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Wave motion
Anyone having watched water waves rippling outward from the point where a stone was thrown in, should have noticed how effortlessly waves can propagate along the water's surface. Wherever we see water, we see its surface stirred by waves. Indeed, witnessing a lake or sea flat like a mirror, is rather unusual. Yet, as familiar we are with waves, we are unfamiliar with how water particles can join forces to make such waves.
Waves are oscillations in the water's surface. For oscillations to exist and to propagate, like the vibrating of a guitar string or the standing waves in a flute, there must be a returning force that brings equilibrium. The tension in a string and the pressure of the air are such forces. Without these, neither the string nor the flute could produce tones. The standing waves in musical instruments bounce their energy back and forth inside the string or the flute's cavity. The oscillations that are passed to the air are different in that they travel in widening spheres outward. These travelling waves have a direction and speed in addition to their tone or timbre. In air their returning force is the compression of the air molecules. In surface waves, the returning force is gravity, the pull of the Earth. Hence the name 'gravity waves' for water waves.
In solids, the molecules are tightly connected together, which prevents them from moving freely, but they can vibrate. Water is a liquid and its molecules are allowed to move freely although they are placed closely together. In gases, the molecules are surrounded by vast expanses of vacuum space, which allows them to move freely and at high speed. In all these media, waves are propagated by compression of the medium. However, the surface waves between two media (water and air), behave very different and solely under the influence of gravity, which is much weaker than that of elastic compression, the method by which sound propagates.
The specific volume of sea water changes by only about 4 thousands of 1 percent (4E-5) under a pressure change of one atmosphere (1 kg/cm2). This may seem insignificant, but the Pacific Ocean would stand about 50m higher, except for compression of the water by virtue of its own weight, or about 22cm higher in the absence of the atmosphere. Since an atmosphere is about equal to a column of water 10m high, the force of gravity is about 43 times weaker than that of elastic compression.
Surface tension (which forms droplets) exerts a stress parallel to the surface, equivalent to only one 74 millionth (1.4E-8) of an atmosphere. Its restoring force depends on the curvature of the surface and is still smaller. Nevertheless it dominates the behaviour of small ripples (capillary waves), whose presence greatly contributes to the roughness (aerodynamic drag) of the sea surface, and hence, to the efficiency with which can generate larger waves and currents. (Van Dorn, 1974)
If each water particle makes small oscillations around its spot, relative to its neighbours, waves can form if all water particles move at the same time and in directions that add up to the wave's shape and direction. Because water has a vast number of molecules, the height of waves is theoretically unlimited. In practice, surface waves can be sustained as high as 70% of the water's depth or some 3000m in a 4000m deep sea (Van Dorn, 1974).
Note that the water particles do not travel but only their collective energy does! Waves that travel far and fast, undulate slowly, requiring the water particles to make slow oscillations, which reduces friction and loss of energy.
In the diagram some familiar terms are shown. A floating object is observed to move in perfect circles when waves oscillate harmoniously sinus-like in deep water. If that object hovered in the water, like a water particle, it would be moving along diminishing circles, when placed deeper in the water. At a certain depth, the object would stand still. This is the wave's base, precisely half the wave's length. Thus long waves (ocean swell) extend much deeper down than short waves (chop). Waves with 100 metres between crests are common and could just stir the bottom down to a depth of 50m. Note that the depth of a wave has little to do with its height! But a wave's height contains the wave's energy, which is unrelated to the wave's length. Long surface waves travel faster and further than short ones. Note also that the forward movement of the water under a crest in shallow water is faster than the backward movement under its trough. By this difference, sand is swept forward towards the beach.
Water waves can store or dissipate much energy. Like other waves (alternating electric currents, e.g.), a wave's energy is proportional to the square of its height (potential). Thus a 3m high wave has 3x3=9 times more energy than a 1m high wave. When fine-weather waves of about 1m height pound on the beach, they dissipate an average of 10kW (ten one-bar heaters) per metre of beach or the power of a small car at full throttle, every five metres. (Ref Douglas L Inman in Oceanography, the last frontier, 1974). Attempts to harness the energy from waves have failed because they require large structures over large areas and these structures should be capable of surviving storm conditions with energies hundreds of times larger than they were designed to capture.
Waves have a direction and speed. Sound waves propagate by compressing the medium. They can travel in water about 4.5 times faster than in air, about 1500m per second (5400km/s, or mach-4.5, depending on temperature and salinity). Such waves can travel in all directions and reach the bottom of the ocean (about 4km) in less than a second. Surface waves, however, are limited by the density of water and the pull of gravity. They can travel only along the surface and their wave lengths can at most be about twice the average depth of the ocean (2 x 4 km). The fastest surface waves observed, are those caused by tsunamis. The 'tidal wave' caused by an under-sea earthquake in Chile in May 1960, covered the 6000 nautical miles (11,000km) to New Zealand in about 12 hours, travelling at a speed of about 900 km/hr! When it arrived, it caused an oscillation in water level of 0.6m at various places along the coast, 1.4m in Tauranga Harbour and 2.4m in Whitianga harbour. Note that tsunamis reach their minimum at about 6000 km distance. Beyond that, the curvature of the Earth bends the wave fronts to focus them again at a distance of about 12,000 km, where they can still cause considerable damage.
The relationship between wave speed (phase velocity) and depth of long surface waves in shallow water is given by the formula
c x c = g x d x (p2 - p1) / p2 or
c x c= g x d for water/air
where c= wave speed, g= acceleration of gravity (9.8066 m/s/s), d= wave depth (or upper layer depth, m), p2= density of water (=1) and p1= density of air (= 0.00125).
The formula states that wave speed increases with wave depth and the relative difference in density.
For an ocean depth of 4000m, a wave's celerity or speed would be about SQR(10 x 4000) = 200 m/s = 720 km/hr. Surface waves could theoretically travel much faster on larger planets, in media denser than water.
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For deep water, the relationship between speed and wavelength is given by the formula:
l = g x t x t / (2 x pi)
l = t x c for all kinds of waves, substitute in above equation: t x c = g x t x t / (2 x pi)
c = g x t / (2 x pi) or t = c x 2 x pi / g or t = c x 0.641 (s)
where t= wave period (sec), f= wave frequency, l= wave length (m) and pi=3.1415...
to calculate c and l from wave period t (in sec): c = t x 1.56 m/s= t x 5.62 km/hr = t x 3.0 knot
l = 1.56 x t x t (metres)
Thus waves with a period of 10 seconds, travel at 56 km/hr with a wave length of about 156m. A 60 knot (110 km/hr) gale can produce in 24 hours waves with periods of 17 seconds and wave lengths of 450m. Such waves travel close to the wind's speed (97 km/hr). A tsunami travelling at 200 m/s has a wave period of 128 s, and a wave length of 25,600 m.
These two diagrams show the relationships between wave speed and period for various depths (left), and wave length and period (right), for periodic, progressive surface waves. (Adapted from Van Dorn, 1974) Note that the term phase velocity is more precise than wave speed.
The period of waves is easy to measure using a stopwatch, whereas wave length and speed are not. In the left picture, the red line gives the linear relationship between wave speed and wave period. A 12 second swell in deep water travels at about 20m/s or 72 km/hr. From the red line in the right diagram, we can see that such swell has a wave length between crests of about 250m.
When the 12s swell enters 10m shallow water (follow the green curve for 10m), its speed will halve to 10m/s (left graph) and so will its wave length (right graph). But the height of the wave increases by a similar factor (not shown here).
