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In
fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) a ...
, wave shoaling is the effect by which surface waves, entering shallower water, change in
wave height In fluid dynamics, the wave height of a surface wave is the difference between the elevations of a crest and a neighboring trough. ''Wave height'' is a term used by mariners, as well as in coastal, ocean and naval engineering. At sea, the te ...
. It is caused by the fact that the
group velocity The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope'' of the wave—propagates through space. For example, if a stone is thrown into the middl ...
, which is also the wave-energy transport velocity, changes with water depth. Under stationary conditions, a decrease in transport speed must be compensated by an increase in
energy density In physics, energy density is the amount of energy stored in a given system or region of space per unit volume. It is sometimes confused with energy per unit mass which is properly called specific energy or . Often only the ''useful'' or extrac ...
in order to maintain a constant energy flux. Shoaling waves will also exhibit a reduction in
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
while the
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
remains constant. In other words, as the waves approach the shore and the water gets shallower, the waves get taller, slow down, and get closer together. In shallow water and parallel depth contours, non-breaking waves will increase in wave height as the
wave packet In physics, a wave packet (or wave train) is a short "burst" or "envelope" of localized wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of diff ...
enters shallower water. This is particularly evident for
tsunami A tsunami ( ; from ja, 津波, lit=harbour wave, ) is a series of waves in a water body caused by the displacement of a large volume of water, generally in an ocean or a large lake. Earthquakes, volcanic eruptions and other underwater exp ...
s as they wax in height when approaching a
coast The coast, also known as the coastline or seashore, is defined as the area where land meets the ocean, or as a line that forms the boundary between the land and the coastline. The Earth has around of coastline. Coasts are important zones in n ...
line, with devastating results.


Overview

Waves nearing the coast change wave height through different effects. Some of the important wave processes are
refraction In physics, refraction is the redirection of a wave as it passes from one medium to another. The redirection can be caused by the wave's change in speed or by a change in the medium. Refraction of light is the most commonly observed phenomen ...
,
diffraction Diffraction is defined as the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a s ...
,
reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...
,
wave breaking In fluid dynamics, a breaking wave or breaker is a wave whose amplitude reaches a critical level at which large amounts of wave energy transform into turbulent kinetic energy. At this point, simple physical models that describe wave dynamics o ...
, wave–current interaction, friction, wave growth due to the wind, and ''wave shoaling''. In the absence of the other effects, wave shoaling is the change of wave height that occurs solely due to changes in mean water depth – without changes in wave propagation direction and
dissipation In thermodynamics, dissipation is the result of an irreversible process that takes place in homogeneous thermodynamic systems. In a dissipative process, energy ( internal, bulk flow kinetic, or system potential) transforms from an initial form to ...
. Pure wave shoaling occurs for long-crested waves propagating
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...
to the parallel depth
contour line A contour line (also isoline, isopleth, or isarithm) of a function of two variables is a curve along which the function has a constant value, so that the curve joins points of equal value. It is a plane section of the three-dimensional gr ...
s of a mildly sloping sea-bed. Then the wave height H at a certain location can be expressed as: :H = K_S\; H_0, with K_S the shoaling coefficient and H_0 the wave height in deep water. The shoaling coefficient K_S depends on the local water depth h and the wave
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
f (or equivalently on h and the wave period T=1/f). Deep water means that the waves are (hardly) affected by the sea bed, which occurs when the depth h is larger than about half the deep-water
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
L_0=gT^2/(2\pi).


Physics

For non-
breaking wave In fluid dynamics, a breaking wave or breaker is a wave whose amplitude reaches a critical level at which large amounts of wave energy transform into turbulent kinetic energy. At this point, simple physical models that describe wave dynam ...
s, the
energy flux Energy flux is the rate of transfer of energy through a surface. The quantity is defined in two different ways, depending on the context: # Total rate of energy transfer (not per unit area); SI units: W = J⋅s−1. # Specific rate of energy transf ...
associated with the wave motion, which is the product of the
wave energy Wave power is the capture of energy of wind waves to do useful work – for example, electricity generation, water desalination, or pumping water. A machine that exploits wave power is a wave energy converter (WEC). Waves are generated by win ...
density with the
group velocity The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope'' of the wave—propagates through space. For example, if a stone is thrown into the middl ...
, between two wave rays is a
conserved quantity In mathematics, a conserved quantity of a dynamical system is a function of the dependent variables, the value of which remains constant along each trajectory of the system. Not all systems have conserved quantities, and conserved quantities are ...
(i.e. a constant when following the energy of a
wave packet In physics, a wave packet (or wave train) is a short "burst" or "envelope" of localized wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of diff ...
from one location to another). Under stationary conditions the total energy transport must be constant along the wave ray – as first shown by
William Burnside :''This English mathematician is sometimes confused with the Irish mathematician William S. Burnside (1839–1920).'' __NOTOC__ William Burnside (2 July 1852 – 21 August 1927) was an English mathematician. He is known mostly as an early res ...
in 1915. For waves affected by refraction and shoaling (i.e. within the
geometric optics Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
approximation), the rate of change of the wave energy transport is: :\frac(b c_g E) = 0, where s is the co-ordinate along the wave ray and b c_g E is the energy flux per unit crest length. A decrease in group speed c_g and distance between the wave rays b must be compensated by an increase in energy density E. This can be formulated as a shoaling coefficient relative to the wave height in deep water. For shallow water, when the
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
is much larger than the water depth – in case of a constant ray distance b (i.e. perpendicular wave incidence on a coast with parallel depth contours) – wave shoaling satisfies Green's law: :H\, \sqrt = \text, with h the mean water depth, H the wave height and \sqrt /math> the
fourth root In mathematics, a radicand, also known as an nth root, of a number ''x'' is a number ''r'' which, when raised to the power ''n'', yields ''x'': :r^n = x, where ''n'' is a positive integer, sometimes called the ''degree'' of the root. A root ...
of h.


Water wave refraction

Following Phillips (1977) and Mei (1989), denote the phase of a wave ray as :S = S(\mathbf,t), \qquad 0\leq S<2\pi. The local wave number vector is the gradient of the phase function, :\mathbf = \nabla S, and the
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit ti ...
is proportional to its local rate of change, :\omega = -\partial S/\partial t. Simplifying to one dimension and cross-differentiating it is now easily seen that the above definitions indicate simply that the rate of change of wavenumber is balanced by the convergence of the frequency along a ray; :\frac + \frac = 0. Assuming stationary conditions (\partial/\partial t = 0), this implies that wave crests are conserved and the
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
must remain constant along a wave ray as \partial \omega / \partial x = 0. As waves enter shallower waters, the decrease in
group velocity The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope'' of the wave—propagates through space. For example, if a stone is thrown into the middl ...
caused by the reduction in water depth leads to a reduction in
wave length In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
\lambda = 2\pi/k because the nondispersive shallow water limit of the
dispersion relation In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given t ...
for the wave
phase speed The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for exampl ...
, :\omega/k \equiv c = \sqrt dictates that :k = \omega/\sqrt, i.e., a steady increase in ''k'' (decrease in \lambda) as the
phase speed The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for exampl ...
decreases under constant \omega.


See also

* * * * * * * * *


Notes


External links


Wave transformation at Coastal Wiki
{{authority control Coastal geography Physical oceanography Water waves Oceanographical terminology