# Chapter 11: Waves

## Guide to Reading and Learning

To most of us waves are simply the endlessly varying undulations of the ocean surface that cause breakers to crash on the beach one day but lap gently on it the next. To some of us they are fun things to play in at the beach or to experience while surfing. To others they are nuisances or things that can make us sick if we choose to sail on the oceans. Waves are all these and much more. Did you know that there would be no barrier islands or even sandy beaches to visit if it were not for waves? Did you know that it is thought that a destructive tsunami wave was at least partially responsible for the extinction of dinosaurs? Did you know that every several thousand years there is a tsunami so large that it would dwarf even the enormously destructive Indonesian tsunami that killed more than 220,000 people in 2004? Did you know that every year many people drown simply because they do not know enough about waves to save themselves or keep out of harm’s way? This information and more is contained in Chapter 11 and elsewhere in this text. However, this chapter must start with some basic information about waves and how they are created; as you read the chapter, be patient, for the importance of this material will be revealed later.

When we sit on the beach and watch the ocean, the waves seem to be almost random on some days—some large, some smaller—but on other days the waves appear much more uniform. This is because the waves that break on the beach are actually the sum of perhaps several waves that come from different directions or at different speeds. We learn that each of these separate waves is actually one of two types of regular smooth oscillating motions. Most are progressive waves, in which the water surface at any point oscillates up and down as the wave motion passes by. Some are standing waves, where the water surface rocks back and forth just like a teeter-totter. These wave motions are all easily described by some simple dimensions and their relationships. Most ocean waves are created by wind stress on the ocean surface, but others can be created by impacts such as earthquakes and moving vessels. Once the motion has been started the waves travel over the ocean surface with very little loss of energy until they reach shallow water and break.

Large area, strong wind storms that last for a long time make longer waves than smaller, weaker and shorter storms, but all storms create waves that have a range of different wavelengths. We learn that the speed of travel of a wave over the ocean surface depends on its wavelength, and this allows waves from a single storm to sort themselves out by wavelength as they travel away from the storm. Long-wavelength waves travel fastest and, if you are visiting the shore when the local winds are calm, you may be able to observe that the wavelength of the smooth-swell waves that arrive at the beach becomes shorter as the day goes on. If the storm that created them was far away this progression takes a long time. If the storm was closer, the wavelength shortens more rapidly. Because the period of time between wave crests is related to wavelength, you can see the wavelength change if you simply time the interval it takes for, say, ten waves to crash in succession on the beach and then repeat your observation several hours later. This is one of the many fascinating observations of wave behavior that you can make yourself, especially when you have learned the basic information set out in this chapter. After reading this chapter, take a trip to the beach or even just to a pond or lake armed with a small stone or two to make your own waves.

All mariners are aware of the power of ocean waves. Many a vessel has sunk after an encounter with waves from a massive storm. In some instances, unusually high waves can seem to appear when all previous waves were much smaller, or unusually steep-sided waves can occur in specific regions where the waves and currents are opposed. Wave heights may also be increasing as global climate changes. All this means that studying waves has major practical value to safety at sea and to the commerce and recreation that it sustains.

Waves that travel into shallow water interact with the seafloor. This interaction slows the waves but also eventually makes them steep enough to break; then the wave motion is lost. These interactions are important to all of us because the energy of breaking waves erodes coastlines and moves sand along the coast to create and sustain beaches. One of the seemingly odd observations that you may have made at the beach is that waves almost always break nearly simultaneously along a beach regardless of whether the beach is straight or curved. On a straight beach the waves always appear to approach from almost directly offshore, even though the location of most storms may be far to the right or left of this offshore direction. This is the result of wave refraction, a very simple process that depends on waves slowing down as they enter shallow water.

Once inshore, waves break, sometimes with great force. You may have been cautioned to watch out for “undertow”, “rip tides”, or “rogue” or “sleeper” waves. Scientists do not use any of those terms because they do not describe the phenomena that actually occur. However, the phenomena that you are cautioned about are all quite simple. Once you have read this chapter and understand what actually happens, you will be able to avoid or escape these hazards. Perhaps you can also pass this knowledge on to your friends and family so they too may enjoy the shore safely.

Most people are shocked to learn that tsunamis travel as fast as jet planes and that, at sea, away from the coast, you would never be able to detect a tsunami as it passed you on its way to wreaking havoc on some distant shore. Giant tsunamis have occurred in the past, before recorded history, and will occur again. When such a giant tsunami does occur it likely will be the most devastating natural disaster that humans have ever witnessed.

At the end of this chapter we investigate the nature of standing waves, which are more important in Chapter 12. We also briefly discuss Rossby waves, which are large-scale meanders in the jet stream and ocean currents and Kelvin waves, which are very long period waves that can flow across the equatorial oceans and in rotary fashion around the ocean basins. Little is known of Kelvin waves other than their importance in climate phenomena such as El Niño.

## Chapter 11 Essential to Know

Critical Concepts used in this chapter

## 11.1 Complexity of Ocean Waves

• Waves seen at the shore often appear to arrive in complex patterns of varying height and period. This is because the observed waves are the sum of a number of simple waves of different wavelengths.

## 11.2 Progressive Waves

• The wave form of progressive waves moves across the surface of the water. These waves can be characterized by their wavelength, period, frequency, wave speed, wave height, and wave steepness, which are related by simple formulae.
• Wave period (time between successive crests) is the inverse of wave frequency (number of crests that pass per unit time).
• Wave speed equals wavelength (distance between two crests) divided by period or frequency times wavelength.
• Wave amplitude equals twice the wave height (vertical distance between crest and trough).

## 11.3 What Is Wave Motion?

• In deep water, each water molecule within a progressive wave moves in a circular path and only energy, not water, is transported forward.

Wave Energy

• In the wave motion, energy is continuously converted between kinetic energy and potential energy.
• The total energy per unit area of a progressive wave is proportional to the wave height squared. When wave height is doubled, wave energy is quadrupled.

Restoring Forces

• Once the water surface is displaced from its mean level, a restoring force tends to return the surface to level and this causes the wave form to move.
• The principal restoring force on very short waves, called capillary waves, is surface tension. For very long waves, the Coriolis effect provides a restoring “force”. However, for most ocean waves, the principal restoring force is gravity.
• Displacement of the ocean surface causes a horizontal pressure gradient to be formed between points under the wave crest and points both back and forward toward the trough. Thus, half of the wave’s energy is transported forward and half is transported backward.

## 11.4 Making Waves

Forces That Create Waves

• Most ocean waves are created by the action of winds on the sea surface, and these waves generally have periods up to about 30 seconds. Waves with longer periods (up to about two hours) can be caused by earthquakes, seafloor slumps, and turbidity currents. These are called tsunamis. Tides are waves with periods of about 12 or 24 hours that are caused the gravitational attraction between the Earth, moon, and sun.

Wind Waves

• Winds blowing across the ocean surface can create or build waves primarily through the shear stress (friction) between the moving air and the water surface.
• Initially very short wavelength capillary waves are formed. These have rounded crests and V-shaped troughs.
• Capillary waves die out very quickly due to surface tension unless the winds continue to blow.
• Once capillary waves have been formed winds can build the waves by pushing on the elevated windward side of the wave and by creating a low pressure area on the side of the wave away from the direction that the wind blows.  This mechanism is similar to the lift created by airflow over an airplane wing.

## 11.5 Sea Development and Wave Height

• Winds create and build waves into a sea that has a spectrum of waves of different heights and wavelengths. The waves are initially small in both wavelength and height but progressively build to greater heights and longer wavelengths.
• As the waves build their shape chnage progressively until they become  trochoidal—pointed crests and rounded troughs.
• Because winds are always variable in time and space, waves of various heights, wavelengths, and directions of travel are formed in windy areas. This confused sea surface state is called a “sea,” as opposed to a “calm” (when there are no waves), and a “swell” when the waves are generally smooth and mostly of the same wavelength.
• In deep water, when waves reach a height of about one-seventh of their wavelength they become oversteepened and unstable and break to form whitecaps.

Factors Affecting Maximum Wave Heights

• Maximum wave heights are determined by wind speed, duration, and fetch (distance over which the wind blows). The maximum heights are only reached when the wind blows long enough. However, if the winds continue to blow beyond this time, wave heights do not increase further. This is called a fully developed sea.

Maximum Observed Wave Heights

• Areas of persistent wave activity occur in the trade wind and westerly wind zones
• The highest waves in the Pacific Ocean are generally higher than those in other oceans due to the large fetch within the east–west wind belts.
• High waves are also developed in the westerly wind belt around Antarctica, where the fetch is belt is not limited by landmasses.
• The highest wind wave reliably reported was about 34 m, but waves that are 40 m and higher have been detected indirectly under major hurricanes.
• Satellites cannot measure the height of individual waves because their sensor beam angles, although very narrow, are still too wide for the sensors to distinguish individual waves.

Effects of Currents on Wave Height

• Waves become steeper when they flow into an area where the current flows in the opposite direction to the wave direction. Large, steep sided waves in such areas can cause even large ships to break apart and sink.

Are Wave Heights Increasing?

• There is strong evidence that wave heights have increased, at least in some areas, in recent decades. It is not known whether this is due to a natural cycle or to greenhouse-induced global climate change.

## 11.6 Wave Dissipation

• Most waves lose energy very slowly due to internal friction and can travel very large distances with little loss of energy. Capillary waves are the exception as they quickly lose energy due to the action of surface tension..
• Storm waves normally all travel away from the storm in the same general direction. However, they tend to spread somewhat laterally. Total energy is not changed but the lengthening of the crests as the wave spreads reduces wave height and the energy per unit length.
• Most wave energy is dissipated as heat when the wave breaks on a shoreline. Although very large, the amount of heat released is not sufficient to cause a significant temperature change in the water because of water’s very high heat capacity.

## 11.7 Deep-Water Waves

• In deep water (depth greater than half the wavelength [L/2]), wave speed depends on wavelength.
• The diameter of the circle in which water within the wave moves is equal to the wave height at the surface and decreases with depth below the wave. At depth of one-ninth of the wavelength, the circle is reduced to one half of the diameter at the surface. Wave motion is essentially reduced to zero at a depth below the surface equal to one half the wavelength; this is called the depth of no motion.
• The speed of a deep water wave is equal to 1.25 times the square root of the wavelength, or 1.5625 times the period squared.

Wave Trains

• The horizontal pressure gradient established under a wave causes half of the wave energy to be transferred forward to the preceding wave (or still water surface if there is no preceding wave) and half to be transferred backward to the following wave (or still-water surface behind the wave if there is no following wave).
• Waves travel in trains in which the leading wave in the train is continuously destroyed and replaced by another at the rear. The speed of the train is half of that of the individual waves.

Wave Interference

• Waves travel independently of each other, but interference between waves of different wavelengths and directions can cause the wave pattern at any one location to be complex.

Wave Dispersion

• Longer wavelength waves travel faster than shorter wavelength waves so they move progressively further ahead ahead of shorter wavelength waves, as the waves all move away from the storm.. The waves are sorted by this process, called wavelength dispersion. Wavelength dispersion increases the separation between waves of different wavelengths with increasing distance from their common area of origin.
• At remote locations the longest waves from a storm arrive first, followed by progressively shorter waves.

## 11.8 Waves in Shallow Water

Interaction with the Seafloor

• As waves enter shallow water, they interact with the seafloor. The resulting friction slows the wave and distorts the water orbits into elongated ellipses. Wave speed and wavelength are both reduced, while period remains the same. Wave height initially decreases slightly but then increases rapidly as the wave enters water shallower than L/20.
• In shallow water (less than L/20) the speed of waves is dependent only on the water depth. The speed is equal to 3.13 times the square root of the depth.

Wave Refraction

• Waves entering shallow water are refracted, as one section of the wave crest enters the shallow water earlier and is slowed down before the adjacent section. This causes the wave crest to turn and approach the beach more nearly parallel to it..
• Wave refraction causes wave energy to be focused on headlands and spread along a longer length of shore within bays.

Breaking Waves

• Waves break at the shore when they reach a depth of water of about 1.2 times the wave height.
• Breakers can be spilling, plunging, collapsing, or surging breakers depending on beach slope and seafloor roughness. Generally, spilling breakers are formed on shallow-slope beaches and plunging, collapsing, and surging breakers on respectively steeper-sloping beaches.

Surfing

• Surfers must position their boards in the breaking wave so that they are within the upward-moving part of the wave’s rotary motion. This ensures that their tendency to fall down the wave due to gravity is balanced with the upward pressure of water in the wave motion.

Surf Drownings and Rip Currents

• When waves break, water moves up then flows back down the beach. The forward surge of the wave or the backwash can knock people off their feet, and successive waves can be strong enough to prevent them from standing up again. This combination causes many beach drownings. Drowning can easily be avoided by using the wave action to wash your body up the beach, then digging hands or feet into the sand to slow the backwash; each wave moves you further up the beach to safety.
• In shallow water there is a small forward transport of water with each wave. This leads to accumulation of water on the beach. The accumulated water flows offshore in narrow rip currents spaced at intervals along the beach. These currents can be too fast to swim against but are narrow and can be escaped by swimming parallel to the beach for a few strokes.

## 11.9 Tsunamis

• Tsunamis are caused by earthquakes or landslides. They have very long wavelengths (10–30 minutes) and travel and are refracted as shallow-water waves. They move at speeds of several hundred kilometers per hour. Tsunamis are generally no more than a meter or two high in deep water, but can build rapidly to heights of tens or, in extreme cases, hundreds of meters when they enter shallow water.  .
• Because tsunami waves have a very long wavelength they never become steep enough to break. However, at the shore they create fast, turbulent currents. Because the tusnami’s period is more than ten minutes, these currents flow continuously inshore for five or more minutes before reversing and flowing offshore, also for many minutes. The large volumes of swiftly flowing water in these currents can cause massive damage to shoreline structures and harbors.
• Either a crest or a trough of a tsunami can reach a particular shoreline first. Most tsunamis consist of a series of waves that may last many hours. The second and subsequent waves are often higher and more destructive than the first wave..

## 11.10 Internal Waves

• Internal waves form on pycnoclines. They have much longer wavelength than surface-wind waves and break as they pass over the continental shelf.

## 11.11 Rossby and Kelvin Waves

• Rossby waves, often called planetary waves, occur in both the oceans and atmosphere as a result of the interaction between geostrophic flows and the latitudinal gradient of the Coriolis effect. Rossby waves flow east to west but may be carried west to east by air or water currents. In the atmosphere they may be seen as meanders of the jet stream. In the oceans, Rossby waves have wavelengths of hundreds to thousands of kilometers and wave heights of only a few centimeters.
• Kelvin waves travel faster than Rossby waves. They travel directly west to east across the equator and along the coastal margins, counterclockwise in the Northern Hemisphere, and clockwise in the Southern Hemisphere. Both Rossby and Kelvin waves are associated with oscillations such as the El Niño Southern Oscillation (ENSO).

## 11.12 Standing Waves

• Standing waves are formed where wave motion is blocked by a land barrier. They are oscillations of the sea surface in which there is one or more node at where the sea surface oscillates up and down but there are no currents associated with the wave motion. Standing waves also have one or more antinode at which the sea surface level does not change but there are reversing currents associated with the wave oscillation.
• Standing waves form tuned oscillations in basins in which the length of the basin is the same as the wavelength, or some multiple thereof.

Critical Concept Reminders:

CC.5 Transfer and Storage of Heat by Water (p. 290)

• Water’s high heat capacity allows large amounts of heat to be stored in the oceans with little change in temperature. Thus, when waves break and release their energy as heat, they cause very little change in the temperature of the water. To read CC5 go to page 15CC in your textbook.

CC.9 The Global Greenhouse Effect (pp. 288–289)

• Major climate and climate related changes may be an inevitable result of our burning fossil fuels. The burning of fossil fuels releases carbon dioxide and other gases into the atmosphere, where they accumulate and act like the glass of a greenhouse retaining more of the sun’s heat. One of these climate related changes is the possibility that average wave heights in the oceans will increase, which would add to existing safety issues for ocean vessels and increase shoreline erosion rates. To read CC9 go to page 22CC in your textbook.

CC.12 The Coriolis Effect (pp. 283, 305)

• Water masses move freely over the Earth surface while the solid Earth itself is constrained to move with the Earth’s rotation. This causes moving water masses, including some long period waves, to appear to follow curving paths across the Earth’s surface. The apparent deflection, called the Coriolis effect, is to the right in the Northern Hemisphere and to the left in the Southern Hemisphere. It is at a maximum at the poles, reduces at lower latitudes, and becomes zero at the equator. To read CC12 go to page 32CC in your textbook.

CC.13 Geostrophic Flow (p. 305)

• Water and air masses flowing on horizontal pressure gradients are deflected by the Coriolis effect until they flow across the gradient such that the pressure gradient force and Coriolis effect are balanced, a condition called geostrophic flow. The interaction of geostrophic currents and the change of the magnitude of the Coriolis effect with latitude are the cause of Rossby waves that flow east to west across the oceans and atmosphere. The meanders in the jet stream seen in many weather maps are Rossby waves. To read CC13 go to page 43CC in your textbook.

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