A KEEN-EYED OBSERVER WALKING along a sea beach can find many interesting things having to do with the movement of water onto and away from the shore. They include tiny crisscrosses etched into the sand by the surf as it spends itself on the shore and large-scale cusps dug out by the incoming waves. In addition one sees the ceaseless and highly varied movements of the surf itself.

One thing to notice is that whereas the waves in deep water move in many directions, the waves coming onto a beach have a uniform direction of movement that is more or less perpendicular to the shoreline. The explanation starts with the fact that the waves are slowed as they move into the shallow water near the shore and begin to "feel" the bottom. A wave coming into shallow water at an angle that is not perpendicular to the shoreline is slowed first at the inshore end while the remainder of the wave front is still traveling relatively fast. The result is that the wave is slewed so that its direction of travel becomes perpendicular to the shoreline. In other words, the wave is refracted.

Another thing to notice is that the waves in deep water have a fairly uniform undulating shape, whereas waves: in shallow water display a variety of shapes. Four groups of breaking waves can be identified. First, in what is called spilling, the wave retains its usual shape but a layer of foam spills down the for. ward slope. Second, a plunging breaker: develops when the crest of the wave overruns the lower section and drops in front of the base in a sheet. (Surfboarders value this kind of breaker. In "shooting the pipeline" the surfer maneuvers his board to run under the sheet of falling water.) Third, a collapsing wave breaks into foam and turbulence forward of the crest. Fourth, in surging, the wave quietly ascends the sloping shore with little turbulence.

I have examined beaches and breaking waves in two places: along the ocean near Charleston and Kiawah Island in South Carolina and along Lake Erie near Cleveland. In South Carolina I had the aid of Mary Golrick. Along the shore we found that many waves showed several of the breaking characteristics simultaneously. For example, part of the wave could be spilling while another stretch was plunging, or a spilling wave could begin to collapse or surge. Even at a particular spot the appearance changed from wave to wave, presumably because of variations in the height of the waves and the speed of the wind. At some places the shape of the waves was more consistent, probably because the configuration of the bottom determined what type of breaker would predominate.

Often a wave rolling up the beach collapsed or plunged, creating a nearly vertical wall of highly turbulent water. For a time such a wave would move up the beach fast and noisily, but then it would abruptly lose its energy and become slower. These waves are examples of the solitary waves called bores or hydraulic jumps, which I described in this department for April, 1981.

What happens is that as a surge of water approaches the beach it begins to move faster than a wave is physically able to travel over the surface of water. This motion is called supercritical flow. It is relieved by a shock transition to subcritical flow brought about by an elevation of the water level, which enables the wave to move faster. The transition develops because the supercritical flow is unstable to the many perturbations introduced by irregularities in the sand bottom. It is the shock transition that comes loudly up the beach as a vertical wall. The energy of the wall is dissipated against the rise of the beach.

Even the relatively docile surging waves Golrick and I observed turned into bores as they advanced on the beach. The vertical wall, however, was often no more than a centimeter in height. Just ahead of it one sees a shallower layer of water the vertical wall seems to be pushing along. In the shallower layer we could see small waves separated by half a centimeter or less, that is, they had a wavelength of the same dimensions. The small waves appeared to be stationary with respect to the vertical wall. Although they could have been formed by the wind or irregularities in the beach, they are part of the instability associated with supercritical flow and so are part of the bore.

A wave that had reached its upper limit on the beach seemed to recede faster than it had risen. We were not able to, time the movement accurately enough to verify this impression, but the action of the running water around our feet suggested that the impression is correct. A receding wave removed more sand around my feet than an incoming one.

Sometimes the receding water displays the sudden change in depth that characterizes a bore. A layer of water running down a sloping beach is of course accelerated by gravity. The flow can become supercritical and thus unstable to perturbations introduced by the sand bottom. Waves generated by the perturbations create the sudden increase in water depth that marks the shock transition to subcritical flow.

It was interesting to see an outgoing flow meet an incoming wave. Even with a strong incoming bore the result was often a standoff as the backwash dissipated the bore's energy. The bore was reduced to a quiet surge unless it was overtaken from the rear by a new bore.

An intriguing feature of the water in a backwash is the appearance of capillary waves. Such waves are governed by surface tension rather than by gravity. Golrick and I had trouble tracking them because they move fast and also because they are hard to see unless the light is reflected from them at a certain angle. My impression is that capillary waves seldom have more than three or four crests. The waves move faster than the line of water retreating from the shore does, and they often move in a different direction. I do not know what causes capillary waves, but I would guess they result from a gust of wind or from an irregularity in the sand.

Except in shallow areas along a beach a given parcel of water associated with a wave has no net motion. In deep water such a parcel moves in a vertical circle as a wave passes over it. Closer to the beach the movement is elliptical with the short axis vertical. Along the beach however, a parcel in an advancing or retreating sheet of water hops. The hopping movement picks up grains of sand that are then carried along with the moving sheet. Watching a mild backwash, I could follow the grains as they were picked up, carried and dropped.

A backwash running over a small object buried in the sand gouges a short trough seaward of the object. The trough behind a shell I buried in sand was about two centimeters long. Apparently the outgoing water grew turbulent enough seaward of the shell to lift grains of sand and carry them for that distance.

Waves can leave a variety of patterns in sand. At low tide Golrick and I found irregularly spaced marks that were concave toward the ocean. The marks delineated the upper reaches of waves that had come in at high tide. Each runup leaves a high-reach line on the beach because it has carried sand to that point before receding. While the tide is coming in most of these lines are erased by backwash, but while the tide is going out a series of unerased swash marks remains on the beach.

Seaward of a swash mark on some beaches one is likely to find smaller diamond-shaped markings left by the backwash. Willard Bascom discusses them in his book Waves and Beaches: The Dynamics of the Ocean Surface, a valuable source of information for any student of beach physics. He says the marks are made by the backwash on a beach with a fairly steep slope. On a beach with a gentle slope the backwash does not move fast enough to dig these tiny valleys, but even there you can find small, irregular lines that point more or less toward the ocean.

Many small domes can be found on a sand beach. They are made by the water sinking into the sand. Before the water arrives the spaces between grains of sand are mostly filled with air. The sinking water pushes air upward through a small opening. You can usually spot such an opening by the bubbles the escaping air makes in the adjacent water.; A dome forms when successively deeper layers of sand are affected by the sinking water. One incoming wave wets a layer of sand near the surface. If more water enters from later waves, the air in the deeper reaches of sand is trapped by the wet top layer of sand. Water moving into the spaces between grains of deeper sand increases the pressure of the trapped air, which pushes the surface of the sand up into a dome. On both the damp area of beach left by a receding tide and the area still being washed by waves one finds systems of sand waves with wavelengths of sever al centimeters. They are complex structures because they are created by moving water and then they alter the movement of the water. Bascom describes two types. In one type the moving water causes the crests of the sand waves to oscillate, with individual grains of sand being transported back and forth in the trough of the wave. In the second type the crests are almost stationary. As water sweeps over them, swirls are set up in the troughs. You can see the deposition of sand grains in a swirl as the re ceding water becomes thinner. Water left in a trough usually drains along it, since the sand waves are seldom exactly perpendicular to the slope of the shore. A breeze will send scores of capillary waves along a trough of water. The troughs make even shallow sand waves easy to see because their crests stand above the flowing water. The shape of sand waves depends on how fast the surf is moving. Shapes range from fine ripples to large dunes. The dunes modify the flow of water. When the waves of water are out of phase with the dunes, the water flowing over a dune breaks into a vortex that digs into the valley between the dune and the next dune. Because of the flow the dunes are irregularly shaped. When waves of moving water are in phase with dunes, the shape of the dunes is smoother and rounder.

On a larger scale a typical beach usually has several variations in height superposed on the general slope toward the water line. The structure is best seen at low tide. At the top of the beach are dune ridges that define the upper boundary of the beach. Below them is a smaller elevation of sand called a berm. It is the line the water reaches at high tide. Another ridge may lie between the berm and the ocean at low tide. Between the ridge and the berm may be a small pool of water, which is called a runnel. A terrace of sand may run from the ridge to the low-tide line. The creation and destruction of these features are major concerns for engineers studying the erosion of beaches. A severe storm is likely to change the features drastically;- Even in the absence of storms the features are slowly modified by the tides.

The beaches along Lake Erie do not show the effects of tide and appear to be affected more by storms than by a gradual movement of sand up onto the shore. On one shore near Cleveland the lake floor drops off sharply only about a meter from the line representing the upper reach of incoming waves. The ridge of sand left by those waves is narrower but more prominent than the wave-built ridge on Kiawah Island. I found no runnels along the lake, although the highwater debris inshore of the ridge shows that at times the waves reach higher than the ridge.

Many shorelines exhibit cusps, which constitute one of the strangest and prettiest sights to be seen on a beach. A cusp is a thin sheet of water forming a horizontal curve as it washes up the slope of the beach. Between each cusp and the next one is a section of beach forming a curve in the opposite direction. The diameter of a cusp can range from less than a meter along the shore of a lake to 100,000 meters or more for major shoreline features along an ocean. The large-scale features are often called giant cusps or shoreline rhythms. Sometimes they are so irregularly spaced that they go unnoticed. Elsewhere the periodicity of the cusps is so precise that the shoreline looks artificial.

According to Bascom, the cusp pattern is maintained by the interaction of two waves that impinge consecutively on the shore. The first one enters an existing cusp carrying a load of suspended sand. Then the wave divides so that the sand is carried to the inner border of the cusp. During the backwash the water flows to the center of the cusp, where there is a troughlike configuration that carries it back to the sea.

At the mouth of the trough the incoming second wave is arrested by the backwash, whereas at the horns of the cusp the new wave is essentially undeterred. It is therefore able to bring in more sand and to deposit it along the inner border of the cusp. The process continues with succeeding waves.

The source of the periodicity of cusp formations along a shore was unknown until recently. Now a number of workers,

including Anthony J. Bowen of Dalhousie University and Robert T. Guza and Douglas L. Inman of the Scripps Institution of Oceanography, have ascribed the periodicity to edge waves, which are created along a shoreline by other waves coming in from deep water.

An edge wave varies sinusoidally in height on an axis parallel to the shore, as is shown in the upper illustration on the right. Points of maximum and minimum height are antinodes. between them are nodes, where the water level does not vary. (The illustration shows only the variations in height due to the edge waves. On a sea beach the variations caused by the incident deep-water waves are superposed on the edge waves to yield the observed differences in water height.)

Seen in cross section along an axis perpendicular to the shore an edge wave has several modes representing variations in its amplitude. They are shown in the lower illustration below. In the simplest mode the amplitude of the edge wave drops off continuously toward the deep water; this is the zero mode. I shall focus on it because it seems to have the most to do with the features that are due to edge waves.

Edge waves are sometimes difficult to see on a natural beach because the waves coming in from deep water vary so much in shape, strength and direction and also because of irregularities along the bottom. To simplify matters Guza and Inman experimented with edge waves formed in a wave basin at Scripps. Their beach was a sloping section of concrete. The deep-water waves were generated by a plunger.

At the nodes of the edge waves the runup of water onto the shore was the same for each wave coming in from deeper water. At a given antinode the runup varied periodically, being at a maximum when the incoming wave was in phase with the high-water level of the edge wave and at a minimum when the two were out of phase.

Since the period of the edge wave's variations in height was twice the period of the waves from deep water, the observers saw an alternation between constructive and destructive interference. If one deep-water wave arriving at a particular antinode of the edge wave interfered constructively with the edge wave, yielding a large runup, the next interfered destructively and the runup was small. Since this periodic interference took place at every antinode along the shore, the borderline between wet and dry concrete on the shoreline was formed into a pattern of cusps centered on the antinodes. The horns of the cusps marked the nodes.

To simulate an erodable beach Guza and Inman covered the concrete with a thin layer of fine sand, adding more as the edge waves developed. At the antinodes of the edge waves the sand was eroded by the swash of water. Each maximum runup lifted and suspended the sand and then redeposited it in the area of the nodes. Soon the characteristic cusp formation appeared in the sand. With time the horns of the cusps extended toward the deeper water and grew higher.

As more sand was added a cusp was likely to show a buildup of sand down the center, where the flow of water was less vigorous. Each such cusp soon split into two cusps, thereby doubling the periodicity of cusps along the shore. This arrangement of a small periodicity laid down on a larger dominant pattern can be seen along some natural beaches.

As even more sand was added the pattern of cusps began to disintegrate. Apparently the sand formations themselves began to interfere with the waves coming in from deep water and with the ability of those waves to generate edge waves. Probably what happens on a real beach is that the incoming waves pump energy into edge waves until the edge waves have redistributed a certain amount of sand. Then the transfer of energy ceases and the patterns in the sand disappear.

Bibliography

EDGE WAVES AND BEACH CUSPS. Robert T. Guza and Douglas L. Inman in Journal of Geophysical Research, Vol. 80, No. 21, pages 2997-3012; July 20 1975.

BREAKING WAVES. E. D. Cokelet in Nature, Vol. 267, No. 5614, pages 769-774; June 30,1977.

EDGE WAVES AND SURF BEAT. Anthony J. Bowen and Robert T. Guza in Journal of Geophysical Research, Vol. 83, No. C4, pages 1913-1920; April 20, 1978.

WAVES AND BEACHES: THE DYNAMICS OF THE OCEAN SURFACE. Willard Bascom. Doubleday Anchor Books, 1980.

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