Chapter 19: Ocean Waves

Sit on the shore of the ocean and what do you see? Waves. Surf. Foam. Close your eyes and listen. What do you hear? The crashing of waves. What do you feel? The vibrations of surf pounding against the beach. Draw in a deep breath. What do you taste and smell? The salty air of an ocean alive with waves. Waves stimulate all of our senses simultaneously. Perhaps that’s why they have fascinated mankind since the dawn of time. 

If any one phenomenon of the sea can define it, it might be waves. They rock ships, wash upon their decks, and crack against their hulls. Where waves come into contact with the seafloor and reach an unstable height, a wave begins to break—this is surf. High surf crashes against lighthouses, piers, and buildings, and occasionally destroys them.

But rather than fear the destructive force of waves, humans fashioned planks from trees and paddled them out to meet them head-on. Surfing—the use of various types of boards (or your body) to ride waves—certainly ranks as one of the more exhilirating sports. A surfer on a wave experiences a rush of moving energy unlike any other on Earth. And people watching from the beach get to marvel at the speed and agility of a surfer cutting up the waves or whooshing out of a barrel. Though its historical roots date back hundreds of years in Polynesia and Peru (Warshaw 2011), surfing in modern times generates billions of dollars in surfing activities, fashions, vacations, movies, magazines, décor, and a lot more (Statista 2022).

To explore these topics, we follow the timeless words of oceanographer Willard Bascom (1916–2000) and his classic book, Waves and Beaches, first published in 1964 (updated: Bascom and McCoy 2021). We’ll add other references as needed. But now, let’s catch some waves. Cowabunga!

19.1 Energy in Motion

The waves that we perceive moving across the surface of the ocean actually represent energy in motion. The rise and fall of a wave traces the passing of a pulse of energy. The water itself hardly moves, except in a circular orbit that edges ever so slightly in the direction of the wave. It is energy that moves rapidly through the ocean surface, moving the water upward and downward as it passes. Thus, we may define a surface wave as the physical expression of energy moving forward at the surface of the ocean.

Waves occur inside the ocean too. Where water layers have different densities, waves can occur along their boundaries. A wave that propagates along the boundary of water layers with different densities beneath the ocean surface is known as an internal wave. Though only detected using scientific instruments, internal waves are the largest waves in the ocean. Their heights may reach hundreds of feet—far larger than the largest waves ever measured on the ocean surface, including tsunami. Thus, we must broaden our definition of an ocean wave to the physical expression of energy moving forward at the interface between two fluids of different density. After all, the atmosphere is a fluid too, so the waves at the surface of the ocean—the ones that we are most familiar with—fit this definition as well as the internal waves.

Moving waves are known as progressive waves because they move, or progress, forward. Where two progressive waves pass through each other in opposite directions, they may create a standing wave. These waves appear to stand still. A standing wave results from wave interference, the interaction of the energies of two or more waves to create a wave that is higher or lower than the individual waves. Under the right circumstances, a standing wave may persist for quite some time. You are most likely to notice them where a wave approaches a seawall and bounces off it, a phenomenon known as wave reflection. The interference of the incoming wave with the reflected wave creates a standing wave. Look for them next time you’re at the beach or walking along a seawall.

19.2 The Source of a Wave’s Energy

The energy for most ocean waves that we observe comes from the wind. The wind is known as the disturbing force because it acts upon the surface of the ocean to create an imbalance. It disturbs the ocean from its resting state, the still-water level or sea level. While the wind is the most common disturbing force in the ocean, underwater earthquakes, landslides (above and below the water), and meteorite impacts can act as disturbing forces and create waves. Even the gravitational forces that cause tides are a disturbing force because tides are a type of progressive wave. 

The disturbing force is counteracted by the restoring force, which returns the surface to a resting sea state. For the most part, gravity—the attraction between masses—acts as the restoring force. The downward pull of gravity on the water’s surface restores it to its resting level, and so nearly all waves may be classified as gravity waves because gravity is the restoring force. However, for extremely small waves (less than 1.73 cm), surface tension—the “sticky” forces of water molecules at the surface of a liquid—is the restoring force.

19.3 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.

19.4 Types of Waves

Oceanographers and mariners use a variety of ways to classify waves—none official that I’m aware of—but we’re stuck with what the experts define, so here we go. 

The shortest waves—those with wavelengths less than 1.73 centimeters—are called capillary waves because they are small, like the capillaries in your body. Capillary waves are most easily recognized as the slight ruffling of the ocean’s surface as a wind blows across it. The term “capillary” actually refers to the surface tension of water in a tube that causes the surface of a column of water to form a slight bulge or depression. Sometimes capillary waves are referred to as ripples—a better name, perhaps, but confusing in that the term has also been applied to waves slightly larger than capillary waves.

Capillary waves are the most abundant wave in the world ocean. It takes only the slightest amount of wind to cause them. Their shape is distinctive, with rounded crests and V-shaped troughs. You can easily make your own by blowing on a bowl of water. It’s said they resemble cats’ paws on water. As an owner of four cats, I really don’t see the resemblance, but I guess a thirsty cat’s habit of gently pawing a bowl of water to gauge the distance between its face and the water surface may have something to do with it. 

Be that as it may, these very short waves behave differently from how larger ocean waves behave because they are under the control of surface tension. When the wind stops, capillary waves disappear immediately as surface tension restores the ocean’s surface. Another interesting thing about capillary waves is that as the wind blows harder, their wavelengths become shorter, just the opposite of larger waves. Despite their small size, capillary waves play an important role in the formation of larger waves because the roughness they create over the ocean’s surface permits a faster exchange of wind energy into the ocean.

Sailors and ocean folks refer to wind-generated waves, or wind waves, by a variety of names. Each roughly corresponds to the wave’s size. Ripples—a bit larger than capillary waves—and wind chop—short and steep irregular waves—represent the waves you see when the wind begins to blow or has been blowing for a short time. Fully developed seas refer to the maximum wave size for a given wind speed over a period of several hours or a few days. Swell refers to the waves that travel out beyond the winds that generate them. Swell is what you see and experience as the rise and fall of the sea surface before they inter the seafloor. Swell moves the fastest of any wind-generated wave and produces the largest and fastest surf, ideal for big wave surfers, defined as surfers who ride waves greater than 20 feet (6.2 m) in height.

Oceanographers and physicists use a more formal classification system for waves based on their forward motion. To understand this system, we need to take a ride on the freeway.

19.5 The Timing of Waves

The time element of progressive waves turns out to be quite important for surf prediction. When watching waves from the end of a pier, you are likely to notice wave crests passing the end of the pier one after another. If you watch the water level along one of the pier pilings, you’ll see the water rise as the crest approaches and fall as the crest passes. After a few to several seconds, it happens again. Seconds later, it happens again, and so on. The time between successive crests is the wave period. 

One way to understand wave periods is to consider a freeway full of cars barreling down the road. Each car maintains a certain time between it and the car ahead of it. You can measure the time between you and the car ahead of you by asking a passenger in your car to start a stopwatch at the moment the car ahead of you crosses under an overpass. Stop the stopwatch right when you pass under the overpass and you have the time that separates you and the car ahead of you. Think of cars as wave crests, and you have the idea.

Of course, wave periods can be measured using more sophisticated equipment. Ocean-deployed weather buoys digitally record the rise and fall of a float as waves pass by.  These real-time wave-monitoring systems are critical for alerting scientists and authorities when potentially destructive big waves are approaching a shoreline. Of course, surfers take an interest in these buoy measurements as well.

Wave periods serve as the basis for a more formal classification system of waves. Capillary waves have the shortest periods, less than 0.1 seconds. Waves with periods of 0.1 to 1 second—what we might call ripples—are classsified as ultragravity waves. The kind of waves we experience most often on the ocean—ordinary gravity waves—vary from 1 to 30 seconds. Waves with periods from 30 seconds to 5 minutes—infragravity waves—originate from interactions between shorter period waves in coastal waters. These waves likely play a role in sediment transport and other nearshore processes (Cook 2018). Long-period waves—waves with periods from 5 minutes to 12 hours—include storm waves and tsunami. Beyond that, we venture into tides and other kinds of very long period waves. 

Wave periods serve as a proxy for surf conditions. In general, the longer the wave period, the faster and more energetic the wave. Savvy surfers check surf reports, internet, app- or even phone-based tools that report information on wave periods, weather, and other surf-related factors. In Southern California wave periods of 10 seconds or more are considered pretty rocking. Wave periods of more than 20 seconds are insane! Some of the most epic surf days have wave periods of more than 20 seconds. Fortunately, such long-period waves are rare. They create very dangerous conditions along the coast and can tear up a coastline and damage property in just a few hours.

Though not wind-driven, tsunami—long-period waves generated by vertical motions of the seafloor during shallow earthquakes—have very long periods, on the order of five minutes to an hour. That’s why it’s important to remain out of harm’s way after the first arrival of a tsunami. There are more waves to come! Tsunami may travel the ocean for days.

19.6 The Speed of Waves

Wave periods and wavelengths vary according to the wave speed—the forward motion of a wave crest over time. Following Knauss and Garfield (2017), their relationship may be expressed mathematically as wave speed (S) equals wavelength (L) divided by wave period (T), or:

S  = L / T

(Eq. 19.1)

These equations are easily rearranged to:

L = S × T or T = L / S

(Eqs. 19.2 and 19.3)

These equations should make sense to you in that the crest of a progressive wave has a particular length to travel in a particular time period—the wave period. It’s important to note that the wavelength and period (and thus wave speed) vary in proportion to each other. The longer the period, the longer the wavelength, and the faster the speed of the wave. For example, a wave with a period of 2 seconds will have a wavelength of about 20 feet (6 m) and a speed of 6 knots (nautical miles per hour) while a 14-second wave will have a wavelength of 1,000 feet (305 m) and a wave speed of more than 40 knots.

One interesting fact about wave periods is that once a wave forms, its period remains constant. This is best observed along a shoreline where waves are approaching straight on. Interactions of the waves with the seafloor cause them to slow down. As a result, the waves begin to bunch up because their wavelengths shorten to compensate for the slower wave speed.

Because wave period is simple to measure—just measure the time between one crest and the next—some rules of thumb have been developed to estimate wavelength and wave speed from wave period. For example, the speed of a wave in knots is approximately three times the wave period. So a wave with a 7-second period has a speed of roughly 21 knots. Wavelength in feet can be estimated by taking the square of the period (in seconds) and multiplying the result by 5. For a 7-second wave, the square is 49, so its length would be 49 × 5, or 245 feet. These approximations help sailors or surfers who want to calculate when a given set of waves will arrive at a location any known distance from the source (e.g., Smith 1973).

19.7 Wave Orbitals

One other aspect of waves, mentioned above, deserves our attention. The water particles in a wave move in a circular orbit—the wave orbital—as the energy of the wave passes through the surface of the ocean. This is really important to recognize because the water is not moving at the speed of the wave! 

If you have ever sat in the water on a raft, or watched a seabird floating on the water, you’ve noticed that you (or the bird) move up and forward as the wave approaches and down and backward as the wave passes. If you can imagine a dot on the surface of the ocean, the path of the dot will trace a circle as the waves pass by. The rotation of the dot is in the same direction of the wave at the crest; at the trough the dot is moving in the opposite direction. 

The diameter of the circle traced by the dot will be equal to the height of the wave. However, water particles below the surface will trace progressively smaller circles. Wave orbitals decrease in diameter from the ocean surface to a depth of one-half the wavelength of the wave. At this depth, called the wave base, the orbital motion is negligible (about 1/23 the surface diameter; Bascom and McCoy 2021). 

The vertical distance from the surface to the depth where particle motions cease defines the boundaries of the energy of the wave. Not surprisingly, the energy of a wave is concentrated at the surface and diminishes with depth. For a wave with a wavelength of 20 feet (6 m), this depth would be 10 feet (3 m). For a 1,000-foot wave (305 m), the energy of the wave would be present in waters shallower than 500 feet (152 m). Surfers and scuba divers know the main energy of a wave is at the surface. Scuba divers entering from the beach get shoved back and forth along the seafloor until they reach a depth where the wave energy no longer penetrates. Surfers perform a duck dive, a skillful maneuver to propel a board underneath a wave to avoid being pushed backward or worse (e.g. Brander 2010). Submarines, too, can avoid storm waves by submerging to a depth where the wave energy disappears.

19.8 Wind Stress: A Transfer of Energy

As we learned in Chapter 15, winds—movements of air—are caused by differences in air pressure at different locations in the atmosphere. Air, of course, is a mixture of gases, including nitrogen (N₂), oxygen (O₂), water vapor (H₂O), argon, carbon dioxide, and others. So when we talk about air molecules, we are talking about these molecules collectively. 

Wind represents molecules of air in motion. These moving molecules contain a form of energy known as kinetic energy—energy in motion. Lots of things have kinetic energy: the air around you, winds, a flying insect, a hurled rock, an automobile speeding down the highway, or people at a rave, for example. The ripples that you notice when the wind interacts with the ocean surface result from collisions between air molecules and water molecules. Literally, an air molecule bangs into a water molecule.

The transfer of energy from the wind to the ocean surface along the horizontal plane of the ocean (parallel to the ocean surface) is known as surface wind stress (e.g., Jones et al. 2001). It’s the surface wind stress that sets the ocean in motion. Most of the energy transferred by surface wind stress makes waves, but a small amount of it causes currents (movements of water, the ocean analog of wind). Some energy is also dissipated as heat.

The banging of molecules transfers energy from the moving molecule to the unmoving molecule. The energy transfer sets the previously unmoving molecule in motion. The amount of energy transferred from the moving molecule to the unmoving molecule depends on momentum—the product of a molecule’s mass and its velocity. The larger a molecule (the greater its mass), the greater its momentum. Alternatively, or additionally, the faster a molecule (the greater its velocity), the greater its momentum. A good way to think about momentum is to think about the difference between an insect hitting your windshield at 75 mph or a rock hitting your windshield at that speed. The insect, with little mass, makes a splat, but no harm done other than a dirty windshield. A rock, on the other hand, with a larger mass and greater momentum, can put a crack in your windshield. If the rock is a meteorite, fuhgeddaboudit! (Your car is toast.)

When a moving molecule of air bangs into an unmoving molecule of water, a transfer of momentum occurs from the moving molecule to the unmoving molecule—an act of physics known as momentum transfer. The formerly unmoving molecule gains kinetic energy and begins to move. Of course, the now-energized water molecule is going to transfer momentum to surrounding water molecules because there will be collisions and transfers of kinetic energy from moving water molecules to other water molecules. So kinetic energy is passed from molecule to molecule within the upper ocean.

At the same time, some of the energy transferred through momentum causes vertical displacement of water molecules. The lifting or lowering of water molecules at the surface will change their potential energy, which is energy contained in an object or system due to its position or placement. When raised above the resting sea level, such as at the crest of a wave, a water particle gains potential energy in proportion to its height above the resting level. When that water particle moves downward, it releases that potential energy as kinetic energy. The same is true when lowering a water particle below the resting state. At the trough of a wave, the buoyancy of the water beneath the water particle will force it upward. As it moves, it will release potential energy.

In general, about half the energy in a wave is in the form of kinetic energy, and the other half is in the form of potential energy. Because the orbital motions of water particles represent kinetic energy, and the height of a wave determines the diameter (and circumference) of the wave orbital, and because the height of a wave determines the potential energy in a wave (the displacement of a water particle above or below the resting level), the total energy in a wave—kinetic plus potential—can be described in terms of the wave height (e.g., Knauss and Garfield 2017), as:

E = 1/8 × ρ × g × H² 

(Eq. 19.4)

Total energy in a wave equals 1/8 times the water density (ρ, pronounced rho) times acceleration due to gravity (g, which is 9.8 meters per second squared ) times the height of the wave squared (H²).

The take-home message here is that a 6-foot wave has four times the energy of a 3-foot wave. A 12-foot wave has 16 times the energy of a 3-foot wave. Waves are extremely powerful, and now you know why.

19.9 The Wind-Stressed Sea Surface

When the wind begins to blow over the surface of a calm ocean, there is a transfer of momentum from the air molecules to the water molecules. The first waves that we notice are capillary waves, the ripples that move like a shadow across the ocean’s surface. Often the wind is uneven or moving slightly upward and downward, so looking out across an expanse of ocean, you see several patches of capillary waves moving, disappearing, and appearing again somewhere else. When the wind stops, the capillary waves disappear as the surface tension of water immediately restores the ocean surface to its level state.

However, if the wind keeps blowing, enough energy is transferred across the air–sea interface to generate gravity waves. Unlike capillary waves, gravity waves are not counteracted by surface tension; they are counteracted by gravity. These waves travel outward from the location where they are formed. Waves of a decent size can travel thousands of miles across the ocean. One of the earliest studies of ocean wave propagation detected waves in Southern California that originated in the Indian Ocean (e.g., Snodgrass et al. 1966).

The prolonged blowing of the wind continues the transfer of energy into the ocean, and the waves continue to grow. Ripples turn into wavelets, which turn into chop and larger waves. But there are limits to the size of waves that can be produced under a given set of wind conditions.

19.10 Factors Affecting Wave “Size”

First, and perhaps most obvious, the wind speed plays a role in determining the size of waves that may be produced on the ocean. The faster the winds, the faster the transfer of energy to the ocean’s surface, and the bigger and more energetic the waves that are formed. Makes sense, right? There’s more energy transferred to the upper ocean.

The length of time that the wind blows, wind duration, is also important. A short, sharp wind, no matter how fast, isn’t going to produce very big waves. You can sneeze on an aquarium—a sneeze has been clocked at about 40 mph—but you’re still not going to produce much in the way of waves. It takes time. So the longer the wind blows, the bigger and more energetic are the waves formed. But wait! There’s more.

The distance over which the wind blows in the same direction, something called the fetch, also makes a difference. Long fetches mean that a larger area of the ocean receives energy from the wind; the ocean surface becomes more energetic. The greater the fetch, the bigger and more energetic the waves that are formed. Fetch explains why hurricanes with the fastest winds don’t always generate the biggest waves. Typically, the size of the wind field in a hurricane is smaller and the fetch shorter and more variable (because winds rotate around a hurricane) than the wind field in a large storm system. Low-pressure systems over the ocean may have fetches of hundreds of miles and wind speeds near hurricane strength. As a result, storms at sea tend to produce the largest waves observed in the ocean. Wind speed, wind duration, and fetch contribute to the formation of ocean waves. They act in combination to deliver energy to the ocean surface. 

Some forces remove energy from a wave. Progressive waves must contend with friction from the atmosphere as they move forward. Friction drains the energy from a wave. Strong winds may create whitecaps, the broken tops of wave crests that appear white because of the bubbles and foam present within them. Whitecaps remove energy from a wave. Other factors within the waves tend to dissipate their energy as well. At some point the energy transferred to the ocean at a given wind speed is balanced out by the energy lost by the waves. The steady state between energy-intensifying factors and energy-dissipating factors produces a fully developed sea. When this occurs, continued blowing of the wind at the same speed will produce no additional gains in wave height or energy. For these reasons waves always move at speeds slower than the wind speeds.

19.11 The Beaufort Wind Force Scale

The concept of a fully developed sea has important implications for sailors. It means that there are limits to how bad the ocean will get for a given wind speed. How long it takes to reach a fully developed sea depends on the wind speed, duration, and fetch. A fully developed sea may take only a few hours under light winds, but it can be two to three days for a fully developed sea to develop in strong winds. 

Recognition of the relationship between wind speeds and a fully developed sea state led to development of the Beaufort wind force scale, a visual guide for determining wind speed from sea state. Adapted from existing wind scales by Sir Francis Beaufort (1774–1857) and formally adopted by the Royal Navy in 1838 (e.g., Wheeler and Wilkinson 2004), the Beaufort scale allows a mariner to estimate the wind speed from wave conditions. Sea and atmospheric (and land) conditions are scaled from 0 to 12 “forces,” where Beaufort force 0 represents the calmest air, and Beaufort force 12 signals a hurricane. For example, knowing that a Beaufort force 5 is in effect, a ship’s captain can prepare the ship and crew and decide whether it’s safe to sail or wait for the seas to subside.

19.12 Dispersion of Waves

Once formed, gravity waves keep on truckin’, even after the wind has subsided. The most energetic waves—the longest waves with the longest wave periods—move away from their point of origin faster than shorter waves. Thus, ocean swell with the longest wavelengths and periods separates from the slower waves with short wavelengths and periods, a process called wave dispersion. Most times, short-wavelength waves dissipate near the storm center. But fast-moving ocean swell will continue across the ocean until it meets an obstacle, such as a seamount, an island, or a distant beach.

Because swell sorts out according to its speed, waves of the same speed tend to travel together in a group called a wave train. Depending on the conditions that create swell, wave trains may have anywhere from 3 to 15 waves associated with them. Surfers know a wave train as a wave set, or simply a set—a series of similarly sized waves that arrive at the shore in close progression, one right after the other. Wave trains are the origin of sets. However, it’s simply not true that the seventh wave or the fifth wave or the third wave of a set is the largest. For a number of reasons, there will always be a wave or two that are bigger than the others in a set, but there is no mathematical consistency to which wave it will be. Sorry, dudes.

One of the interesting things about wave trains is that the train moves at half the speed of the individual waves. It’s not terribly important that you know why, but it has to do with the distribution of energy in the wave train. Just like a Tour de France cyclist, the lead wave does all the work. 

In any wave the kinetic energy goes into moving water particles in their orbits. As they move, they lift the sea surface and create potential energy. The potential energy of the wave is released when the crest drops, and the released potential energy supplies energy to making wave orbitals—generating kinetic energy—and so on. In a wave train, the first wave has to lift the sea surface on its own. It uses kinetic energy to do that. But the second wave is the recipient of the potential energy generated by the first wave; the raised sea surface falls and releases energy to the second wave. In a wave train, the leading wave diminishes and disappears, but a new wave appears at the end of the train, because potential energy is released by the last wave in the train. In short, this backward trading of energy that goes on in a wave train results in the first wave disappearing and a new wave appearing at the end of the train, so that the forward progress of the entire train is slower. As a result, the group speed—the speed of the wave train—is half the speed of the individual waves.

One way to visualize a wave train is to imagine a conga line of dancers where the lead dancer exits and a new dancer joins the line at the rear when the line advances two steps. Even though individuals advance by two steps, the whole line only makes forward progress one step at a time. Try it at home with friends. Do the conga wave train. (“Come on, shake your body, baby, do the conga wave train!”)

Though swell moves hundreds to thousands of miles across the ocean, there is some loss of energy along the way. Frictional forces will dissipate some of the energy. But energy also dissipates by elongation of the wave front, the “edge” of the wave perpendicular to its direction of travel. The loss of energy due to expansion of the wave front is called spreading loss. Most of the energy generated by a storm, some 90 percent, will travel outward in a cone with an angle of about 30° to 45°. So the greater the distance you are from the vertex (the location where the two rays of the angle intersect), the greater the distance between the two rays, or sides, of the angle.

 You can easily see this by making a peace sign with your hand. The distance between the knuckles of your fingers is shorter than the distance between your fingertips. Because swell has a fixed amount of energy, the widening of the wave front causes a reduction in the energy at any point along the wave front. The energy per unit length of wave front decreases with distance from the point of origin of the wave.

Close to the point of origin, spreading loss is minimal. Farther away from the point of origin, spreading loss can be significant. Some 60 percent of the energy of swell (per unit length of wave front) may be lost within the first 500 miles (805 km). That’s why killer surf for Hawaii might not be so epic once it reaches the US West Coast. Spreading loss reduces the size of the swell as it travels outward. The width of the wave front depends on the width of the fetch that created it, the fetch width. Typical storm systems across the ocean may range from 600 to nearly 2,000 miles in horizontal diameter (1,000–3,000 km; e.g., Bierly 2005). Swell-generating fetch will be on a similar scale. Thus, wave fronts for a typical ocean swell may extend hundreds of miles, if not a thousand miles or more, in length. What that means is that waves from a single storm system can affect the entire coastline of the western United States.

Storm systems rarely remain in one place. Generally, storm systems move west to east in both hemispheres, in the direction of the jet streams. Thus, the effective fetch of a storm may be much greater. The swell generated over a period of days will travel outward for the duration of the storm. Large storm systems at great distance may send waves for several days toward coastlines (e.g., Butt 2021).

The movement of storm systems also affects the swell direction—the direction the swell is coming from. Swell direction is identified according to directions on a compass rose, an illustration that depicts the cardinal directions—north, east, south, and west—and the ordinal directions, the points between the cardinal directions—northeast, southeast, southwest, and northwest. Compass directions are always given in degrees of a circle, where north is 0°, east is 90°, south is 180°, and west is 270°. The range of compass directions from which swell may arrive is known as the swell window. Surf forecasters calculate swell windows to determine which beaches may offer the best surfing conditions on a given day. 

In California, swell windows may change from northwest to southeast, depending on the time of year. Storms in the North Pacific often occur in the winter and generate swell from west to northwest. In summer, winter storms in the Southern Hemisphere produce swell from the southwest to southeast. Hurricanes off the coast of Mexico swing into action in the summer and occasionally send swell our way. Of course, California has a tough time competing with Hawaii, which has to be one of the best places for surfing in the world. Situated in the middle of the Pacific, Hawaii can experience swell from just about any direction. As a result, there are surfable waves nearly year-round in Hawaii.

Swell direction also proves critical for places where islands or other land features may interfere with swell. Islands can block swell or change its direction. Gaps between the islands mean that some spots are going off, while other spots are snoozy. Because of the Channel Islands, it’s not uncommon for San Diego to experience big waves while Orange County is quiet, or vice versa. The islands produce what is called a swell shadow—a region over which swell is blocked from a particular direction. 

19.13 Deep- and Shallow-Water Waves

Ultimately, the swell reaches shallower water. Remember the orbitals in a wave that extend to a depth of about one-half the wavelength? Waves traveling in water depths deeper than one-half the wavelength—like ocean swell—are called deep water waves. Their progress is unimpeded by the seafloor. But as waves approach water depths less than one-half the wavelength, the wave orbitals begin to interact with the seafloor. The orbitals at the bottom of the wave are unable to complete their orbits, and they assume a more elliptical path. When the seafloor begins to interfere with the wave orbitals, the wave is said to “feel bottom.” It’s at this point that the life of a deep water wave ends.

Waves traveling in water depths less than 1/20 of their wavelength are classified as shallow water waves. Now, let’s take a minute to think about what that means. If a wavelength is 100 feet, then 1/20 of its wavelength is 5 feet because 100 / 20  = 5. So that means that when a wave with a wavelength of 100 feet reaches a water depth of 5 feet or shallower, it is officially designated as a shallow water wave. What is the depth of “shallow water” for a 1,000-foot-wavelength wave? (Hint: divide 1,000 by 20. Simple, huh?)

The behavior of shallow water waves is much different from that of deep water waves. In addition to the Eq. 19.1 above (S = L / T), we can express the speed of a deep water wave as (e.g., Knauss and Garfield 2017):

S_deep = 1.56 × T 

(Eq. 19.5)

where S is the speed of the wave and T is the wave period. S here gives the speed at which some part of the wave—the crest, for example—is moving forward. (Officially, it’s called the wave’s phase speed.) It’s this relation that tells us that longer-period waves travel faster. Deep water waves disperse (see wave dispersion above) because they separate according to their speed.

Shallow water waves behave differently. The speed of shallow-water waves depends solely on the water depth, according to:

S_shallow = √(g × z)

(Eq. 19.6)

The speed of a shallow water wave equals the square root of the gravitational constant (g, or 9.8 m/s²) times water depth (z).

What that means is that the speed of a shallow water wave depends solely on depth. Neither wavelength nor wave period makes any difference. For this reason, shallow water waves are nondispersive—they do not separate according to wavelength or wave period.

One surprising thing about shallow water waves is that they include some waves you would never suspect—tsunami, for example. The wavelength of a large tsunami can be up to 300 miles (482 km). The deepest place in the world ocean, the Mariana Trench, is only 6.79 miles deep (10.9 km). That means tsunami act like shallow water waves everywhere in the ocean. Their speed, then, is governed by the depth of the water. What is the speed of a tsunami traveling across the Mariana Trench? It’s over 700 mph. Cowabunga, indeed!

For the sake of completeness, waves between wavelengths ½ L and 1/20 L are called intermediate (or transitional) waves. They have a different set of equations that govern their behavior, which, because of their mathematical complexity, we are not going to discuss here.

19.14 The Making of Surf

Interesting things happen when waves enter shallow water. When a wave begins to feel bottom, it begins to interact with the seafloor. Friction causes the wave to move slower. The geometry of wave orbitals becomes flattened as their vertical motion is hindered. With nowhere to go but up, the energy of the wave is directed upward. As a result, the wave gains height. Viewed from the shore or a pier, you can see the swell get bigger as it approaches the shore.

Eventually, the water particles in a wave can’t even complete an orbit, and they simply move back and forth. If you’ve ever dived beneath the waves or sat on the bottom as a scuba diver, you can attest to the strength of the back-and-forth motion of waves in shallow water. At some point in the wave’s progress into shallow water, the crest topples forward—sometimes with fury—and the wave breaks. This usually happens at a point when the wave height becomes too steep.

Oceanographers define the ratio between a wave’s height (H) and its wavelength (L) as wave steepness, which we’ll symbolize as Z (e.g., Bascom 1980):

Z = H / L

(Eq. 19.7)

In general, waves break when their height exceeds their wavelength by about 1/7 (0.142; Michell 1883; but see Perlin et al. 2013). A 7-foot wave will break when its height reaches 1 foot. Generally that happens when the water depth becomes less than about 1.3 times the wave height (e.g. Butt 2014). So a 3-foot wave will break when the water depth reaches 3.9 feet.

19.14.1 Types of Breaking Waves

Breaking waves are generally called surf or breakers. And as everyone who watches waves knows, no two breakers are alike. This is partly due to the characteristics of the waves themselves, but the slope of the seafloor also makes a difference. A gently sloping nearshore will slow down the wave gradually, and its energy will be expended along a greater distance in its progress toward the shore. A steeply sloping nearshore will permit a wave to approach a beach at near top speed. The differences between the waves breaking on a gradual versus a steep beach slope can be dramatic.

In general surf may be classified into one of four categories depending on the characteristics of its break. A wave whose lip trickles down the front of the wave as it approaches is called a spilling breaker. The top of the crest of the wave spills down the front of the wave. This type of wave generally occurs on a gently sloped bottom. Most of the small waves you’ll see in Florida are of this type. A wave that creates a tube with a lip that shoots over the face of the wave in a kind of waterfall is called a plunging breaker. This is the kind of wave most sought after by surfers. The waves at Pipeline in Hawaii and Teahupo’o in Tahiti are great examples of plunging breakers. These locations produce the much-coveted barrels. Plunging breakers can be found on shallow- to intermediate-sloped seafloors. More steeply sloped seafloors produce collapsing breakers, where the entire wave face disintegrates into foam. The steepest nearshore bottoms produce waves that don’t even break; they slide up onto the beach in a swoosh. Swooshy waves are called surging breakers (e.g., Butt 2014).

The speed and shape of the incoming wave, the dynamics of the nearshore that change the slope and depth of the seafloor, tides, and winds, among other factors, ultimately determine the way the wave breaks. Many waves are a combination of these types. For example, in Southern California, we often see breakers that spill and plunge. And really, when you’re out enjoying the waves, it doesn’t matter what you call them. They’re simply fun!

19.14.2 Wave–Seafloor Interactions

Often waves approach the beach at an angle, meaning the wave front doesn’t arrive parallel to the shoreline. This can occur because waves generated far out to sea come from different directions and because the direction a beach faces depends on where it’s located. Most beaches in Orange County face southwest. (The Huntington Beach pier points to about 220°, per Google Earth.) Further up the coast in the South Bay, Hermosa and Manhattan Beaches face almost due west. When waves approach at an angle, one part of the wave front may enter shallower water earlier than the rest. The part entering shallow water slows down, but the part in deeper water continues at full speed. The result is a bending of the wave, something known as wave refraction. 

As a wave approaches a beach and refracts, the wave front will bend and align with the depth contours of the bottom—the isobaths. You can actually figure out which parts of the nearshore are deeper and which are shallower simply by watching the approaching waves. As waves approach the shore at an angle, the wave front bends and becomes more parallel to the beach, but rarely becomes perfectly parallel. If you watch carefully, you will see that one part of the wave begins to break before the other parts. The places where the wave breaks first are the shallowest parts of the nearshore. If the depth of the nearshore has a constant slope (no spots deeper or shallower than others), the wave will break right to left or left to right, depending on whether it’s coming from the right or the left of the direction the beach is facing, respectively.

Headlands—points of land that jut out into the ocean—are a great place to observe wave refraction. A wave approaching a headland straight on will slow down in the shallower water surrounding the headland, but because the extremities of the wave front are traveling faster, the wave will refract. The result is a wave front that bends toward the headland. If the wave approaches the headland at an angle, it will bend around the headland and produce something known as a point break. Point breaks produce some of the best surfing in Southern California. Dana Point once had the best point break in the world, until someone decided to build a jetty. (I hate it when that happens.) Similarly, a wave entering a bay will slow down at its extremities so that the wave bends and takes on the shape of the bay as it nears the shore. 

Between islands or inside boat harbors, boat owners have to worry about something called wave diffraction, the lateral shifting of wave energy along the wave front. It gets complicated because it depends on the wavelength of the approaching wave relative to the width of the opening, and constructive and destructive interference can play a role, but wave diffraction will cause a wave front to arc and expand as it enters the opening. This can result in waves coming from an unexpected direction in what is supposed to be a safe harbor. Between islands the effects can be dramatic as wave patterns become fanlike and superimpose on each other.

When waves approach an underwater reef or seamount, wave refraction may cause the energy of the wave to converge on the reef or seamount. This concentration of energy is known as wave focusing (e.g., Butt 2014). Extreme surf spots such as Jaws, Mavericks, Cortes Bank, and Shark Park produce epic waves as a result of wave focusing around an underwater obstacle. Ranging in size from 50 to 100 feet (15–30 m; e.g., Smith 2006), these are waves to die for—literally, as most of us would be dead if caught in a wave of this magnitude. Next time you’re standing next to an eight-story building, look up. The wall next to you is about the same height as a wave you might experience at one of these places. Except that in an extreme wave, the wall is moving 20 to 30 mph and falling down on top of you. Yikes!

One Southern California extreme wave spot—featured in Bruce Brown’s classic film The Endless Summer (1966)—serves as a perfect example of wave reflection—the change in direction of a wave front when it encounters an obstacle such as an island, beach, seawall, or jetty. If swell period and swell direction are ideal, the Wedge, at the south end of Newport Beach on the Balboa Peninsula, can produce 20-foot (6 m) waves (e.g., Surfline 2023). Waves at the Wedge result from the interaction of incoming waves with waves reflected off the Newport Inlet jetty. When waves approach from the SSW, they bounce off the jetty, which extends from the shore about 1,800 feet (549 m), and reflect 90° (NNW), where they interfere with the incoming waves. Where the crests of the incoming and reflected waves meet (constructive interference), a towering “wedge” of water is produced, hence the name. Next to Hawaii’s Banzai Pipeline, the Wedge rates as one of the best places to watch extreme wave surfing. On a summer day with a large SSW swell, hundreds of onlookers will gather at the Wedge to watch bodyboarders, bodysurfers, and even board surfers take their chances on the waves. 

All of the science aside, ocean waves captivate our attention, whether we’re at the beach or simply watching a surf cam from home. Waves reflect the moods of the ocean—perhaps our own, as well. As the fictional teen “surfer girl” Gidget puts it, “Surfing is out of this world. . . . It positively surpasses every living emotion I’ve ever had” (Wendkos 1959).

 19.15 Chapter References

Bascom, Willard. 1964. Waves and Beaches: The Dynamics of the Ocean Surface. Garden City: Anchor Press. https://archive.org/details/wavesbeaches00doub 

———. 1980. Waves and Beaches: The Dynamics of the Ocean Surface. Revised, updated, and enlarged. Garden City: Anchor Press. https://archive.org/details/wavesbeachesdyna0000basc

Bascom, Willard, and Kim McCoy. 2021. Waves and Beaches: The Powerful Dynamics of Sea and Coast. Ventura: Patagonia. https://www.patagonia.com/product/waves-and-beaches-the-powerful-dynamics-of-sea-and-coast-book/BK855.html

Bierly, Greg. 2005. “Extratropical Cyclones.” In Encyclopedia of World Climatology. Edited by J. E. Oliver. Dordrecht: Springer. https://doi.org/10.1007/1-4020-3266-8_80

Brander, Rob. 2010. Dr. Rip’s Essential Beach Book: Everything You Need to Know About Surf, Sand, and Rips. Sydney: University of New South Wales Press. https://www.scienceofthesurf.com/books

Brown, Bruce. 1966. The Endless Summer. Cinema V. Film. 95 minutes. https://www.imdb.com/title/tt0060371/

Butt, Tony. 2014. Surf Science: An Introduction to Waves for Surfing. Honolulu: University of Hawaii Press. https://uhpress.hawaii.edu/title/surf-science-an-introduction-to-waves-for-surfing/

———. 2021. “Hurricane Larry: How to Track Swells and Calculate Storm Distance.” Magicseaweed. September 7, 2021. https://magicseaweed.com/news/hint-you-can-use-our-swell-charts-to-tell-how-far-away-a-storm-is/11738/

Center for Western Weather and Water Extremes (CW3E). 2021. “CW3E Event Summary: 19-26 October 2021.” October 23, 2021. CW3E.  https://cw3e.ucsd.edu/cw3e-event-summary-19-26-october-2021/

Dally, William R. 2001. “The Maximum Speed of Surfers.” Journal of Coastal Research 29: 33–40. https://www.jstor.org/stable/25736203

do Carmo, José Simão Antunes. 2022. “Dominant Processes that Amplify the Swell Towards the Coast: The Nazaré Canyon and the Giant Waves.” Research, Society, and Development 11(11): e578111133804. https://doi.org/10.33448/rsd-v11i11.33804

Google Earth. 2023. “Huntington Beach Pier.” Google Earth, version 9.181.0.1. Accessed January 31, 2023. https://earth.google.com/web/search/Huntington+Beach+Pier,+Huntington+Beach,+CA

Jones, Ian S. F., Y. Volkov, Y. Toba, S. Larsen, and N. E. Huang. 2001. “Overview.” In Wind Stress Over the Ocean. Edited by Ian S. F. Jones and Yoshiaki Toba, 1-34. Cambridge: Cambridge University Press. https://doi.org/10.1017/CBO9780511552076.002

Knauss, John A., and Newell Garfield. 2017. Introduction to Physical Oceanography. Long Grove: Wavelend Press. https://www.waveland.com/browse.php?t=504

Michell, J. H. 1893. “XLIV. The Highest Waves in Water.” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 36(222): 430-437. https://doi.org/10.1080/14786449308620499

Perlin, Marc, Wooyoung Choi, and Zhigang Tian. 2013. “Breaking Waves in Deep and Intermediate Waters.” Annual Review of Fluid Mechanics 45: 115–145. https://doi.org/10.1146/annurev-fluid-011212-140721

Pilastro, Eleonora. 2022. “Sebastian Steudtner Surfs Giant Wave and Smashes World Record.” Guiness World Records. May 25, 2022. https://www.guinnessworldrecords.com/news/2022/5/sebastian-steudtner-surfs-giant-wave-and-smashes-world-record-705874

Smith, Craig B. 2006. Extreme Waves. Washington, DC: Joseph Henry Press. https://doi.org/10.17226/11635

Smith, F. G. Walton. 1973. The Seas in Motion: Waves, Tides, and Currents—How They Work; Their Causes and Effects. New York: Thomas Y. Crowell. https://archive.org/details/seasinmotion00fgwa

Snodgrass, F. E., K. F. Hasselmann, G. R. Miller, Walter Heinrich Munk, and W. H. Powers. 1966. “Propagation of Ocean Swell across the Pacific.” Philosophical Transactions of the Royal Society A 259(1103): 431–497. https://doi.org/10.1098/rsta.1966.0022

Statista. 2022. Surfing—Statistics and Facts. Statista. https://www.statista.com/topics/9833/surfing/#topicOverview

Surfer Today. 2023. “Big Wave Surfer Márcio Freire Dies in Nazaré.” Surfer Today. January 6, 2023. https://www.surfertoday.com/surfing/big-wave-surfer-marcio-freire-dies-in-nazare

Surfline. 2023. “The Wedge.” Surfline. Accessed January 31, 2023. https://www.surfline.com/surf-report/the-wedge/5842041f4e65fad6a770882b

Warshaw, Matt. 2011. The History of Surfing. San Francisco: Chronicle Books. https://www.chroniclebooks.com/products/the-history-of-surfing

Wendkos, Paul. 1959. Gidget. Columbia Pictures. Film. 95 minutes. https://www.imdb.com/title/tt0052847/

Wheeler, Dennis, and Clive Wilkinson. 2004. “From Calm to Storm: The Origins of the Beaufort Wind Scale.” Mariner’s Mirror 90(2): 187–201. https://doi.org/10.1080/00253359.2004.10656896