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The energy in waves

Blame the Sun. Blame the Sun for all your surfing-related problems—because that giant nuclear fireball, 150 million km (93 million miles) away, is surely to blame.

Like almost everything else on Earth, surfing is solar-powered: the energy that shoots you over the sea comes indirectly from the Sun. And what a lot of energy beams our way: theoretically, up to 1000 watts of solar power per square feet of land. But the interesting thing is not where this energy comes from, but where it goes. Earth’s lopsided tilt means the planet cooks unevenly in sunlight, like a spit-roasting joint we’ve stupidly propped at the wrong angle. The tropical bits can blacken and char to the point of forest fires while the polar caps (for now, at least) stay locked in ice. Because energy likes to even itself out, Earth has a turbulent atmosphere and an equally dynamic ocean. And where the howling winds meet the tumbling water, we get waves. Lots of waves.

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What are waves anyway?

You know the answer to this question both in theory and in practice. In theory, because you can still remember flipping through the pages of your old school textbook: amplitude, wavelength, and frequency—those are officially the measure of waves. Now you’re older and a surfer, and you spend a significant part of your life bouncing up and down the sea surface, waves mean something different: breaking waves make your day. They’re no longer two-dimensional scientific abstractions—wiggly black lines drawn on white paper—but colourful, three-dimensional memories, vividly tied to places and times, burned in your memory till the day you die. Every science book explains waves the same way: as energy in motion, shooting from place to place while the medium (water, air, or whatever it might be) goes nowhere. But every surfer—even every beach-bound wave watcher—knows better than to reduce practical waves to a simple theory. Because, in glorious surfing reality, every wave is slightly different from every other wave that has ever broken in exactly the same place.

The waves surfers care about happen at the interface between the atmosphere and oceans, although they’re not the only waves you’ll find in either the air or the water. High above your head, there are waves shooting through the sky; that’s one of the reasons you’ll sometimes see cool, repetitive patterns formed in clouds (instead of the random lumpy cotton-wool you might be used to). There are also waves that travel deep underwater, but—intriguingly—often visible from high up in space.

How do waves form?

We all know the simple answer is “when the wind blows across the sea”, so the energy that was in the air is systematically transferred to the water. But how does that transfer take place? Is it like stirring a cup of coffee, except with friction from the wind dragging the water surface and tugging it along? It’s not hard to think of all kinds of ways the wind might stir up the water—but what does the science say?

Credit: Wind makes waves, so it's no surprise that the biggest waves are in the windiest places. This image of wave heights around the world was snapped from space by the TOPEX/Poseidon satellite in 1992. The red and yellow areas show that wave hotspots occur in places like the Roaring Forties where the winds are strongest. Photo courtesy of NASA Jet Propulsion Laboratory (public domain).

Imagine the surface of the ocean is flat and glassy with not a wave in sight. Peer close enough and the water-air interface you see is no different from the mirror surface of a pond, where cunning insects float and scamper on invisible skin. Water has surface tension, just like a drum, and if you deform it slightly, the pulling force between neighboring water molecules will spring it rapidly back again. This is the first key bit of science in a glorious sequence of events that build the waves for surfing. Because as the wind starts to blow over water, it creates minuscule ripples called capillary waves, barely a foot high. The water’s own elasticity—surface tension—tries to destroy them immediately by tugging them back into place.

But with a steady wind blowing, it’s already too late: the water surface is roughening up. Now friction kicks in and the wind can get more of a grip, systematically building up the ripples to make wind chop and swell that will eventually clean itself up into perfectly formed surf in a voyage that could last hundreds or thousands of miles. Once waves grow beyond capillary size, surface tension can’t stop them. Now they’re at the mercy of the most persuasive long-range force in the Universe: gravity. Where surface tension does its best to rid the ocean of puny capillary waves, gravity is responsible for wrecking every surfer’s fun by cleaning away the bigger waves: it’s the force that determines the life and death of every ocean wave as it constantly tries to smooth out the sea.

A swell party

If you’ve ever played at making surf in your bored, Sunday afternoon bath-tub, by flipping your hand back and forth in the water, you’ll have figured out that there are three ways to make bigger waves: you can flip your hand faster, further, or for longer. The wind in a storm zone works exactly the same way when it’s making waves. If it blows faster, longer, or over a greater distance (technically called the fetch), it creates bigger waves. Why? Because bigger waves need more energy to create them (you have to lift more water up against the force of gravity, for one thing) and a faster wind blowing for longer, or over a bigger area of sea, is the way to get that energy into the water. That’s one key reason why open coastlines are so much better for surfing. The simple rule is that it takes energy, time, and distance to make great surf. The 20ft waves that delighted southern California surfers in August 2014, courtesy of Hurricane Marie, had had 800 miles to get their act together.

That begs another interesting question: just how big can waves ever be? If a hurricane blew for weeks or months over a long enough fetch of open water, would we get ridiculously big waves? “Yes” is the simple answer, but there’s still a scientific limit to how much waves can grow. Like houses of cards, waves are unstable structures that gravity is determined to collapse, sooner or later—with the added complication that they’re moving in the turbulent interface between the atmosphere and the ocean. Seven decades of oceanographic research has determined that waves don’t build beyond a certain steepness: the ratio of their length (measured between one wave crest and the one following behind) to their height (measured from crest to trough, or maximum to minimum) can never be more than seven to one. Waves break on the shore when the rising slope of the beach (or reef) increases their steepness beyond that critical ratio; out in the open ocean, the same limit applies, and we get white caps forming as gravity forces excessively steep waves into premature collapse.

In practice, when the wind blows across the water in a perfect-surf-creating storm, we reach an equilibrium. The wind keeps on adding more energy to the water, but the waves keep collapsing. At this point, we have what the oceanographers call a fully developed sea. The waves are as big as they’re ever going to get. All they have to do now is get themselves to the shore, where the surfers are waiting.

The birth of surf science

Surfing is essentially a 20th-century invention, and so is surf science. But who first had the idea to turn the wonder of waves into a scribble of mathematics —and why?

Surf science owes its birth to military manoeuvres. The pioneers of surf forecasting, Norwegian oceanographer Harald Sverdrup (head of the famous Scripps Institution) and his young American student Walter Munk, figured out how to predict wave heights from the wind speed, fetch, and duration while working for the US military during World War II. Fortunately, they also had loads of data to test their theory and quickly honed their equations enough to make accurate predictions. Although no-one knew it at the time, this crucial work was used by the Allied forces to select the best days for the famous beach landings. It was first used to pick a calm day for an assault on North Africa on 8 November 1942 and, subsequently, for the D-Day landings in Europe in June 1944. Surfing science, in other words, changed history.

Sverdrup and Munk completed their work in 1943, but it remained classified until after the War, finally appearing in March 1947 as US Navy Hydrographic Office Publication Number 601, “Wind, Sea, and Swell: Theory of Relations for Forecasting”. Later refined and extended by Charles Bretschneider, the revised theory became known as the SMB (Sverdrup, Munk, Bretschneider) model. Though it’s only a basic explanation of how the wind makes waves, it’s still widely referred to today, but it’s now been superseded by decades of more detailed research.

How much energy is there in waves?

It’s worth quantifying the energy in waves for all kinds of practical reasons. From an environmental point of view, it tells us how feasible it is to build things like renewable wave-energy systems—and whether we can harvest more energy from the hidden heat in ocean water (the temperature difference between the ocean surface and its depths) than from its mechanical energy (the back-and-forth, up-and-down movement of water caused by tides and waves). From a surfing point of view, asking this question tells us exactly which waves are rideable and what you can do on them. Everyone knows you can’t catch a small wave on a surfboard (or even a boogie board)—and the simple scientific reason for that is that a rideable wave needs to contain a minimum amount of energy to lift your body against the force of gravity and accelerate you to its own speed.

Go figure

Let’s try to guesstimate how much energy there is in a medium-sized wave crashing down on top of us as we stand in the surf. I must emphasize that this is “back-of-the-envelope” physics and not in any sense rigorous or correct oceanography. It’s what’s technically referred to as “just a bit of fun”! I’m not going to attempt to use the complex equations that are really needed to do this properly. For now, let’s see how far we can get with the kind of basic science we learn at school.

We know the total energy is the sum of the wave’s kinetic energy (because the water is moving) and potential energy (because the crest is lifted up above the mean water level). Let’s not get too bogged down, though: let’s simplify everything as much as we can to the point of basic, school-level physics.

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Suppose we have a 1m width of a wave that closes out completely, with the water coming to an impossible, screeching halt (so effectively it loses all its kinetic and potential energy when it breaks). According to Willard Bascom (one of the founding fathers of surf science), the speed of a decent surfing wave is about 25mph or 11 m/s. For easy sums, let’s assume the wavelength (the distance between one wave peak and the next one) is 2m and the amplitude (the height of the wave) is also 2m, so the volume of water above the mean sea level that we’re interested in (the dark grey bit in the figure is roughly 1 x 1 x 0.6 = 0.6m3 = 600 liters, which weighs about 600kg. Simplifying very greatly indeed (I know, I know… but bear with me), that gives us potential energy of mgh = 600 x 10 x 1 = 6000 joules and kinetic energy of ½mv2 = 300 x 11 x 11 = 37,000 joules, making a grand total of about 43,000 joules—or call it 50 kilojoules to keep things simple. This is a rough estimate of how much useful (non-heat) energy there is per foot of a simple breaking wave—and the actual value is likely to be less than this because of all the simplifications I made (a real wave isn’t this steep; it doesn’t stop completely when it breaks; it has an ever-changing, irregular volume; its total mass is not all concentrated at exactly the same height, and so on).

Catching waves

Do the numbers tell us anything useful? Suppose you weigh 70kg (not including the weight of your board). If you want to travel at 11m/s, you need kinetic energy of ½mv2 = 35 x 11 x 11 = 4235 joules. To ride a meter above the ocean surface, you’ll also need potential energy of mgh = 70 x 10 x 1 = 700 joules, so you’ll need about 5000 joules of energy altogether. Let’s say it takes you 5 seconds to catch the wave. The power your muscles and the wave need to supply for you to start surfing is the total energy needed to be divided by the time it takes, so that makes an average power of about 1000 watts to reach 5000 joules in 5 seconds—as much as a typical clothes washing machine. Could you get that from a 1ft wave? Maybe yes, maybe no. My estimate of about 50 kilojoules was for the total energy in 1ft width of a wave, which sounds like it’s 10 times more than enough—but, remember, you wouldn’t get all that energy from the wave (it keeps moving and doesn’t break) and you’re not tapping into a 1ft width of water (maybe only the width of your board).

Kids, lucky things, can catch smaller waves than adults because they weigh about half as much and they can accelerate faster. The potential and kinetic energy of a surfer are both linearly related to body mass, so if you have less mass, you need correspondingly less energy—making it more likely the wave will sweep you along. By the same token, if a wave is big enough, you can (theoretically) surf it in or on whatever you like.

How exactly does paddling help? If you’re paddling as you catch a wave, you’ve already given your body a certain amount of kinetic energy and momentum, so any oncoming wave has to provide you with less of the total energy you need to get moving: paddling, put very simply, gives you a head-start in terms of kinetic energy and momentum. It doesn’t help you with potential energy: unless you’re lying prone on a bodyboard, you’ve still got to get upright!

But different waves deliver very different force even if they contain the same amount of energy. Why? Waves that break faster produce more force, which is why a plunging wave that closes out in a shore-dump is more dangerous than a wave that peels gradually across its width. Simple physics tells us why: if two waves contain exactly the same amount of energy but one breaks five times faster than the other, it can (theoretically) deliver five times the force (because F=mv/t and if t is five times smaller, f is five times greater).