher frequency.

  We can also adjust the amplitude—how strong/loud the wave is—by multiplying the function by another number, such as 2, giving us the function 2sin(2x). The resulting graph on page 236 shows a sine wave with higher crests.

  The sine and cosine are the simplest waveforms, equivalent to pure musical notes, or a light wave of a single color. Different kinds of waves can interfere with each other, mixing together to form more complex waveforms. 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 wat
ers.

  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 Laboratory, who explained that it’s easier to establish directionality along San Diego’s far more sheltered coastline than it is in Hawaii, where wave directionality isn’t clear at all—the waves are literally coming in from all directions at once. So the San Diego algorithms don’t apply; scientists can’t make the same set of underlying assumptions. But if Garces’ hunch turns out to be right, infrasound could end up being a very useful tool for oceanographic monitoring in that region.