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In fluid dynamics, **wave shoaling** is the effect by which surface waves entering shallower water change in wave height. It is caused by the fact that the group velocity, 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 order to maintain a constant energy flux.^{ [2] } Shoaling waves will also exhibit a reduction in wavelength while the frequency remains constant.

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 and **hydrodynamics**. Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation,

In fluid dynamics, the **wave height** of a surface wave is the difference between the elevations of a crest and a neighbouring trough. Wave height is a term used by mariners, as well as in coastal, ocean and naval engineering.

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.

In shallow water and parallel depth contours, non-breaking waves will increase in wave height as the wave packet enters shallower water.^{ [3] } This is particularly evident for tsunamis as they wax in height when approaching a coastline, with devastating results.

When waves travel into areas of **shallow water**, they begin to be affected by the ocean bottom. The free orbital motion of the water is disrupted, and water particles in orbital motion no longer return to their original position. As the water becomes shallower, the swell becomes higher and steeper, ultimately assuming the familiar sharp-crested wave shape. After the wave breaks, it becomes a wave of translation and erosion of the ocean bottom intensifies.

In physics, a **wave packet** 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 different wavenumbers, with phases and amplitudes such that they interfere constructively only over a small region of space, and destructively elsewhere. Each component wave function, and hence the wave packet, are solutions of a wave equation. Depending on the wave equation, the wave packet's profile may remain constant or it may change (dispersion) while propagating.

A * tsunami* or

Waves nearing the coast change wave height through different effects. Some of the important wave processes are refraction, diffraction, reflection, wave breaking, 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. Pure wave shoaling occurs for long-crested waves propagating perpendicular to the parallel depth contour lines of a mildly sloping sea-bed. Then the wave height at a certain location can be expressed as:^{ [4] }^{ [5] }

In physics, **refraction** is the change in direction of a wave passing from one medium to another or from a gradual change in the medium. Refraction of light is the most commonly observed phenomenon, but other waves such as sound waves and water waves also experience refraction. How much a wave is refracted is determined by the change in wave speed and the initial direction of wave propagation relative to the direction of change in speed.

**Diffraction** refers to various phenomena that occur when a wave encounters an obstacle or a slit. It is defined as the 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 secondary source of the propagating wave. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1660.

**Reflection** is the change in direction of a wavefront at an interface between two different media so that the wavefront returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves. The *law of reflection* says that for specular reflection the angle at which the wave is incident on the surface equals the angle at which it is reflected. Mirrors exhibit specular reflection.

with the shoaling coefficient and the wave height in deep water. The shoaling coefficient depends on the local water depth and the wave frequency (or equivalently on and the wave period ). Deep water means that the waves are (hardly) affected by the sea bed, which occurs when the depth is larger than about half the deep-water wavelength

**Frequency** is the number of occurrences of a repeating event per unit of time. It is also referred to as **temporal frequency**, which emphasizes the contrast to spatial frequency and angular frequency. The

In physics, the **wavelength** is the **spatial period** of a periodic wave—the distance over which the wave's shape repeats. It is thus the inverse of the spatial frequency. Wavelength is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. Wavelength is commonly designated by the Greek letter *lambda* (λ). The term *wavelength* is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids.

For non-breaking waves, the energy flux associated with the wave motion, which is the product of the wave energy density with the group velocity, between two wave rays is a conserved quantity (i.e. a constant when following the energy of a wave packet 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 in 1915.^{ [6] } For waves affected by refraction and shoaling (i.e. within the geometric optics approximation), the rate of change of the wave energy transport is:^{ [5] }

In fluid dynamics, a **breaking wave** is a wave whose amplitude reaches a critical level at which some process can suddenly start to occur that causes large amounts of wave energy to be transformed into turbulent kinetic energy. At this point, simple physical models that describe wave dynamics often become invalid, particularly those that assume linear behaviour.

**Energy flux** is the rate of transfer of energy through a surface. The quantity is defined in two different ways, depending on the context:

- Rate of energy transfer per unit area.
- Total rate of energy transfer.

In physics, **ray tracing** is a method for calculating the path of waves or particles through a system with regions of varying propagation velocity, absorption characteristics, and reflecting surfaces. Under these circumstances, wavefronts may bend, change direction, or reflect off surfaces, complicating analysis. Ray tracing solves the problem by repeatedly advancing idealized narrow beams called *rays* through the medium by discrete amounts. Simple problems can be analyzed by propagating a few rays using simple mathematics. More detailed analysis can be performed by using a computer to propagate many rays.

where is the co-ordinate along the wave ray and is the energy flux per unit crest length. A decrease in group speed and distance between the wave rays must be compensated by an increase in energy density . This can be formulated as a shoaling coefficient relative to the wave height in deep water.^{ [5] }^{ [4] }

For shallow water, when the wavelength is much larger than the water depth – in case of a constant ray distance (i.e. perpendicular wave incidence on a coast with parallel depth contours) – wave shoaling satisfies Green's law:

with the mean water depth, the wave height and the fourth root of

Following Phillips (1977) and Mei (1989),^{ [7] }^{ [8] } denote the phase of a wave ray as

- .

The local wave number vector is the gradient of the phase function,

- ,

and the angular frequency is proportional to its local rate of change,

- .

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;

- .

Assuming stationary conditions (), this implies that wave crests are conserved and the frequency must remain constant along a wave ray as . As waves enter shallower waters, the decrease in group velocity caused by the reduction in water depth leads to a reduction in wave length because the nondispersive shallow water limit of the dispersion relation for the wave phase speed,

dictates that

- ,

i.e., a steady increase in *k* (decrease in ) as the phase speed decreases under constant .

- ↑ Wiegel, R.L. (2013).
*Oceanographical Engineering*. Dover Publications. p. 17, Figure 2.4. ISBN 978-0-486-16019-1. - ↑ Longuet-Higgins, M.S.; Stewart, R.W. (1964). "Radiation stresses in water waves; a physical discussion, with applications" (PDF).
*Deep-Sea Research and Oceanographic Abstracts*.**11**(4): 529–562. doi:10.1016/0011-7471(64)90001-4. - ↑ WMO (1998).
*Guide to Wave Analysis and Forecasting*(PDF).**702**(2 ed.). World Meteorological Organization. ISBN 92-63-12702-6. - 1 2 Goda, Y. (2010).
*Random Seas and Design of Maritime Structures*. Advanced Series on Ocean Engineering.**33**(3 ed.). Singapore: World Scientific. pp. 10–13 & 99–102. ISBN 978-981-4282-39-0. - 1 2 3 4 Dean, R.G.; Dalrymple, R.A. (1991).
*Water wave mechanics for engineers and scientists*. Advanced Series on Ocean Engineering.**2**. Singapore: World Scientific. ISBN 978-981-02-0420-4. - ↑ Burnside, W. (1915). "On the modification of a train of waves as it advances into shallow water".
*Proceedings of the London Mathematical Society*. Series 2.**14**: 131–133. doi:10.1112/plms/s2_14.1.131. - ↑ Phillips, Owen M. (1977).
*The dynamics of the upper ocean (2nd ed.)*. Cambridge University Press. ISBN 0-521-29801-6. - ↑ Mei, Chiang C. (1989).
*The Applied Dynamics of Ocean Surface Waves*. Singapore: World Scientific. ISBN 9971-5-0773-0.

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In classical mechanics, a **harmonic oscillator** is a system that, when displaced from its equilibrium position, experiences a restoring force *F* proportional to the displacement *x*:

In mechanics and physics, **simple harmonic motion** is a special type of periodic motion or oscillation where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.

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