"The waves you ask about are just one example of so-called bedforms that occur when a fluid flows over loose sedimentary material. They occur all over the world under many different conditions and produce a magnificent variety of shapes and patterns. Some of them remain stationary, such as the diamond-shaped patterns sometimes seen on a dry beach. Others, such as sand dunes in the desert, move in the direction of the prevailing wind or current.
"There are several possible mechanisms leading to the particular bedforms, and this is an area of legitimate scientific controversy. The most accepted explanation is that the flow of the overlying fluid (water or air) interacts with the moving sediment grains in a manner that results in a stable shape, or bedform. The specific shape therefore depends on the density and viscosity of the fluid, its speed above the sand and the nature of the sediment (that is, its characteristic size, shape and density).
"Some of the interesting early observations and explanations of bedform patterns are provided by Ralph A. Bagnold in his book The Physics of Blown Sand and Desert Dunes (Chapman and Hall, 1984 [reprint of 1941 edition])."
Robert S. Anderson, associate professor of earth sciences at the University of California at Santa Cruz, provides a more thorough overview of this deceptively complicated phenomenon:
"Ripples in sand, found on both beaches and dunes, are one of nature's most ubiquitous and spectacular examples of self-organization. They do not result from some predetermined pattern in the wind that is somehow impressed on the surface, but rather from the dynamics of individual grains in motion across the surface. They arise whenever wind blows strongly enough over a sand surface to entrain grains into the wind. The subsequent hopping and leaping of these grains is called saltation. Saltating grains travel elongated, asymmetric trajectories: Rising relatively steeply off the bed, their path is then stretched downwind as they are accelerated by drag forces. They impact the sand surface centimeters to tens of centimeters downwind, typically at a low angle, around 10 degrees. It is this beam of wind-accelerated grains impacting the sand surface at a low angle that is responsible for ripples.
"An artificially flattened sand surface will not remain flat for long. (Try it on the beach or on the upwind side of a dune and see for yourself.) Small irregular mottles in the sand surface, perhaps a couple centimeters in wavelength, rapidly arise and grow once the wind starts to blow hard enough to initiate saltation. They then slowly organize themselves into more regular waves whose low crests are aligned perpendicular to the wind direction and begin to march slowly downwind. Typical ripple spacing is about 10 centimeters, whereas the typical height of the crests above the troughs is a few millimeters. The pattern is never perfect, but instead the ripple crests occasionally split or terminate, generating a pattern that looks remarkably like one's fingerprint. In cross section, the ripples are asymmetric, having low-angle upwind (stoss) faces and steeper downwind (lee) faces. Interestingly, the larger grains tend to accumulate on the crests of the ripples, leaving the troughs enriched in smaller grains.
"How does a low-angle beam of impacting saltating grains result in such downwind-marching waves of sand? This topographic pattern arises from a spatial pattern of deposition and erosion on the sand surface. In order to march downwind, deposition must be occurring on the downwind faces of the ripples, and erosion must be occurring on the upwind faces. Erosion occurs where there are more grains leaving the surface than are arriving, and deposition occurs where more grains are arriving than are leaving. Once saltation is well under way, by far the majority of grains in motion are actually blasted off the sand surface by the impacts of other saltating grains, rather than being carried along directly by fluid forces of the wind. And whereas the grains of sand that hit your ankles or accumulate in your pants cuffs travel long trajectories, most of the grains in motion are traveling in very short hops, only a few millimeters in length. The few grains traveling in long trajectories impact the sand surface with sufficient energy to force a large number of sand grains to hop.
"So, what sets the pattern of deposition and erosion? Because of their low-impact angle, the intensity of the bombardment by energetic, long trajectories is highest on the upwind sides of ripples. This results in many more grains being launched off from the stoss surface than are landing there, causing erosion. Conversely, the lee of the ripple is inclined so steeply that it does not get bombarded; the crest of the ripple in essence shields the lee face from bombardment. The lee face therefore becomes a zone of net accumulation; the many grains blasted from the upwind face and crest of the ripple land here, while few grains are blasted from the surface. This process explains why even small initial bumps in the sand surface will grow and why ripples move downwind.
"And what controls the wavelength? Indirectly, the wind speed, in the following way. The wind speed determines the impact angle of the longer trajectories: the greater the wind speed, the more the saltation trajectories are stretched out, and the lower the resulting impact angles. A lower-impact angle results in a longer bombardment shadow behind each ripple crest and hence a longer wavelength.
"This is a long-winded answer, perhaps, but one I that hope helps others to understand these wonderful patterns in nature."
Walt Hoagman, a Sea Grant educator at the Michigan State University Extension in northeast Michigan on the shores of Lake Huron, fills in some concrete details:
"Regular, wavelike ridges on a beach are called sand ripples or ripple marks. A ripple is simply a small wave, having a period of three seconds or less. Sand ripples, however, do not have easily discernible periods (they do have periods, but they are on the order of days). Wind velocities of three to 15 kilometers per hour will produce water waves having periods up to three seconds. The maximum size of a fully developed water wave in a 15 kilometer-per-hour wind is 0.55 meter, and the average size is around .30 meter (one foot). Most water ripples are much smaller, however, coming from the more common winds that blow at between three and 11 kilometers per hour.
"Water waves are fairly easy to understand. They arise when pressure from the wind starts a pressure hump, which passes on its energy to adjacent water molecules in the direction of the wind. As the wind speed increases, the waves become regular and 'march' along the surface at predictable speeds and with predictable heights. Water waves away from shore become unstable when the wind speed exceeds 1.3 times the waves' speed. At that point, they begin to get steep and begin to break forward, producing whitecaps. The upwind slope then becomes shallower (flatter) than the downwind slope. The higher the wind in relation to the wave speed, the steeper the waves get, especially on the downwind side.
"Wind over sand behaves somewhat differently, but some of the same basic forces are at work to create regular spaced ripples. Because sand does not transmit energy to adjacent sand particles when wind blows over it, the wind must rise over a stationary hump. In so doing, it is compressed vertically, so it must speed up. The top of the hump (sand wave) and the windward slope get the greatest velocity. The valley between (where the wind can expand again) has the slowest wind; a rolling vortex, which acts as a vacuum to lift sand from the valley, arises there. A wind, or the wash along a shore, must be prolonged and continuous from one direction in order for sand ripples to form. In the Great Lakes region, sand ripples are best seen or felt underwater. They are fairly rare on the dry beach around here, probably because the local winds are very erratic and the sand is less sticky than ocean sand.
"The frequency of any wave formation is determined by the overall strength of the wind or, in the case of sand ripples underwater, by the strength of the waves. With dry sand, a strong wind begins to wipe out ripples and make larger and regular berms, spaced fairly evenly perpendicular to the wind. These are in reality large ripples. Continued, strong winds can change these rises into undulating dunes that often go far inland.