Theory and principles of waves, how they work and what causes them

by Dr J Floor Anthoni 2000 | www.seafriends.org.nz/oceano/waves.htm

When the wind blows across the water, it changes the water’s surface, first into ripples and then into waves. Once the surface becomes uneven, the wind has an ever increasing grip on it. Storms can make enormous waves, particularly if the wind, blows in the same direction for any length of time. In this chapter, you can learn what waves are and how they behave. Learn to understand the principles behind all surface waves.

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 3×3=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 harvest 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 are also unpredictable, like winds for harvesting energy.

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 (due to ‘spreading’). 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.

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.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).

How wind causes water to form waves is easy to understand although many intricate details still lack a satisfactory theory. On a perfectly calm sea, the wind has practically no grip. As it slides over the water surface film, it makes it move. As the water moves, it forms eddies and small ripples. Ironically, these ripples do not travel exactly in the direction of the wind but as two sets of parallel ripples, at angles 70-80º to the wind direction. The ripples make the water’s surface rough, giving the wind a better grip. The ripples, starting at a minimum wave speed of 0.23 m/s, grow to wavelets and start to travel in the direction of the wind. At wind speeds of 4-6 knots (7-11 km/hr), these double wave fronts travel at about 30º from the wind. The surface still looks glassy overall but as the wind speed increases, the wavelets become high enough to interact with the air flow and the surface starts to look rough. The wind becomes turbulent just above the surface and starts transferring energy to the waves. Strong winds are more turbulent and make waves more easily.

The rougher the water becomes, the easier it is for the wind to transfer its energy. The waves become steep and choppy. Further away from the shore, the water’s surface is not only stirred by the wind but also by waves arriving with the wind. These waves influence the motion of the water particles such that opposing movements gradually cancel out, whereas synchronising movements are enhanced. The waves start to become more rounded and harmonious. Depending on duration and distance (fetch), the waves develop into a fully developed sea.

Anyone familiar with the sea, knows that waves never assume a uniform, harmonious shape. Even when the wind has blown strictly from one direction only, the resulting water movement is made up of various waves, each with a different speed and height. Although some waves are small, most waves have a certain height and sometimes a wave occurs which is much higher.

When trying to be more precise about waves, difficulties arise: how do we measure waves objectively? When is a wave a wave and should be counted? Scientists do this by introducing a value E which is derived from the energy component of the compound wave. In the left part of the drawing is shown how the value E is derived entirely mathematically from the shape of the wave. Instruments can also measure it precisely and objectively. The wave height is now proportional to the square root of E.

The sea state E is two times the average of the sum of the squared amplitudes of all wave samples.

The right part of the diagram illustrates the probability of waves exceeding a certain height. The vertical axis gives height relative to the square root of the average energy state of the sea: h / SQR( E ) . For understanding the graph, one can take the average wave height at 50% probability as reference.Fifty percent of all waves exceed the average wave height, and an equal number are smaller. The highest one-tenth of all waves are twice as high as the average wave height (and four times more powerful). Towards the left, the probability curve keeps rising off the scale: one in 5000 waves is three times higher and so on. The significant wave height H3 is twice the most probable height and occurs about 15% or once in seven waves, hence the saying “Every seventh wave is highest”. Click here for a larger version of this diagram.

When the wind blows sufficiently long from the same direction, the waves it creates, reach maximum size, speed and period beyond a certain distance (fetch) from the shore. This is called a fully developed sea. Because the waves travel at speeds close to that of the wind, the wind is no longer able to transfer energy to them and the sea state has reached its maximum. In the picture the wave spectra of three different fully developed seas are shown. The bell curve for a 20 knot wind (green) is flat and low and has many high frequency components (wave periods 1-10 seconds). As the wind speed increases, the wave spectrum grows rapidly while also expanding to the low frequencies (to the right). Note how the bell curve rapidly cuts off for long wave periods, to the right. Compare the size of the red bell, produced by 40 knot winds, with that of the green bell, produced by winds of half that speed. The energy in the red bell is 16 times larger!
Important to remember is that the energy of the sea (maximum sea condition) increases very rapidly with wind speed, proportional to its fourth power. The amplitude of the waves increases to the third power of wind speed. This property makes storms so unexpectedly destructive.

The biggest waves on the planet are found where strong winds consistently blow in a constant direction. Such a place is found south of the Indian Ocean, at latitudes of -40º to -60º, as shown by the yellow and red colours on this satellite map. Waves here average 7m, with the occasional waves twice that height! Directly south of New Zealand, wave heights exceeding 5m are also normal. The lowest waves occur where wind speeds are lowest, around the equator, particularly where the wind’s fetch is limited by islands, indicated by the pink colour on this map. However, in these places, the sea water warms up, causing the birth of tropical cyclones, typhoons or hurricanes, which may send large waves in all directions, particularly in the direction they are travelling.

As waves enter shallow water, they slow down, grow taller and change shape. At a depth of half its wave length, the rounded waves start to rise and their crests become shorter while their troughs lengthen. Although their period (frequency) stays the same, the waves slow down and their overall wave length shortens. The ‘bumps’ gradually steepen and finally break in the surf when depth becomes less than 1.3 times their height. Note that waves change shape in depths depending on their wave length, but break in shallows relating to their height!How high a wave will rise, depends on its wave length (period) and the beach slope. It has been observed that a swell of 6-7m height in open sea, with a period of 21 seconds, rose to 16m height off Manihiki Atoll, Cooks Islands, on 2 June, 1967. Such swell could have arisen from a 60 knot storm.

The photo shows waves entering shallow water at Piha, New Zealand. Notice how the wave crests rise from an almost invisible swell in the far distance. As they enter shallow water, they also change shape and are no longer sinus-like. Although their period remain the same, their distance between crests and their speed, diminish.Not quite visible on this scale are the many surfies in the water near the centre of the picture. They favour this spot because as the waves bend around the rocks, and gradually break in a ‘peeling’ motion, they can ride them almost all the way back to the beach.

Going back to the ‘wave motion and depth‘ diagram showing how water particles move, we can see that all particles make a circular movement in the same direction. They move up on the wave’s leading edge, forward on its crest, down on its trailing slope and backward on its trough. In shallow water, the particles close to the bottom will be restricted in their up and downward movements and move along the bottom instead. As the diagram shows, the particle’s amplitude of movement does not decrease with depth. The forward/backward movement over the sand creates ripples and disturbs it.

Since shallow long waves have short crests and long troughs, the sand’s forward movement is much more brisk than its backward movement, resulting in sand being dragged towards the shore. This is important for sandy beaches.

Note that a sandy bottom is just another medium, potentially capable of guiding gravity waves. It is about 1.8 times denser than water and contains about 30-40% liquid. Yet, neither does it behave like a liquid, nor entirely like a solid. It resists downward and sideways movements but upward movements not as much. So waves cannot propagate over the sand’s surface, like they do along the water’s surface, but divers can observe the sand ‘jumping up’ on the leading edge of a wave crest passing overhead (when the water particles move upward). This may help explain why sand is so easily stirred up by waves and why burrowing organisms are washed up so readily.

Surf breakers are classified in three types:

Spilling breakers

result from waves of low steepness (long period swell) over gentle slopes. They cause rows of breakers, rolling towards the beach. Such breakers gradually transport water towards the beach during groups of high waves. Rips running back to sea, transport this water away from the beach during groups of low waves. When caught swimming in a rip, do not attempt to swim back to shore because such rips can be very strong (up to 8 km/hr). Swim parallel to the beach towards where the waves are highest. This is where water moves towards the beach. The next group of tall waves should assist you to swim back to shore. However, when launching (rescue) boats, this is best done in a rip zone.

Plunging breakers

result from steeper waves over moderate slopes. The slope of a beach is not constant but may change with the tide. Some beaches are steep toward high tide, others toward low tide. A plunging breaker is dangerous for swimmers because its intensity is greatly augmented by backwash from its predecessor. This strong backwash precludes easy exit from the breaker zone, particularly for divers. Often a steep bank of loose sand prevents one from standing upright. In order to exit safely, wait for a group of low waves.

Surging breakers

occur where the beach slope exceeds wave steepness. The wave does not really curl and break but runs up against the shore while producing foam and large surges of water. Such places are dangerous for swimmers because the rapidly moving water can drag swimmers over the rocks.

Spilling breakers are a familiar sight on most beaches. They arise from long waves breaking on gently sloping beaches. There are several rows of breakers.

Plunging breakers can occur on steeply sloping beaches. There is only one row of breakers.

Surging breakers surge over steeply sloping (but not vertical) beaches or rocks. Waves break one at a time.
Photos Van Dorn, 1974

Wave groups
Part of the irregularity of waves can be explained by treating them as formed by interference between two or more wave trains of different periods, moving in the same direction. It explains why waves often occur in groups. The diagram shows how two wave trains (dots and thin line) interfere, producing a wave group of larger amplitude (thick line). Such a wave group moves at half the average speed of its component waves. The wave’s energy spectrum, discussed earlier, does not move at the speed of the waves but at the group speed. When distant storms send long waves out over great distances, they arrive at a time that corresponds with the group speed, not the wave speed. Thus a group of waves, with a period of 14s would travel at a group velocity of 11m/s (not 22 m/s) and take about 24 hours (not 12 hr) to reach the shore from a cyclone 1000 km distant. A group of waves with half the period (7s) would take twice as long, and would arrive a day later. (Harris, 1985)Most wave systems at sea are comprised of not just two, but many component wave trains, having generally different amplitudes as well as periods. This does not alter the group concept, but has the effect of making the groups (and the waves within them) more irregular.

Anyone having observed waves arriving at a beach will have noticed that they are loosely grouped in periods of high waves, alternated by periods of low waves.

Adapted from Van Dorn, 1974.

Wave reflection
Like sound waves, surface waves can be bent (refracted) or bounced back (reflected) by solid objects. Waves do not propagate in a strict line but tend to spread outward while becoming smaller. Where a wave front is large, such spreading cancels out and the parallel wave fronts are seen travelling in the same direction. Where a lee shore exists, such as inside a harbour or behind an island, waves can be seen to bend towards where no waves are. In the lee of islands, waves can create an area where they interfere, causing steep and hazardous seas.

When approaching a gently sloping shore, waves are slowed down and bent towards the shore.

When approaching a steep rocky shore, waves are bounced back, creating a ‘confused sea’ of interfering waves with twice the height and steepness. Such places may become hazardous to shipping in otherwise acceptable sea conditions.

When wave fronts approach a gently sloping beach on an angle, they slow down in the shallows, causing them to bend towards the beach. If the beach slopes gently enough, all breakers will eventually line up parallel to the beach.

When a beach is steep, the wave fronts get bent and then reflected back. Sometimes part of the energy is absorbed and the remaining energy reflected.

This drawing shows how waves are bent around an island which should be at least 2-3 wave lengths wide in order to offer some shelter. It causes immediately in the lee of the island (A) a wave shadow zone but further out to sea a confusing sea (B) of interfering but weakened waves which at some point (C) focuses the almost full wave energy from two directions, resulting in unpredictable and dangerous seas. When seeking shelter, avoid navigating through this area.Recent research has shown that underwater sand banks can act as wave lenses, refracting the waves and focussing them some distance farther. It may suddenly accelerate coastal erosion in localised places along the coast.

Drawings from Van Dorn, 1974.