# Water wave

In water waves is surface waves at the interface between water and air, or an internal wave at the interface between two different densities water layers in the isopycnic (layered) ocean. According to Walter Munk , this means all water level deflections with periods from tenths of a second to hours ( tidal waves ).

Classification of ocean waves according to Munk: names, stimulating forces and relative amplitudes
Steep water waves are characterized by sweeping valleys and pointed ridges. The picture shows a wave running from left to right shortly before the overturning.
Audio recording of ocean waves rolling on land
Movement of small ocean waves

At wavelengths smaller than 4 mm, the surface tension of the water determines the properties of the capillary waves , in which the viscosity of the water also causes strong dissipative effects. At wavelengths greater than 7 cm, the inertia, the force of gravity and the resulting changes in pressure and movement are decisive for the properties of the gravity wave .

## Wave formation

Video of ocean waves hitting rocks
Video of a wave, clearly visible the water retreating in front of the wave, breaking the wave and then running out of the beach

Stones and flow obstacles thrown into the water create waves, while moving ships are accompanied by a bow wave . Seaquakes can cause tsunamis . No further reference should be made to the latter or tidal waves at this point, but rather, preferably , surface waves of the sea generated by wind as a function of the water depth will be treated.

### Wave formation by wind

The mechanism of wave formation by wind is the Kelvin-Helmholtz instability . In the area in which the swell occurs, a distinction must be made between the following influencing variables:

• the strike length ( fetch ) F = exposure distance of the wind on the water surface,
• the wind speed U and
• the duration of the wind as the so-called maturing time of the sea.${\ displaystyle D _ {\ mathrm {min}}}$

Their interaction determines the size of the waves and their shape. The larger one of these influencing variables, the larger the waves. In shallow seas, the water depth has a limiting influence.
The resulting swell is characterized by:

• the wave heights,
• the wavelengths,
• the periods and
• the direction of the wave advance (based on the north direction).

In a given sea area there are waves with different bandwidths of heights and periods. The following characteristic data are defined for the wave forecast:

• the significant wave height and${\ displaystyle H_ {S} = H_ {1/3}}$
• the significant wave period .${\ displaystyle T_ {S} = T_ {1/3}}$

Both relate to the waves observed over a given period of time and, as statistical values, represent mean values ​​for the third of the highest waves in the collective.

## Structure and properties

Geometry of a trochoidal deep water wave: To define the wave height H, the wave length L, the calm water level, the horizontal and the vertical wave asymmetry.

### Wave height, wave length, wave steepness

Water waves differ in shape from the regular sinusoidal shape . Their shape is asymmetrical both horizontally and vertically. The part of the wave that is above the calm water level is called the wave crest . The position of the highest deflection is the crest of the wave . The part of the wave that lies below the calm water level is the wave trough . The wave height is the sum of the amounts of both neighboring maximum deflections:

${\ displaystyle H = H_ {o} + H_ {u}}$

The amount of the maximum positive water level deflection exceeds the maximum negative water level deflection the smaller the water depth becomes. In the case of waves in shallow water, the height of the wave crest can be up to 3/4 of the total wave height H, while the wave trough H / 4 is below the calm water level. The wavelength, (symbol ), is the sum of their unequal partial lengths of the ridge area and the valley area related to the calm water table, see picture on the right. It is ${\ displaystyle L}$

${\ displaystyle L_ {B}}$< and${\ displaystyle L_ {T}}$ ${\ displaystyle \ qquad}$
${\ displaystyle L = L_ {B} + L_ {T}}$.

The quotient of wave height and wave length is an important indicator for assessing the stability of waves and is called wave steepness S.

${\ displaystyle S = H / L}$.

According to Stokes (1847), the theoretical limit value applies to waves above a water depth . In fact, the waves break already at . On the open ocean, there are wave steepnesses between . For the shallow water area, nature measurements have confirmed Miche's formula (1944), which also takes into account the limiting effect of the seabed. ${\ displaystyle d> L / 2}$${\ displaystyle S _ {\ mathrm {max}} = 1/7}$${\ displaystyle S = 1/10}$${\ displaystyle 1/100

${\ displaystyle {\ text {Limit steepness:}} \ quad \ max \ left ({\ frac {H} {L}} \ right) = 0 {,} 142 \, \ tanh {\ left ({\ frac {2 \ pi d} {L}} \ right)}}$

Since the 19th century, the asymmetrical shape of natural water waves has been described by Gerstner (1804) and above all by Stokes (1847) with increasing mathematical effort. Regardless of this, the linear wave theory according to Airy- Laplace (1845), which is based on the regular sine shape, is still often used for practical estimations .

### Orbital motion

Trochoidal deep water wave : instantaneous directions of the orbital velocity at different positions on the wave surface.${\ displaystyle w = {\ frac {2 \ cdot \ pi \ cdot r} {T}} = {\ frac {\ pi \ cdot H} {T}}}$
Stokes deep water wave: Orbital orbits of water particles starting at two positions with a distance of half a wavelength.

According to the wave theories of Gerstner and Airy-Laplace, the water particles are moved approximately on circular paths (orbital paths) over great water depths when passing a wave, the radii of which in the flow field below the water surface to a depth that corresponds to about half the wavelength, according to an exponential law decrease to zero. The cycle period is the period of revolution that corresponds to the advance of the wave by a full wavelength . So the orbital velocity at the water surface is: ${\ displaystyle T = 1 / f}$${\ displaystyle L}$

${\ displaystyle w = {\ frac {2 \ pi \, r} {T}}}$.

And the wave advance speed is ${\ displaystyle c_ {w}}$

${\ displaystyle c_ {w} = {\ frac {L} {T}}}$.

In contrast, according to Stokes' theory, the trajectories of the water particles are not closed after a wave period . According to this theory, the circular orbital movement is superimposed by a horizontal drift velocity U in the direction of the wave advance velocity c, which is called the mass transport velocity . In the adjacent animation, the red dots indicate the current positions of the massless particles that move with the flow velocity. The light blue lines are the trajectory of these particles and the light blue points indicate the particle positions after each wave period. The white dots are liquid particles moving in the same direction. Note that the wave period of the liquid particles near the free surface is different from that with respect to a fixed position (indicated by the light blue dots). This is due to the Doppler effect .
(to be added for limited water depth)

## Dispersion and group velocity

c ( L , d )
c ( f , d )

### Gravity waves

While the wave progression speed ( phase speed ) applies to all types of waves, the dispersion relation also applies to gravity waves , which contains the water depth as a variable in addition to the wavelength${\ displaystyle c = L / T}$${\ displaystyle L}$${\ displaystyle d}$

(1) ${\ displaystyle c = {\ sqrt {{\ frac {g \, L} {2 \ pi}} \ tanh {\ left ({\ frac {2 \ pi d} {L}} \ right)}}}}$
${\ displaystyle \ pi}$: Circle number
${\ displaystyle g}$: Gravitational acceleration

The two figures on the right show the dependence of the phase velocity on the wavelength or frequency. The dependence on the water depth is also given. Gravity waves do not occur as individual monochromatic waves, but always as a superposition of waves with neighboring frequencies. As a result, wave packets or wave groups occur, which move with the group speed${\ displaystyle d}$

(2) ${\ displaystyle c _ {\ mathrm {g}} = cL {\ frac {\ mathrm {d} c} {\ mathrm {d} L}}}$

move around. Depending on the sign of the differential quotient , the group speed is smaller, larger or equal to the phase speed. Correspondingly, a distinction is made between normal dispersion , anomalous dispersion and dispersionless wave propagation. In the case of gravity waves, the dispersion is negative: normal dispersion is present (as opposed to capillary waves). ${\ displaystyle {\ tfrac {\ mathrm {d} c} {\ mathrm {d} L}}}$

#### Approximation: The wavelengths are small relative to the water depth (deep water waves)

For waters with a depth of more than half a wavelength ( ) approaches the value 1 in (1), the phase velocity becomes independent of the water depth: ${\ displaystyle d> L / 2}$${\ displaystyle \ tanh (x)}$${\ displaystyle c}$

(3) for${\ displaystyle c \ approx {\ sqrt {\ frac {gL} {2 \ pi}}}}$${\ displaystyle L <2d}$

or with c = L / T:

${\ displaystyle c = L \, f \ approx {\ sqrt {\ frac {gL} {2 \ pi}}}}$

Designates the period with the frequency , it follows from (3): ${\ displaystyle T}$${\ displaystyle f = 1 / T}$${\ displaystyle c = L / T}$

(4) ${\ displaystyle {\ frac {1} {f}} = T \ approx {\ sqrt {\ frac {2 \ pi L} {g}}}}$

Long-wave waves propagate faster and have a longer period than short-wave waves. At a wavelength of 1 km, c is about 142 km / h and T about 25 s, at a wavelength of 10 m c is about 14 km / h and T about 2.5 s.

The maximum dispersion is:

${\ displaystyle {\ frac {\ mathrm {d} c} {\ mathrm {d} L}} = {\ sqrt {\ frac {g} {8 \ pi L}}} \ quad {\ text {or} } \ quad {\ frac {\ mathrm {d} c} {\ mathrm {d} f}} = {\ frac {-g} {2 \ pi f ^ {2}}}}$

From (2), the group velocity is to ${\ displaystyle c _ {\ mathrm {g}}}$

${\ displaystyle c _ {\ mathrm {g}} \ approx {\ frac {1} {2}} \, c}$

Because of this dispersion relation, the composition of wave packets changes in such a way that the longer waves leave the area of ​​their generation faster than the shorter ones and thus arrive earlier at distant locations . As the short-period waves are also dampened more strongly, storm waves in distant areas are perceived as long-period swells .

#### Approximation: The wavelengths are large relative to the water depth

At wavelengths that are greater than the water depth ( ), one speaks of shallow water waves . With them, the speed of propagation depends only on the depth , but not on the wavelength. For small it holds and thus one obtains from (1) ${\ displaystyle L> 20 \ mathrm {d}}$${\ displaystyle d}$${\ displaystyle x}$${\ displaystyle \ tanh (x) \ approx x}$

(5) for .${\ displaystyle c \ approx {\ sqrt {gd}}}$${\ displaystyle d <{\ frac {L} {20}}}$

When the water is deep, these waves can reach very high speeds. This is the reason why tsunamis spread very quickly in the open ocean. At the same time, the speed of propagation is independent of the wavelength. Therefore, a wave packet of a shallow water wave hardly diverges as it propagates. The phase velocity is the same as the group velocity:

${\ displaystyle {\ frac {\ mathrm {d} c} {\ mathrm {d} L}} = 0 \ quad {\ text {or}} \ quad {\ frac {\ mathrm {d} c} {\ mathrm {d} f}} = 0}$
${\ displaystyle c \, = \, L \, f \, = \, {\ sqrt {g \, d}}}$
${\ displaystyle c _ {\ mathrm {g}} \, = \, c \ ;.}$

### Capillary waves

At wavelengths shorter than a few centimeters, the surface tension determines the speed of propagation. The following applies to capillary waves :

${\ displaystyle c \, = \, L \, f \, = \, {\ sqrt {\ frac {2 \ pi \ eta} {\ rho L}}} \, = \, \ left ({\ frac { 2 \ pi \ eta f} {\ rho}} \ right) ^ {1/3}}$

Therein mean the surface tension and the density of the liquid. The dispersion of capillary waves is less than zero and therefore abnormal${\ displaystyle \ eta}$${\ displaystyle \ rho}$

${\ displaystyle {\ frac {\ mathrm {d} c} {\ mathrm {d} L}} \, = \, {\ frac {- \ left (2 \ pi \ eta L \ right) ^ {- 1 / 2}} {2L}} \ quad {\ text {or}} \ quad {\ frac {\ mathrm {d} c} {\ mathrm {d} f}} \, = \, {\ frac {2 \ pi \ eta} {3 \ rho}} \, \ left ({\ frac {2 \ pi \ eta f} {\ rho}} \ right) ^ {- 2/3}}$

## Wave effects

### reflection

Circular waves are reflected at the edge and overlap
Keel of a ship.
Waves of water run parallel to the beach

Wave reflection at progressive water waves signifies the throwing back a portion of its energy ( wave energy ) on a structure ( breakwater , seawall , embankment ) or in places where the configuration of the natural seabed changes (strong). According to the law of reflection in optics, another part of the wave energy is simultaneously transmitted and the remaining part isdissipated or absorbedby the processes of wave breaking , fluid and ground friction, etc., cf. wave transformation , wave absorption .

### refraction

Under refraction is dependent on the water depth change of the wave traveling direction in shallow water waves (waves with wavelengths which are significantly greater than the water depth) is understood. It is caused by a wave speed that varies from place to place, which in shallow water waves depends on the depth. On gently sloping beaches, their effect means that wave fronts increasingly bend parallel to the shoreline and the observer on the beach sees the waves (not necessarily breaking) coming towards them. As with the refraction of light, Snellius 'law of refraction based on Huygens' principle is also applicable here.

### Diffraction

Under diffraction is diffraction of wavefronts understood at the ends of islands or on the edges of buildings. As with the diffraction of light at edges, Huygens' principle can also be used here . In the case of protective structures ( breakwaters and piers ), the diffraction of the wave fronts means that part of the energy of the incoming waves also gets behind the protective structure or in the area of ​​a port entrance that is to be protected by moles against the effects of waves.

### Breaking waves

Wave breaking describes the critical degree of wave transformation at which the surface tension at the crest of the wave is overcome, the orbital movement loses its characteristic shape and water emerging from the wave contour falls into the front slope. With regard to their geometry, about four crusher shapes can be distinguished.

### Examples of the behavior of waves when they hit a beach

Example 1 : breaking waves

If a wave approaches a slowly rising bank , the speed of propagation of the wave front decreases with decreasing water depth. The following waves roll over the wave front until they too are slowed down. The wavelength decreases, as a result of the conservation of energy, the wave height increases until the wave breaks.

Example 2 : refraction

If a wave front approaches a slowly rising bank at an inclined angle, the waves slow down in the flat area. Those further outside maintain their speed. Similar to the refraction of light on glass, the wave front rotates until it runs parallel to the beach line.

## Interface waves

Surface waves on a lake

In the considerations above, only the parameters of one medium are included. This assumption is justified for surface waves from water in air, since the influence of the air is negligible due to the low density.

The expanded version of equation (3) takes into account the density of both phases, denoted by and${\ displaystyle \ rho _ {\ mathrm {1}}}$${\ displaystyle \ rho _ {2}}$

${\ displaystyle c ^ {2} = {\ frac {\ rho _ {\ mathrm {1}} - \ rho _ {2}} {\ rho _ {\ mathrm {l}} + \ rho _ {2}} } \ cdot {\ frac {gL} {2 \ pi}}}$

And for capillary waves:

${\ displaystyle c ^ {2} = {\ frac {2 \ pi \ eta} {L (\ rho _ {\ mathrm {1}} + \ rho _ {2})}}}$

## Special waves

Surf waves are breaking waves near the beach. The breakwater criteria determine the maximum possible wave height H (vertical distance between wave trough and crest) in surf zones (= breaker height). Measurements in nature have shown that crusher heights can very well be greater than the local water depth.

Tsunamis are triggered by seaquakes. They are characterized by a very long wavelength and, on the high seas, by small amplitudes of less than one meter. The speed of propagation of tsunamis follows the relationship (5), because the wavelength of several 100 km is significantly greater than the depth of the oceans. Tsunamis spread (at an average sea depth of 5 km) at a speed of 800 km / h. Near the coast, the speed decreases while the altitude increases at the same time. The damage they cause when they run into flat coasts is devastating.

Tidal waves are waves caused by the tide .

In the stratification of light fresh water on heavy salt water, surface waves can be observed, the effects of which on ships are known as dead water . If a ship enters the zone, it can create bow waves on the surface of the salt water layer if the draft is sufficient. It clearly loses speed without water waves being visible on the surface of the water.

The Grundsee is a short, steep and breaking water wave, the valley of which extends to the bottom.

Individual, extremely high monster waves can be caused by overlapping , among other things . Strong winds and a countercurrent flow favor this. However, according to the models described above, the maximum possible wave height is limited. When designing ships, it was therefore assumed until the 1990s that waves more than 15 m high were impossible, or at least extremely improbable. This was first refuted by measurements in 1995 ( en: Draupner wave ). In the meantime, satellite observations have proven the existence of monster waves with heights of more than 30 m, which, viewed on a global scale, even occur relatively frequently (daily). The mechanism of their formation is still not fully understood and is the subject of basic physical research.

Wiktionary: Welle  - explanations of meanings, word origins, synonyms, translations
Commons : Water wave  - collection of images, videos and audio files

## literature

• Pohl, Introduction to Physics
• Franz Graf von Larisch-Moennich, Sturmsee und Brandung, Verlag von Velhagen and Klasing, 1925
• Petra Demmler: Das Meer - Wasser, Eis und Klima Verlag Eugen Ulmer, 2011. ISBN 3-8001-5864-7 , Creation of wind seas, swell, freak waves, tidal waves, storm surges and tsunamis; popular science presentation

## Individual evidence

1. ^ Andreas Mielke: Seminar: Theoretical Mechanics. (No longer available online.) Archived from the original on January 3, 2017 ; Retrieved January 3, 2017 .
2. http://www.bbc.com/earth/story/20170510-terrifying-20m-tall-rogue-waves-are-actually-real
3. ^ Zoe Heron: Freak Wave. BBC Horizon, - 2002 imdb