Ocean Waves

[aboobacker]

Waves

--------------------------------------------------------------------------

A wave is a progression of energy from one point to another

The wave does not move forward only the energy in it progresses

Wave is almost friction free


Properties of waves

Wavelength: Distance from one crest to the next
Period : Time taken for one wave to pass a fixed point
Frequency : Number of waves per second that pass a fixed point
Velocity : Speed with which the waves are moving past a fixed point


Orbital motion

The size of the orbit of the water particles increases with wavelength
The orbit size decreases rapidly with depth

Orbit size decreases to 1/23 of surface value at a depth equal to 1/2 of wavelength

Only "feel" waves to a depth of 1/2 of their wavelength

Classification of waves: according to the way they are formed or destroyed

Major formation forces:

*Wind

*Atmospheric pressure

*Landslides or other earth movements

*Gravitational attraction

=>Wind waves have the most energy in surface ocean

Restoring forces: Try to flatten out the waves

1.Surface tension for very small waves (<0.6 inches)

2.Gravity, for everything bigger

Deep and shallow water waves

Relationship between wavelength and water depth determines wave characteristics

Deep water wave: water depth >1/2 wavelength orbits die away above bottom

Shallow water wave: water depth <1/20 wavelength orbits are flattened at the bottom

Transitional wave: water depth >1/20 but <1/2 wavelength
Wave "feels" bottom

Gravity and seismic waves have very long wavelengths are always shallow water

waves regardless of ocean depth

Wave velocity

1.Deep water waves

Velocity of wave energy through water determined by wavelength

Longer waves move faster

Use period as is easier to measure than length

Speed (m/sec) = 1.56 x wave period

Typical 8 second trade wind wave moves at 12.4 m/sec=28 mph

2.Shallow water waves

Velocity of wave is related to water depth

Speed (m/sec) = 3.1 x square root (depth)

Typical 20 minute seismic wave moves at 470 mph

When a deep water wave moves into shallow water it slows down

Trade wind wave (8 second) 28 mph in deep water in 1 metre deep water
speed is 3.1 m/sec=7 mph


Wind wave formation

1.Wind attempts to "stretch" surface skin of ocean
2.Surface tension: capillary wave
3.Wind deflected upwards, adds energy to wave pushes it forward
4.Low pressure behind wave contributes to forward motion
5.Continued wind, wave period and height grow together
6.Waves are peaked in areas of formation, rounded swell away from formation regions

Wave progression

*Longest waves move away from storm fastest
*Form wave trains
*Leading waves "excite" still water ahead of wave train
*New waves forms behind wave train
*Wave train travels at half the speed of the individual waves within it

Maximum development of wind waves is the result of 3 factors:

1.Wind strength
2.Wind duration
3.Uninterrupted ength of ocean that wind blows over (Fetch)


*The stronger the wind the longer the duration and fetch needed to fully develop the sea

*Rarely get fully developed seas for strongest winds

*Highest waves found around Antarctica, constant wind, uninterrupted ocean

Wave steepness and dispersal from a storm

1.Maximum wave height in open ocean is 1/7 of wavelength, higher waves
get whitecaps

2.In region of formation seas chaotic

3.Waves sorted by wavelength and speed as move away from formation
region

4.Waves turn to swell as they move away from region of formation height to
length ratio gets smaller

5.When waves overtake each other constructive interference causes very
large waves

6.Distant observer see longest and fastest waves first



Waves approaching the shore

1 As wave train approaches shore "feels" bottom at depth = 1/2 wavelength
2.Wave energy packed into shallower depth, becomes peaked
3.Wave slows, period is constant, wavelength decreases
4.Bottom of wave slows even more as gets shallower, wave crest moves
ahead of base of wave
5.Wave breaks when wave height to water depth ~ 3:4

Types of wave breaks

*Type of wavebreak depends on bottom
*Plunging waves from steeply sloping bottoms
*Spilling wave from gentle slopes
*Abrupt slope change: water surges on to beach

Wave refraction

1.Wave approaching coast at an angle
2.End of wave entering shallow water slows down, rest of wave continues at
full speed
3.Wave bends towards shore (towards the slowest end)

Wave diffraction

*Waves passing through a small gap diffract
*New waves reform at a point in gap and radiate out
*Occurs in harbour entrances and between islands
*Radiated waves between island groups form interference patterns
*Polynesian navigators recognised interference patterns to indicate island
chains beyond the horizon

Internal waves

*There are waves below surface at regions of density gradient, e.g.
pycnocline
*Waves can be large but travel slowly as density gradient is small compared
with one at sea surface
*Sub surface Kelvin waves bring an end to El Nino

__________________________________________________ _______________

Continue.....

[venu]

kudos to aboobacker for that capsule course on waves...would benefit ppl who'd like to know more about oceans, but have no formal grounding..may be we can think of more like that.

[vinu]

Hi Aboobacker

Your contributions on Waves are interesting and usefull indeed.
It gives us a quick overview on surface gravity waves. It will be more
attractive and usefull, if it contains some simple figures
explaining the essential features of the waves.

Also please include the 'Wave number' in 'Properties of waves' category.
It is the 'number of waves' in unit distance or number of waves in (2.Pi)
distance. (k=1/L or k=2.pi/L) where 'L' is the
wave length. The later one represents the 'spatial frequency' of
waves (just like w=2.Pi/T represents the time-frequency of the wave).

It would be more usefull if the above post continues on further discussions
about waves, in a hierarchal order, (like Tide, Tsunamies, coastal
waves, planetary waves etc...).

-Thanks

-

[rocksea]

Hi Vinu & Aboobacker. Appreciate your efforts for better understanding
of the physical, numerical and other aspects of physical oceanography.
Let me give a suggestion. At the end of the notes, it would be useful
to give a brief "reference" or "further reading".

[aboobacker]

Go through the websites

http://www.seafriends.org.nz/oceano/waves.htm
http://www.stormsurf.com/page2/tutorials/wavebasics.shtml
http://www.irbs.com/bowditch/
http://www.boatsafe.com/nauticalknowhow/waves.htm

[vinu]

Phase velocity and Group velocity

In Oceanography, identifying the dimensional property of waves, such as wave length, phase speed, group speed requires careful attention. It may be easily confused while framing a concept on the physical properties of the waves and questions like, What is phase speed, What is group speed, How do they appear together, often comes in our mind. This post will be explaining the answers for the following questions with pictorial illustrations.


1. What is phase speed?
2. What is group speed?
3. How does these combined together and How do they appear in the real Ocean?
4. How do waves propagate with its phase speed and group speed.
5. How can we identify propagation of waves and its group
from the data sets?

1. What is phase speed? What is group speed?

A particular property of Ocean wave is that they are dispersive in nature. That means, waves with different wavelength will have different speeds. This dispersive nature of the wave, or if simply put, a single source of perturbance may generate different dimensional waves and they travel with different speeds, cause distinct speeds for the individual waves and a unique speed for its group. The speed of the individual wave crust with which it propagate within a group is the phase speed of the wave.

These waves travel as a group. Within a group, the waves appears to generate at one side of the group and it propagate to the other side. As it reaches the another side, it disappears, and in compensation, new waves generate in the former end. Thus within a group, the waves at the sides will have amplitudes smaller than the waves at the center. Thus the energy (because energy is proportional to the amplitude of the wave) of the waves will be largest in the center of the group. Thus, on an average, the energy of individual waves are concentrated on the center of its group and so the energy propagation depends not on the individual waves but on the group speed. The group of the wave itself has a propagation with less speed than its constituent waves and it may or may not be in the same direction. The speed of the individual waves are called the phase speed and the speed of the whole group is called the group speed.


3. How does these combined together and How do they appear in the real Ocean?

Mathematically it can be represented as, for any given field of the Ocean, (say, the streamfunction), psi = A(x,y,t)*cos(kx + ly - wt), where A(x,y,t) is the amplitude of the individual waves. Note that, amplitude is a function of space and time, which implies that the waves will be having a different amplitude with space and time. Such a wave can be pictured as shown in Figure.1. The change in amplitude with space is slight and it become an envelope of waves with larger amplitude at the center and smaller amplitude at the ends. Thus the size of the envelope is the size of the group and its speed is the group speed.

Here, let us take, a simple form for A(x,y,t) as follows. Note that the variables are non-dimensional. A(x,y,t) = exp(-((Kx-Cgx.t)^2 - (Ly-Cgy.t)^2)).. This simple form yields an isolated group of waves in space (x,y) whose size is determined by the wave number 'K' and 'L' (of the envelope, not the individual waves) and propagate with a speed of Cgx and Cgy in (x,y) directions, respectively. Here, to keep the symmetry, the 'Kx' and 'Ly' are so chosen that the resulting group is tapered in the center of the space-frame we consider. In this form, the envelope is a hyperbolic function in space which is advected with a speed of Cgx and Cgy. The pictorial representation of the above envelope is shown in Figure. 1(a,b).

http://wwwoa.ees.hokudai.ac.jp/people/vinu/share_pal/phasegroup/1.gif

The figure shows the existence of waves in the Ocean as a group. The cross section of the group along the center is shown separately. The examination of the figure tells that the wave amplitudes are minimum at the ends and maximum at the center. Thus for any given perturbation, let use assume a group of waves of above kind will be formed in the Ocean.

4. How do waves propagate with its phase speed and group speed.

In this excersice, let us assume a group of waves with non-zero x-wavenumber and zero y-wavenumber. These waves travels westward. This convention will be followed in the cases examined here (expect for Case-4).

Case-1: A stationary group (ideal case)

Now let us see, how this group of waves evolve in time. First assume, the group have no motion (i.e. stationary), and only the waves inside the group are propagating. The following figure shows the animation (Figure2) , how the waves propagate, keeping a stationary group.

http://wwwoa.ees.hokudai.ac.jp/people/vinu/share_pal/phasegroup/anim1.gif

The wave crest is shown as megenta and trough is show in blue. It can be seen that, the individual wave crest (or the phase of the waves) propagate westward. Each waves originate at the east and propagate to the west and disappears, and new wave forms at the east despite the group remain stationary.

Now, let us examine the a Hovmuller diagram (space-time diagram) of the above wave propagation inorder to understand, How does a propagation of wave can be pictured. The following figure shows the x-t plot of the waves across the center of the group. The crust and trough of the waves are shown as red and blue, respectively. This is a typical x-t plot for a westward propagating waves. The phase lines are seen such that the individual wave crust (trough) are originating at the eastern side and gradually propagate to the west and disappears at the west. The dark blue line shows the phase propagation and the slope of this line will give the approximate phase speed of the wave. Note that the group is group is stationary in this case.

http://wwwoa.ees.hokudai.ac.jp/people/vinu/share_pal/phasegroup/2.gif


Case-2: A westward moving group

The following animation (Figure.3) shows the same process, but with a group in motion, westward comparatively less speed than the individual waves. A close examination tells that, the individual wave propagation is similar to the above case, but the group is sightly advancing west. Please pay few time to observe the animation. Since the group speed and phase speed are westward, a carefull attention is required to infer the drifting of group westward.

http://wwwoa.ees.hokudai.ac.jp/people/vinu/share_pal/phasegroup/anim2.gif

The following figure (Figure.5) shows the corresponding Hovmuller representation of the wave. Now we can see a noticeable difference between the first case (waves with stationary group). The hovmuller digram shows the westward propagating phase. But at the each time the wave itself is tilted to the west. That is its group propagation. Thus, from the Hovmuller diagram, the lines of constant phase shows the phase speed and line of drifting phase centers shows the group speed. The green line shown connects the centeres of the maximum wave amplitude, and they drift to the west. The slope of this line gives the group speed of the waves. It is noticeable that the phase velocity is larger
than the group velcoity. (Please note the scales chosen here is arbitrary).

http://wwwoa.ees.hokudai.ac.jp/people/vinu/share_pal/phasegroup/3.gif

Case-3: A eastward moving group.

The following animation (Figure.7) is same as above, except for an eastward drifting group. The corresponding Hovmuller diagram (Figure.8) shows the individual phase speeds to the west and lines connecting maximum amplitude drift eastward. It shows the eastward group propagation.

http://wwwoa.ees.hokudai.ac.jp/people/vinu/share_pal/phasegroup/anim3.gif

http://wwwoa.ees.hokudai.ac.jp/people/vinu/share_pal/phasegroup/4.gif

Case-4: A southwestward phase speed, northwestward group speed.
http://wwwoa.ees.hokudai.ac.jp/people/vinu/share_pal/phasegroup/anim4.gif

This is rather a complicated case. The individual waves have non-zero wave number in both x and y direction (westward and southward respectively) and group propagate with a westward and northward velocities. Constricting Hovmuller diagram is rather difficult in this case.

Remarks

Thus as illistrated above, the phase velocity and group velocity can be inferred by looking at the approximate space-time plots of the data. In real Ocean, the Sea Surface Altimetry Data can be ploted in similar way to find the propagation of waves and group. However in the real Ocean, the wave poropagation is complicated in its phase and group velocities. As an example, the Case-4 shown above seems it is diffucult to construct the exact route of the waves.

The illustration here is based only on 2-Dimensional waves. In the real Ocean, in addition to the surface gravity waves, internal waves can be generated in a 3-Dimensional space. It is often observed in the Atmosphere, waves with downward phase propagation and upward group propagation. Identifying such waves are important, becuase the energy of the waves are associated with its group velocity. In the Ocean, the equatorial wave guides provides solutions with eastward propagating group and westward propagating phase. From recent observations on Indian Ocean, Sengupta et al (2004), shown such high-frequency variablity in the equatorial Indian Ocean. Jensen (2003) has shown similar high-frequency wave with eastward group and westward phase in his tracer experiments. The readers are further directed to the Bibliography (http://www.oceanographers.net/forums/forumdisplay.php?f=15) section of our website to find references.


PS: If any of the above figures are missing/distorted/too large to open,
please feed back. The animations may take a few time to download completely.


-Thanks

-

[aboobacker]

Wave Measurement

Waves - disturbances of water - are a constant presence in the world's oceans. Because waves travel all across the globe, transmitting vast amounts of energy, understanding their motions and characteristics is essential. The forces generated by waves are the main factor impacting the geometry of beaches, the transport of sand and other sediments in the nearshore region, and the stresses and strains on coastal structures. When waves are large, they can also pose a significant threat to commercial shipping, recreational boaters, and the beachgoing public. Thus for ensuring sound coastal planning and public safety, wave measurement and analysis is of great importance.

The discussion below is largely based on Part II, Chapter 1 of the Coastal Engineering Manual (CEM), published by the United States Army Corps of Engineers' Coastal and Hydraulics Laboratory. For more details, we recommend referring directly to the CEM.

Wave Generation

Waves are generated by forces that disturb a body of water. They can result from a wide range of forces - the gravitational pull of the sun and the moon, underwater earthquakes and landslides, the movements of boats and swimmers. The vast majority of ocean waves, however, are generated by wind.

Out in the ocean, as the wind blows across a smooth water surface, air molecules push against the water. This friction between the air and water pushes up tiny ridges or ripples on the ocean surface. As the wind continues to blow, these ripples increase in size, eventually growing into waves that may reach many meters in height.

Three factors determine how large wind-generated waves can become. The first factor is wind speed, and the second factor is wind duration, or the the length of time the wind blows. The final factor is the fetch, the distance over which the wind blows without a change in direction. The faster the wind, the longer it blows, and the larger the fetch, the bigger the waves that will result. But the growth of wind-generated ocean waves is not indefinite. After a certain point, the energy imparted to the waters by a steady wind is dissipated by wave breaking (often in the form of whitecaps). When this occurs and the waves can no longer grow, the sea state is said to be a 'fully developed'.

When waves are being generated by strong winds in a storm, the sea surface generally looks very chaotic, with lots of short, steep waves of varying heights. In calm areas far from strong winds, ocean waves often have quite a different aspect, forming long, rolling peaks of uniform shape. For this reason, physical oceanographers differentiate between two types of surface waves: seas and swells. Seas refer to short-period waves that are still being created by winds or are very close to the area in which they were generated. Swells refer to waves that have moved out of the generating area, far from the influence of the winds that made them.

In general, seas are short-crested and irregular, and their surface appears much more disturbed than for swells. Swells, on the other hand, have smooth, well-defined crests and relatively long periods. Swell is more uniform and regular than seas because wave energy becomes more organized as it travel longs distances. Longer period waves move faster than short period waves, and reach distant sites first. In addition, wave energy is dissipated as waves travel (from friction, turbulence, etc.), and short-period wave components lose their energy more readily than long-period components. As a consequence of these processes, swells form longer, smoother, more uniform waves than seas.

Wave Dynamics

Looking out at the water, an ocean wave in deep water may appear to be a massive moving object - a wall of water traveling across the sea surface. But in fact the water is not moving along with the wave. The surface of the water - and anything floating atop it, like a boat or buoy - simply bobs up and down, moving in a circular, rise-and-fall pattern. In a wave, it is the disturbance and its associated energy that travel from place to place, not the ocean water. An ocean wave is therefore a flow of energy, travelling from its source to its eventual break-up. This break up may occur out in the middle of the ocean, or near the coast in the surfzone.

In order to understand the motion and beahvior of waves, it helps to consider simple waves: waves that can be described in simple mathematical terms. Sinusoidal or monochromatic waves are examples of simple waves, since their surface profile can be described by a single sine or cosine function. Simple waves like these are readily measured and analyzed, since all of their basic characteristics remain constant.

(Fig.)

Wave Anatomy:

Still-Water Line - The level of the sea surface if it were perfectly calm and flat.

Crest - The highest point on the wave above the still-water line.

Trough - The lowest point on the wave below the still-water line.

Wave Height - The vertical distance between crest and trough.

Wavelength - The horizontal distance between successive crests or troughs.

Wave Period - The time it takes for one complete wave to pass a particular point.

Wave Frequency - The number of waves that pass a particular point in a given time period.

Amplitude - One-half the wave height or the distance from either the crest or the trough to the still-water line.

Depth - the distance from the ocean bottom to the still-water line.

Direction of Propagation - the direction in which a wave is travelling.


The motion and behavior of simple sinusoidal waves can be fully described when the wavelength (L), height (H), period (T), and depth (d) are known. For instance, in deep water - when the depth is greater than one-half the wavelength - wave speed can be determined from the wave size. In shallow water, on the other hand, wave speed depends primarily on water depth.

Similarly, wave height is limited by both depth and wavelength. For a given water depth and wave period, there is a maximum height limit above which a wave becomes unstable and breaks. In deep water this upper limit of wave height - called breaking wave height - is a function of the wavelength. In shallow water, however, it is a function of both depth and wavelength. (Studies suggest the limiting wave steepness to be H/L = 0.141 in deep water and H/d = 0.83 for solitary waves in shallow water.)
--------------------------------------------------------------------------

Irregular Waves

Although simple waves are readily analyzed, in their perfect regularity they do not accurately depict the variability of ocean waves. Looking out at the sea, one never sees a constant progression of identical waves. Instead, the sea surface is composed of waves of varying heights and periods moving in differing directions. When the wind is blowing and the waves are growing in response, the seas tend to be confused: a wide range of heights and periods is observed. Swell is more regular, but it too is fundamentally irregular in nature, with some variablility in height and period. In fact, highly regular waves can be generated in the laboratory but are rare in nature.

If a recorder were to measure waves at a fixed location on the ocean, a the wave surface record would be rather irregular and random. Although individual waves can be identified, there is significant variability in height and period from wave to wave. Consequently, definitions of wave characteristics - height, period, etc. - must be statistical or probabilistic, indicating the severity of wave conditions.

By analyzing time-series meaurements of a natural sea state, some statistical estimates of simple parameters can be produced. The most important of these parameters is the significant wave height, Hs. Hs (or H 1/3) is the mean of the largest 1/3 (33%) of waves recorded during the sampling period. This statistical measure was designed to correspond to the wave height estimates made by experienced observers. (Observers do not notice all of the small waves that pass by; instead they focus on the larger, more salient peaks.)

Since ocean conditions are constantly changing, measures like significant wave height are short-term statistics, calculated for sample periods that are generally one hour or less. (The majority of CDIP's parameters are calculated for periods from 26 to 30 minutes.) Moreover, it is important to remember that the significant wave height is a statistical measure, and it is not intended to correspond to any specific wave. During the sampling period there will be many waves smaller than the Hs, and some that are larger. Statistically, the largest wave in a 1000-wave sample is likely to be nearly two times (1.86x) the significant wave height!

A number of other wave parameters - like Ta, the average period - are measured to describe natural sea states. Yet even taken together, the basic wave parameters give very limited information about wave characteristics and behavior. A single Hs value may correspond to a wide range of conditions, combining waves from any number of different swells. For this reason, phyical oceanographers have developed analyses that give more detailed, complete meaures of ocean waves.
--------------------------------------------------------------------------

Spectral Analysis

Two main approaches exist for treating complex waves: spectral anlysis and wave-by-wave (wave train) analysis. The more powerful and popular of these two approaches is spectral analysis. Spectral analysis assumes that the sea state can be considered as a combination or superposition of a large number of regular sinusoidal wave components with different frequencies, heights, and directions. This is a very useful assumption in wave analysis since sea states are in fact composed of waves from a number of different sources, each with its own characteristic height, period, and direction of travel.

Mathematically, spectral analysis is based on the Fourier Transform of the sea surface. The Fourier Transform allows any continuous, zero-mean signal - like a time-series record of the sea surface elevation - to be transformed into a summation of simple sine waves. These sine waves are the components of the sea state, each with a distinct height, frequency, and direction. In other words, the spectral analysis method determines the distribution of wave energy and average statistics for each wave frequency by converting the time series of the wave record into a wave spectrum. This is essentially a transformation from the time-domain to the frequency-domain, and is accomplished most conveniently using a mathematical tool known as the Fast Fourier Transform (FFT).

The spectral approach indicates what frequencies have significant energy content, as well as the direction wave energy is moving at each frequency. A wave spectrum can readily plotted in a frequency vs. energy density graph, which can reveal a great deal information about a wave sample and ocean conditions. The general shape of the plot, in fact, reveals a great deal: whether seas or swell predominate, the number of distinct swells present, etc. For example, during strong wind events, the spectrum tends to have a broad central peak. For swell that has propagated a long distance from the source of generation, on the other hand, waves tend to have a single sharp, low-frequency (long period) peak.

Gauging Waves
All of the valuable information produced by spectral wave analysis is based on one thing: a time-series record of sea surface elevations. In general, time series are analyzed over short periods, from 17 to 68 minutes, and are measured at around one sample per second (1 Hz).

There are two main types of sensors used to measure sea surface elevation, pressure sensors and buoys. Pressure sensors are mounted at a fixed position underwater, and they measure the height of the water column that passes above them. As wave crests pass by, the height of the water column increases; when troughs approach, the water column height falls. By deducting the the depth of the sensor from the water column heights, a record of sea surface elevations can be generated.

Buoys ride atop the surface of the ocean. Equipped with accelerometers to record their own movements, buoys rise with the wave crests and fall with the troughs. Since buoys are always floating on the sea surface, by recording their own movements they are in fact recording the movements of the sea surface. Readings from the accelerometers inside the buoys can be used to calculate the buoys' vertical displacements; these values are also a record of sea surface elevation.

A record of sea surface elevations from a single point is enough to generate an energy spectrum. To determine the direction of the waves and generate a directional spectrum, however, more information is needed. One way to generate a directional spectrum is to measure the same parameter - such as pressure - at a series of nearby locations. CDIP's early directional measurements, for instance, were all recorded by square arrays of pressure sensors, measuring 10 meters on a side.

The other way to produce a directional spectrum is by measuring different parameters at the same point. This is the approach used in directional buoys, which measure pitch and roll in addition to vertical heave. Although CDIP has relied on pressure sensor arrays and directional buoys for its directional measurements, other instruments can also be used. For instance, the p-U-V technique uses a pressure gauge and a horizontal component current meter in almost the same location to measure the wave field. Other techniques for directional wave measurement include arrays of surface-piercing wires, triaxial current meters, acoustic doppler current meters, and radars.
Surge and Energy Basin
For measuring sea and swell - wave motions with periods under 40 seconds or so - CDIP's wave gauging is as described above. CDIP's pressure sensors, however, have also been used to measure surge, water level changes with periods between a minute and an hour. Surge is created by atmospheric and seismic forces, and falls in between standard wind wave motion and tidal motions.


(see http://cdip.ucsd.edu/?nav=documents&sub=index&xitem=waves)

[aboobacker]

An interesting link on wave generation, fetch etc.,
see http://meted.ucar.edu/marine/mod2_wlc_gen/print.htm

[aboobacker]

Swell Rating System (SRS)
Quantifying the Effects of Swell Height, Period and Quality on Wave Size

see : http://www.stormsurf.com/page2/papers/category.html#bookmark1

[aboobacker]

Third Generation Ocean Wave Model
Wattana Kanbua

1.0 Introduction
The forecasting of ocean waves dates back to the attempts made during the second world war by Sverderup and Munk to predict waves in the English channel, for the invasion of Normandy in 1945. During this study they introduced the statistical expression, significant wave height, which was intended to express the wave height a trained observer would report for a given sea state. In the fifties, the wave energy spectrum was introduced by Pierson in 1957. This was based on the assumption that the sea surface may be represented as a Fourier series of superimposed waves with different wave lengths and with statistically random phases. The energy spectrum would then represent the mean wave energy at each of these Fourier modes. From the wave energy spectrum, several statistical parameters describing the sea state may be extracted. Examples are significant wave height, mean wave direction, mean wave length period etc. When the wave energy spectrum was introduced, an equation describing the evolution of the spectrum in time could be developed. This equation which express the conservation laws for wave energy in the ocean, and forms the basis for all numerical wave prediction models.
2.0 Theory
Before the wave model WAM will be described, a short summary of the basic theory behind wave modeling will be given.
The waves considered in numerical wave prediction, is assumed to be “short” in the sense that they are not affected by the rotation of the earth. Also the effects of density stratification of the ocean will be assumed to have no impact of the ocean waves. We will start with describing the basic laws of wave motion in fluids, and then give an interpretation of the individual terms in the wave energy equation.
2.1 Wave propagation
When considering the propagation of waves on the surface of a fluid, it is important to recognize the difference between the velocity of one individual wave, and the group velocity, which expresses the velocity of the wave energy for a particular wave length. The surface elevation caused by one Fourier component may be expressed as (for simplicity we assume only one dimension of propagation)
h = sin (k x - w t )
where k = 2p / l is the wave number and l is the wave length. w = 2p / T is the angular frequency and T is the wave period. The speed of one individual wave will in this case be

while the group velocity is defined as the rate of change of angular frequency to the wave number.
......

http://www.tmd.go.th/~marine/wamdoc/thirdwave.html

[kmfkais]

The velocity of idealized traveling waves on the ocean is wavelength dependent and for shallow enough depths, it also depends upon the depth of the water. The wave speed relationship is

<CENTER>http://hyperphysics.phy-astr.gsu.edu/HBASE/waves/imgwav/ocwav.gif</CENTER><CENTER><TABLE cellSpacing=2 cellPadding=2 border=1><TBODY><TR><TD>Calculation (http://hyperphysics.phy-astr.gsu.edu/HBASE/watwav.html#c3)</TD></TR></TBODY></TABLE></CENTER><CENTER>http://hyperphysics.phy-astr.gsu.edu/HBASE/waves/wavpic/wave9.jpg</CENTER>Any such simplified treatment of ocean waves is going to be inadequate to describe the complexity of the subject. It is certainly a subject that has been studied thoroughly, but no single model is going to apply to all cases. Some of the things that have been learned will be summarized here.

It will be presumed that ocean waves obey the basic wave relationship (http://hyperphysics.phy-astr.gsu.edu/HBASE/wavrel.html#c1) c=fl , where c is traditionally used for the wave speed or "celerity". The term celerity means the speed of the progressing wave with respect to stationary water - so any current or other net water velocity would be added to it.

The shape of an ocean wave is often depicted as a sine wave, but the the experimental waveshape is described as a "trochoid". A trochoid can be defined as the curve traced out by a point on a circle as the circle is rolled along a line. The following sketch is adapted from Bascom.

<CENTER>http://hyperphysics.phy-astr.gsu.edu/HBASE/waves/imgwav/trochoid.gif</CENTER>The trochoid shape does approach the sine curve in shape for small amplitudes, but you can perhaps see that the shape is different, with a narrowing of the peaks of the trochoid compared to the sinusoid. This narrowing or steepening of the peak becomes more pronounced as the wave amplitude increases.

The discovery of the trochoidal shape came from the observation that particles in the water would execute a circular motion as a wave passed without significant net advance in their position. The motion of the water is forward as the peak of the wave passes, but backward as the trough of the wave passes, arriving again at the same position when the next peak arrives. (Actually, experiments show a slight advance of the water with the waves, but that advance is small compared to the overall circular motion.)

<CENTER>http://hyperphysics.phy-astr.gsu.edu/HBASE/waves/imgwav/circonwave.gif</CENTER>The illustration above, adapted from von Arx, shows the direction of the water motion at different points along the wave.

<CENTER>http://hyperphysics.phy-astr.gsu.edu/HBASE/waves/imgwav/wavcirc.gif</CENTER>Bascom describes wavetank experiments where the circulation of the water was studied. The circles summarize the total motion of the medium when a full wavelength passes. It was common practice to inject droplets of oil which were whitened with zinc oxide and adjusted to have the same density as the water. The orbits could then be traced out on the sides of the wavetank. It was found that there was a small progression of the orbits in the direction of the wave propagation.

The fact that the water motion at depth flattens into a forward and backward surge is readily observed by scuba divers on shallow reefs. The effect of waves traveling overhead is translated into forward and backward motion of a diver suspended over the reef, and can be seen in the motion of soft corals and vegetation.

All of the description of waves above applies to long-wavelength waves which von Arx calls "gravity waves", implying that they are mainly controlled by gravity and inertia. Their wave speeds increase with wavelength, a behavior that is called "normal dispersion". For waves shorter than 1.73 cm, the surface tension of the water exerts a controlling force and von Arx calls them "capillary waves". Their speed increases as the wavelength gets shorter, a behavior that is called "anomalous dispersion". The minimum wavespeed at wavelength 1.73 cm is 23.1 cm/s. The following dependence of deep-water speed on wavelength is adapted from von Arx.

<CENTER>http://hyperphysics.phy-astr.gsu.edu/HBASE/waves/imgwav/wavcvslength.gif</CENTER>The small ripples or capillary waves are observed first when a fresh wind blows over smooth water. They have rounded crests and v-shaped troughs. Von Arx describes these waves as the way the wind gets a "grip" on the water since they are numerous and move slowly before the wind. Stories are told of seafarers who first observed these capillary waves when the wind was rising and later experienced the swells of the gravity waves, so it was thought that the capillary waves were the first mechanism by which the wind gave energy to the water. As the gravity waves build up, their wavelength tends to lengthen and speed increase until it matches the speed of the wind, at which point they can no longer extract energy from the wind.

The trochoid form that is attributed to ocean waves changes shape as the amplitude increases for a given wavelength.

<TABLE><TBODY><TR><TD><CENTER>http://hyperphysics.phy-astr.gsu.edu/HBASE/waves/imgwav/trochoidpair.gif
Trochoid wave models</CENTER></TD><TD>The shape change is illustrated at left for a single wavelength, but the peak tends to become narrower and steeper as the amplitude increases for a given wavelength. Bascom reports experimental evidence from wavetanks that suggests that a ratio of 1:7 for peak height to wavelength is the maximum and that an angle of 120° is the minimum angle for a peak. Above this ratio the peaks became unstable. The bottom wave sketch is scaled to that 1:7 ratio of peak-to-trough distance compared to wavelength.

</TD></TR></TBODY></TABLE><TABLE><TBODY><TR><TD>http://hyperphysics.phy-astr.gsu.edu/HBASE/waves/wavpic/wavebreak78.jpg </TD><TD>As waves move toward a beach, the shallower water decreases the wavespeed, so the wavelength becomes shorter and the peak heights increase. The wavepeaks become unstable and, moving faster than the water below, they break forward












http://hyperphysics.phy-astr.gsu.edu/HBASE/waves/watwav2.html
</TD></TR></TBODY></TABLE>

[aboobacker]

Waves - Beaufort Wind Scale
Beaufort Wind Force Wind Speed
(Knots) WMO* Wind Classification Wave Height (ft) Sea Conditions
0 Less than 1 Calm Sea surface is smooth and mirror-like.
1 1-3 Light Air Scaly ripples without foam crests.
2 4-6 Light Breeze Small wavelets with glassy appearing crests and no breaking.
3 7-10 Gentle Breeze Large wavelets, crests begin to break and whitecaps are scattered whitecaps.
4 11-16 Moderate Breeze 1-4 Small waves becoming longer and whitecaps are numerous.
5 17-21 Fresh Breeze 4-8 Moderate waves take longer form and there are many whitecaps and some spray.
6 22-27 Strong Breeze 8-13 Larger waves form and whitecaps are common, along with more spray.
7 28-33 Near Gale 13-20 The sea heaps up and white foam streaks off breakers.
8 34-40 Gale 13-20 Moderately high waves of greater length are formed. The edges of crests begin to break into spindrift and foam blown in streaks.
9 41-47 Strong Gale 20 High waves occur, the sea begins to roll and dense streaks of foam form. Spray may reduce visibility.
10 48-55 Storm 20-30 Very high waves form with overhanging crests. The sea is white with densely blown foam. There is heavy rolling and lowered visibility.
11 56-63 Violent Storm 30-45 Exceptionally high waves form and foam patches cover the sea. Visibility is more reduced.
12 64+ Hurricane 45 Exceptionally high waves. The air is filled with foam and the sea completely white with driving spray. Visibility greatly reduced.

* World Meteorological Organization

http://www.onr.navy.mil/focus/ocean/motion/waves4.htm