9.3: What is Wave Motion?

When a wave moves across the ocean surface without breaking, there is almost no net forward motion of the water itself. As we can easily see at the shore, objects such as logs and surfers floating outside the surf zone are not carried inshore with the waves. As waves pass, the floating objects ride up and down on each wave but remain at almost the same location indefinitely. Surfers must paddle forward to “catch” a wave before they can ride it inshore.

If water moved forward with the wave, the world would be an entirely different place. If you have walked or swum out through the surf on a beach, you have experienced what this would be like. When a wave breaks on the beach, the foaming water in the wave crest does move forward faster than the water under the wave is moving back away from shore. Walking or swimming through the surf is difficult because each wave tends to knock you down and carry you back toward the beach until the trough of the wave arrives and carries you seaward as water drains down the beach. However, once you have passed through the surf zone, the same waves that knocked you down in the surf no longer push you toward the beach. Imagine the difficulty ships would have if water on the open ocean were transported in the direction of the waves, in the manner that water in the surf zone is.

Wave Energy

A wave has both kinetic energy and potential energy. Kinetic energy is possessed by water molecules that are moving in the wave, and potential energy, by water molecules that have been displaced vertically against gravity and surface tension.

At the wavelengths of most ocean waves, the total energy (E) per unit area of a wave is approximately

E = 0.125(grH²)

where r is the absolute density of water (in g•cm^–3), g is the acceleration due to gravity (9.8 m•s^–2), and H is the wave height (in m). E is measured in joules per square meter (J•m^–2). Wave energy does increase with wavelength, but this factor is only important for waves of very long or very short wavelength.

Because water density changes very little in the open oceans (Chap. 5) and g is a constant, the total energy of a wave depends primarily on its height. The total energy of a wave is multiplied by a factor of 4 if the wave height is doubled.

Restoring Forces

Instead of moving forward with the wave, each water molecule within a wave moves in an orbital path. In deep water, the orbital path is circular (Fig. 9-3). Only the waveform and energy associated with the wave move forward. For the waveform to move forward, a restoring force must exist that tends to return the ocean surface to its original flat configuration after the water is initially displaced.

Figure 9-3. In a progressive wave, water at the crest is moving forward but has no vertical velocity. After the crest passes, the forward velocity slows, and vertical velocity increases until, halfway between crest and trough, the velocity is entirely vertical. As the wave continues forward, the vertical velocity is slowed, and the water gains velocity in the backward direction. At the trough, the vertical velocity is zero, and the backward velocity is at a maximum. The waveform moves forward, but each particle of water moves in a circular orbit whose diameter equals the wave height (for water at the surface) and returns to its starting point after each wave. You can see this for yourself if you push down on a water bed. The wave travels across the bed, but the plastic “surface layer” returns to its original location after the wave has passed.

The principal restoring forces acting on ocean waves are surface tension and gravity. Surface tension pulls the surface equally in all directions, contracting the surface to its minimum area—a flat plane (Chap. 5). A trampoline provides a good analogy for surface tension. The trampoline surface is depressed and stretched when someone lands on it, but its “surface tension” causes it to snap back to its normal flat configuration, launching the trampoliner into the air.

Gravity acts on water molecules within a wave and causes a pressure gradient to develop beneath the sloping surface of the waveform (Fig. 9-4). The water flows in response to the pressure gradients and tends to flatten the ocean surface. In simpler terms, gravity causes the water to move downwards from the high parts of the wave to fill the depressions and restore the surface to a flat configuration.

Figure 9-4. The horizontal pressure gradients under a wave illustrate how waves move. The pressure difference between locations directly beneath the crest and locations at the same depth beneath the trough tends to move water away from each crest toward the troughs. Between two crests, the pressure gradients are symmetrical and tend to move water toward the troughs. However, the water under each crest has kinetic energy because it is moving forward with the wave motion, and this energy is transferred forward from wave to wave with the wave motion. In front of the leading wave, the pressure gradient tends to move the water forward, displacing the still water surface in front, transferring some of the kinetic energy into potential energy, and beginning the wave motion. Some energy is transferred backwards from the last wave of a group of waves, in much the same way, to form a new wave at the back of the train.

Although interactions of restoring forces with water molecules in the wave are somewhat complicated, the principle is straightforward. Consider a water molecule located at the high point of an elevation of the ocean surface. The water molecule has potential energy because it is elevated above the mean water level. The restoring forces accelerate the molecule downward, and potential energy is converted to kinetic energy.

When the molecule reaches the mean surface level, its initial potential energy has been converted to kinetic energy and its motion is vertically downward. Because it has kinetic energy, it continues its downward motion but is slowed by the restoring forces. As it slows, kinetic energy is converted to potential energy. The molecule will continue in motion until all of its kinetic energy is converted to potential energy, at which point the molecule is at the same distance below the mean surface level as its starting point was above that level.

This process explains why water moves up and down as potential energy is converted to kinetic energy and back. Why, then, do water molecules in a progressive wave move in a circular path and not simply oscillate vertically up and down, falling from crest to trough and then flowing back up to the crest to repeat the cycle? In fact, this simple vertical oscillation can occur at some locations where a vertical barrier exists at the exact location of a wave crest or trough. Where this happens, the wave is not a progressive wave but instead is a standing wave. Standing waves (discussed later in the chapter) behave differently from progressive waves.

In contrast, if no barrier is present, the leading wave of any series of waves is adjacent to an undisturbed water surface. The horizontal pressure gradient under the leading wave moves water molecules toward the undisturbed surface (Fig. 9-4). This movement causes the undisturbed water surface at the leading edge of the wave to be displaced. Thus, the restoring force (gravity) causes a disturbing force that transfers energy to undisturbed water ahead of the leading wave, wave energy is translated forward, and the wave is progressive.

The two restoring forces, gravity and surface tension, act on all waves. Gravity is the principal restoring force for most waves within the range of wavelengths normally present in the oceans, and such waves are often called gravity waves. In contrast, for waves with very short periods (<0.1 s) that can have only very small wave height (for reasons discussed later), gravity is less important and surface tension is the principal restoring force. These small waves are called capillary waves because capillarity is another term for surface tension. For waves with very long periods, such as the tides, an additional restoring force, the Coriolis effect (CC12), which is not actually a “force,” is important (Fig. 9-5).

Figure 9-5. It is estimated that most of the energy associated with waves in the world’s oceans is contained in wind waves that have periods of between 0.2 and 30 s. However, there is also considerable energy associated with tides (periods of 12 h and above) and, despite their infrequent occurrence, considerable energy is also associated with tsunamis and severe storm waves (periods between about 1 min and 1 h). The smallest waves, capillary waves, are created by winds and restored principally by surface tension. Winds are responsible for creating most waves with periods less than about 15 min, and gravity is responsible for restoring all except the shortest-period capillary waves within this range of periods. Longer waves are formed by earthquakes and tidal forces, and the Coriolis effect is the principal restoring force for the longest waves.