9.8: Waves in Shallow Water

On a calm-wind day at the shore, when there is a swell from a distant storm, we observe that offshore in deep water, the waves roll smoothly toward the shore without breaking. Once they reach shallow water, the same waves form breakers. Clearly, something happens in shallow water that alters the behavior of waves.

Interaction with the Seafloor

When waves enter water shallower than half their wavelength (L/2), they begin to interact with the seafloor. The wave motion becomes inhibited because the solid seafloor prevents water molecules from moving in circular orbits. Molecules immediately above the seafloor can move only back and forth. As a result, the orbits of water molecules within the wave are distorted into elongated ellipses (Fig. 9-16). Compression of the orbits and friction between water and seafloor slow the forward motion of the wave.

Figure 9-16. Effects of water depth on waves. (a) In waves traveling across the ocean where the water depth is greater than one-half of the wavelength, the orbits of water particles in the wave are circular and wave speed is determined only by the wavelength. (b) Where the water depth is less than one-half but more than one-twentieth of the wavelength, wave motion is slowed and distorted by friction with the seafloor, and the particle orbits are flattened into ovals. (c) Where the water depth is less than one-twentieth of the wavelength, the particle orbits are flattened so much that they become nearly horizontal, back-and-forth motions. In this case, the wave speed is determined only by the depth. Note that each part of the figure does not depict a single wave moving into shallower water. As explained later in the chapter, wavelength is progressively shortened once a wave enters shallow water and becomes an intermediate or shallow water wave. Also, the wave height decreases somewhat when it is an intermediate wave and then increases rapidly once it becomes a shallow water wave and moves into progressively shallower water.

As water depth decreases between L/2 and L/20, the forward speed of the wave decreases progressively (Fig. 9-17). The equation used to calculate the speed of waves when they are in water depths between L/2 and L/20 is somewhat complex. In these depths, waves are called intermediate waves. In contrast, at water depths of L/20 or less, the wave speed is controlled only by water depth. Waves in water depths less than L/20 are called shallow-water waves. The speed of shallow-water waves is given by the equation:

C = √gD = 3.13 √D

where C is the celerity (m•s^–1), D is the depth (m), and g is the acceleration due to gravity (m•s^–2). Thus, all waves, regardless of their period, travel at the same speed when they are in water of the same depth, provided that the depth is less than L/20. Because wavelength and period are related, longer-period waves will become shallow-water waves in deeper water than short-period waves do (because L/20 is greater when L and T are larger).

Figure 9-17. The speed (or celerity, C) of a deep-water wave is constant and depends on the wavelength (or wave period). For example, a deep-water wave of wavelength 1000 m has a speed of almost exactly 40 m•s^–1. The speed of shallow-water waves is determined only by depth and decreases as water depth decreases. To determine the speed of a shallow-water wave in this figure, simply find the speed on the orange line that corresponds to the water depth. The speed of intermediate waves is determined by both water depth and wavelength. For example, the speed of an intermediate water wave of wavelength 1000 m decreases from almost exactly 40 m•s^–1 when the water is 400 m deep, to a little more than 30 m•s^–1 when the depth equals one-twentieth of the wavelength (50 m) and the wave becomes a shallow-water wave.

As waves enter shallow water and are slowed, their period does not change. Because wave speed is the ratio of wavelength to period (C = L/T) and the period does not change, the wavelength must decrease as the speed decreases (Fig. 9-17). We can easily see why the wavelength decreases if we consider what happens to two successive waves following each other into shallow water. The first wave enters shallow water and is slowed before the second wave. The second wave comes closer to the first wave because it does not slow until it also reaches the shallow water. As they move into shallower water, both waves continue to be slowed, but the first wave continues to be slowed sooner than the second wave, which is always in deeper water. Therefore, the wavelength decreases progressively as waves move into shallower water and wave speed continues to decrease (Fig. 9-17). As wavelength decreases, wave steepness increases (Fig. 9-8).

Wave Refraction

Waves usually approach a shoreline at an angle. Consequently, because one end of the wave-crest line enters shallow water and slows while the rest of the wave is still in deeper water and traveling at its original speed, the wave is refracted, or bent (Fig. 9-18a). Lines perpendicular to the crest lines of successive waves in Figure 9-18a show the path that a specific point on the wave crest follows as it moves inshore. These lines are called wave rays or “orthogonals.” As the wave continues to move inshore, more of the wave enters shallow water and slows. However, the section of the wave that first entered shallow water is still moving into progressively shallower water and slowing further, so wave refraction continues.

Figure 9-18. Wave refraction. (a) Waves generally approach the coastline at an angle. Consequently, part of the wave front reaches shallow water and is slowed before the rest of the wave. It continues to slow as the other parts of the wave move more rapidly toward shore until they, too, are progressively slowed. As a result, the wave front is bent and becomes aligned more closely parallel to the beach. The path followed by any point on the wave crest is called a wave ray. (b) As a wave approaches a bay flanked by headlands whose topography extends onto the offshore seafloor, the first parts of the wave to slow are those over the underwater extensions of the headlands, and the last part to be slowed is that over the deeper water at the center of the bay. Thus, the wave is refracted, and wave energy is concentrated on the headlands and spread out inside the bay. Breaking waves are the strongest at the headlands.

Note that the wave rays bend toward shallower water. As the refraction process continues to bend the wave, the wave crests tend to become aligned parallel to the shore. The refraction is seldom complete, so a wave crest rarely reaches the shore or breaks at precisely the same time along its entire length. However, even if waves approach the shore at a large angle, they are refracted and always reach the shore almost parallel to it (Fig. 9-18a).

If waves move into shallow water where seafloor ridges or depressions are present, the parts of each wave where the water is shallower slow down, while other parts in deeper water do not. Thus, the refraction pattern differs from the simple pattern depicted in Figure 9-18a. In many areas, seafloor topography immediately offshore is a continuation of land topography. For example, many coastlines have horseshoe-shaped bays where valleys reach the shore (Fig. 9-18b). The valley often continues offshore from the center of the bay, and the seafloor falls away in a valley-shaped depression between submerged ridges that extend out from the bay’s headlands. Most bays of this type have a horseshoe-shaped sandy beach at the center and rocky headlands at each end (Fig. 9-18b). As we shall see, the headlands are rocky, and the bay’s interior is sandy because of the wave refraction.

Figure 9-18b shows the locations of successive wave crests as waves move toward a bay. As a wave approaches shore, the first part of the wave to encounter shallow water will be that over the undersea ridge extending out from the headland closest to the direction from which the wave arrives. While this part of the wave slows, the rest of the wave continues at its original speed. Then another part of the wave slows where it encounters the submerged ridge extending from the second headland. Thus, while parts of the wave are slowed at each headland, the rest of it continues at its full speed into the center of the bay. As the wave front enters the bay, the center section travels farther in deep water than the parts that enter the bay at each side. The wave is refracted so that it breaks at almost the same time along the entire length of the beach. Note again that the wave rays always bend toward shallower water.

Refraction redistributes the wave energy. In deep water, the wave has the same amount of energy per unit length along the entire length of its wave crest. As the wave is refracted in a bay, the total length of the wave crest is increased (Fig. 9-18b), and the same wave energy is distributed over this greater length. Lengthening of the wave crest also results in a lowering of the wave height. In contrast, at a headland, refraction reduces the length of the wave crest and, consequently, increases the wave energy per unit area and the wave height.

Refraction focuses wave energy on headlands while spreading wave energy along the beach within the bay. This is why we generally swim on the beach near the center of the bay and not at the headlands. The same wave that breaks gently at the middle of the beach smashes violently at the headland. This is also the reason we should exercise caution while walking on rocks of a headland. The headland is where waves are highest and crash most violently ashore. In addition, on narrow headlands where the bottom topography is appropriate, a single wave with the right period that approaches from the right direction may be refracted to hit the headland simultaneously from two directions (Fig. 9-18b).

The wave refraction that concentrates wave energy at headlands and spreads energy within a bay determines the character of the shore at these locations. At the headlands, where wave energy is focused, sand does not accumulate because it is carried away by the wave action, and the shore is steadily eroded. In contrast, within the bay, gentler wave action transports sand toward the shore, where it builds and maintains a beach (Chap. 11).

In areas where the seafloor has complex topography with offshore rocks, ridges, and depressions, wave refraction can be far more complicated than the simple patterns depicted in Figure 9-18. Sometimes we can tell where such underwater features are by carefully watching wave refraction patterns from a beach or headland.

Breaking Waves

When waves enter shallow water and interact with the seafloor, their height is altered by the interaction. At first, the wave height is reduced slowly as the water depth decreases below L/2. This loss in wave height is caused by flattening of the orbital paths of water molecules in the wave (Fig. 9-16b). However, in water depths of about L/10 and less, the trend reverses and wave height increases rapidly as water depth decreases (Fig. 9-19). The reason is that wavelength decreases as water depth decreases, each wave is “squeezed” by its neighbors, and kinetic energy is converted to potential energy.

Figure 9-19. Wave height changes as a wave moves into progressively shallower water. The wave height declines slowly until the water depth is about one-tenth of the wavelength, then increases rapidly as depth decreases. Most waves become unstable and break before their height is much greater than it was in deep water (usually before the wave height exceeds 1.1 times its height in deep water). However, very long-wavelength waves, such as tsunamis, may be dramatically increased in height before they break.

Wavelength decreases faster than wave height as water depth decreases below L/2. Hence, wave steepness increases (H decreases but L decreases more rapidly, and H/L therefore increases). The speed and wavelength continue to decrease in water shallower than L/20 as the wave becomes a shallow-water wave (Fig. 9-17), but wave height increases (Fig. 9-19), wave steepness increases more rapidly as depth decreases (H increases, L decreases, and H/L increases rapidly), and wave shape is much modified.

Wave steepness increases until the wave becomes unstable and breaks. Wave heights and wavelengths vary among waves reaching the shore at any one time. Waves that follow each other, therefore, become unstable and break at different depths and thus different distances from shore. The area offshore within which waves are breaking is called the surf zone.

Although waves are refracted as discussed above, waves rarely reach the shore from a direction exactly perpendicular to the shoreline, and the seafloor rarely has exactly the same slope along the entire shoreline, so almost always one part of a wave breaks before another. A wave often breaks progressively along the crest of the wave as each part of the crest moves into water shallow enough to cause the wave to break.

Waves break in different ways that depend on several factors, including wave period, wave height, and the slope of the ocean floor. These factors determine how quickly the wave becomes oversteepened and unstable.

Spilling breakers are formed when the seafloor over which the wave is traveling is almost flat. When a wave reaches a water depth of about 1.2 times its wave height, it becomes unstable and the crest begins to tumble down the forward face of the wave (Fig. 9-20a). The forward face fills with churning, turbulent water and air bubbles that we see as a white foam. As the wave continues inshore, wave steepness increases slowly because of the almost flat seafloor, but as water spills off the wave crest, it reduces wave height and steepness. The spillage occurs fast enough to maintain the wave at its critical steepness against the tendency for the shallowing water to increase the steepness. Thus, spilling breakers break progressively as they travel inshore. Wave height is reduced progressively by the spilling action as the water depth decreases, until finally the turbulent wave crest encounters the seafloor and wave motion ends.

Figure 9-20. Breaking waves. (a) The peak of a spilling breaker becomes oversteepened, and it breaks by spilling water from the crest down the front face of the wave. (b) The peak of a plunging breaker travels fast enough to outrun the rest of the wave, and the crest curls over and smashes down on the front of the wave. (c) The base of the front side of a collapsing breaker becomes unstable before the crest, and the front face of the wave collapses. (d) A surging breaker never really breaks. The front face of the wave simply surges up and down on the beach, and the wave energy is partially reflected back out to sea.

Plunging breakers are the spectacular curling waves that many surfers covet (Fig. 9-21). They form when the seafloor slope is moderately steep. When the wave reaches the depth at which it becomes unstable, the bottom part of the wave is slowed more quickly than the upper part can slow or spill. Thus, the bottom of the wave lags behind as water in the wave crest outruns it while still traveling in its wave orbit. As a result, the wave crest curls over in front of the wave and plunges downward until it crashes into the trough preceding the wave (Fig. 9-20b).

Figure 9-21. This surfer is riding in the tube formed by a plunging breaker in the north end of the Banzai Pipeline, the famed North Shore surfing area of Oahu, Hawaii.

Collapsing breakers, which are relatively rare, occur where the seafloor has a steep slope, and the lower part of the wave is slowed so rapidly that the leading face of the wave collapses before the crest arrives. As the wave breaks, foam and bubbles are concentrated at the base of the forward face of the wave, and the crest collapses behind, usually with little splash (Fig. 9-20c).

On very steep shores, waves may appear not to break at all. The waves simply surge up and down a very steep beach with little bubble production (Fig. 9-20d). Surging breakers are rare, but the very small, gentle waves that lap onto some beaches in very calm conditions behave somewhat like surging breakers because even a flat beach has significant slope in relation to the tiny wave height. Surging breakers do not break up in foam and bubbles because most of the wave energy is reflected by the steep shore face.

Waves are reflected off vertical or nearly vertical solid objects such as cliffs or seawalls with little loss of energy, just as light is reflected by a mirror. When waves are reflected, they pass through and interfere with the incoming waves. Wave patterns created by such reflection can be very complex and are of great concern to engineers who build harbors, marinas, and other coastal structures. Waves are also diffracted by solid objects. Diffraction occurs when part of a wave is blocked by a solid object and the edge of the remaining wave spreads out after passing the object (Fig. 9-22).

Figure 9-22. A wave is diffracted when the wave crest encounters a solid object, such as a breakwater. The part of the wave that passes by the end of the breakwater is bent such that the wave spreads out behind the breakwater.

Often, we see waves breaking at some distance from shore. Reduced in height, the same waves continue toward shore until they break again near the water’s edge. This pattern indicates that the seafloor is shallower in the offshore surf zone than it is just inshore of that area. In tropical locations, the shallow area is commonly a fringing coral reef (Chap. 13). In other locations, it is a longshore bar (Chap. 11). Fringing reefs and long-shore bars are aligned parallel to many coastlines. They help to protect coasts from erosion by waves because they dissipate some wave energy, particularly from the highest and most energetic waves, before they reach the shore.

Surfing

To surf a wave, surfers must propel themselves forward on the board to join the wave motion just at the point where the wave becomes unstable and begins to break. They must position themselves on the forward face of the chosen wave and point the board “downhill.” If the surfers “catch” the wave correctly, they will move forward with the wave. In the correct location, surfers are balanced such that their tendency to fall down the wave because of gravity is just offset by the pressure on the board from the water pushing upward in its orbital motion. Therefore, they must place the board where water is moving upward into the breaking crest. Plunging breakers (Figs. 9-20b, 9-21) provide unique conditions for surfing because most of the front face of the wave is traveling upward and then forward to join the curling crest.

Many surfers believe the best surfing is found where waves approach the shore from a large angle. In such places, refraction does not align the waves exactly parallel to the shore before they break. A wave breaks progressively along the crest as each section reaches shallow water. Surfers simply ride laterally along the wave crest. They remain on the section of the wave crest where the forward face of the wave is moving upward in its last orbital motion before breaking. Where plunging breakers break in this way, surfers can ride under the wave crest in the tube formed where the crest plunges forward ahead of the lower part of the wave (Fig. 9-21).

Humans are not the only animals that enjoy riding the orbital motion of waves. Porpoises ride ships’ bow waves by positioning themselves where the water in the bow wave is moving forward in its orbital motion. They often ride this way for hours, “pushed” along by the ship.

Surf Drownings and Rip Currents

Unfortunately, many people drown while playing or swimming in the surf. Drownings are often blamed on “undertow” that supposedly sucks unwary individuals underwater away from the beach. However, “undertow” does not exist. People drown because of two simple phenomena, each of which can be easily avoided or escaped. When large waves break very close to the beach, the force of a wave can easily knock over a person who walks out into the surf. Once knocked over by a wave, an unwary individual may be washed seaward as the water flows back down the beach from the collapsed wave. The person is then at a place where the next wave crashes down. This wave washes the person first toward and then away from the beach as it breaks and the water runs back. In these circumstances, many people simply panic and drown because they cannot recover their balance and stand up.

If you find yourself in this situation, you will not drown if you follow two simple rules. First, grab a breath each time a wave has passed, before you are hit by the next one. Second, use the waves to your advantage. Let a wave carry you inshore. When your inshore motion ceases, do not try to stand up. Instead, either dig your hands and feet into the sand or start crawling toward the beach as you grab that next important breath. As each new wave arrives, let go and allow it to carry you farther inshore. If you take this approach, you will soon be sitting safely on the beach telling others how much fun you had playing in the surf. If you are a good swimmer, an alternative is to swim out beyond the breaking waves, where you can float comfortably and rest before swimming back to shore or calling for help.

The second major cause of beach drownings is rip currents (often incorrectly called “rip tides”). Waves create a small but significant net movement of water in the direction of wave travel (Fig. 9-10b). This net forward movement is increased in the surf zone because water in the wave crest outruns water in the wave trough as the wave breaks. Accordingly, in the surf zone, water is continuously transported toward the beach. However, it cannot simply accumulate on the beach, but must flow back through the surf zone. When the waves approach the beach at an angle, as they usually do, water that is transported onto the beach is also transported along the beach in the direction of the waves. Eventually the water encounters an area where it can flow back out as a rip current through the surf zone against the waves more easily than it can continue to accumulate and flow along the beach (Fig. 9-23a). In such areas, wave heights are generally somewhat smaller because a depression or shallow channel runs offshore from the beach.

Figure 9-23. (a) Rip currents are fast, narrow currents that return water offshore after it has been transported toward the beach that form at intervals along a beach. (b) Rip currents are sometimes visible as foam or discoloration of the water caused by sediments suspended by turbulence in the swiftly moving current. This type of rip current is not in a fixed location on the beach and may appear quickly and dissipate just as quickly. (c) A more permanent type of rip current may form where water runs through a channel on the seafloor. This type can often be seen as a gap in the breaking crests of the waves.

The corridors in which water flows away from the beach are usually narrow and may be spaced well apart. The rip currents that flow offshore through these corridors consist of all the water that was transported onshore over a broader width of beach (Fig. 9-23). Because large amounts of water are moving through the narrow corridors across the surf zone, the flow rate or current through the return corridors can be fast. 

Swimmers who enter a rip current will find themselves carried rapidly out into deeper water by the current. A rip current can be so fast that even the most accomplished swimmer cannot swim against it back to the beach. Tragedies occur when a panicked swimmer tries to fight the rip current, becomes exhausted, and drowns. However, even poor swimmers can easily avert such tragedies if they understand that rip currents are narrow, often only a few meters wide. A swimmer who meets a rip current should turn and swim parallel to the beach. A few strokes will bring the swimmer out of the rip current to an area where it is easy to swim safely ashore. A less desirable alternative is simply to ride with the rip current until it stops, usually only 200 or 300 m offshore, and then signal for help.

The locations of rip currents cannot always be seen easily, particularly by a swimmer in the water, but they are most likely to occur where depressions run down the beach into the surf. Rip currents may reveal themselves in plumes of increased turbidity (Fig. 9-23b), reduced wave heights in the surf zone, or, occasionally, as lines of floating debris or foam moving offshore.