Wave shoaling

Last updated Surfing on shoaling and breaking waves. The phase velocity cp (blue) and group velocity cg (red) as a function of water depth h for surface gravity waves of constant frequency, according to Airy wave theory. Quantities have been made dimensionless using the gravitational acceleration g and period T, with the deep-water wavelength given by L0 = gT /(2π) and the deep-water phase speed c0 = L0/T. The grey line corresponds with the shallow-water limit cp =cg = √(gh). The phase speed – and thus also the wavelength L = cpT – decreases monotonically with decreasing depth. However, the group velocity first increases by 20% with respect to its deep-water value (of cg = 1/2c0 = gT/(4π)) before decreasing in shallower depths.

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.  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.

Contents

In shallow water and parallel depth contours, non-breaking waves will increase in wave height as the wave packet enters shallower water.  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 tidal wave,, also known as a seismic sea 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 explosions above or below water all have the potential to generate a tsunami. Unlike normal ocean waves, which are generated by wind, or tides, which are generated by the gravitational pull of the Moon and the Sun, a tsunami is generated by the displacement of water.

Overview

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 $H$ at a certain location can be expressed as: 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.

$H=K_{S}\;H_{0},$ 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 $f$ (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 $L_{0}=gT^{2}/(2\pi ).$ 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 period is the duration of time of one cycle in a repeating event, so the period is the reciprocal of the frequency. For example: if a newborn baby's heart beats at a frequency of 120 times a minute, its period—the time interval between beats—is half a second. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio signals (sound), radio waves, and light. 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.

Physics When waves enter shallow water they slow down. Under stationary conditions, the wave length is reduced. The energy flux must remain constant and the reduction in group (transport) speed is compensated by an increase in wave height (and thus wave energy density). Convergence of wave rays (reduction of width b{\displaystyle b}) at Mavericks, California, producing high surfing waves. The red lines are the wave rays; the blue lines are the wavefronts. The distances between neighboring wave rays vary towards the coast because of refraction by bathymetry (depth variations). The distance between wavefronts (i.e. the wavelength) reduces towards the coast because of the decreasing phase speed. Shoaling coefficient KS{\displaystyle K_{S}} as a function of relative water depth h/L0,{\displaystyle h/L_{0},} describing the effect of wave shoaling on the wave height – based on conservation of energy and results from Airy wave theory. The local wave height H{\displaystyle H} at a certain mean water depth h{\displaystyle h} is equal to H=KSH0,{\displaystyle H=K_{S}\;H_{0},} with H0{\displaystyle H_{0}} the wave height in deep water (i.e. when the water depth is greater than about half the wavelength). The shoaling coefficient KS{\displaystyle K_{S}} depends on h/L0,{\displaystyle h/L_{0},} where L0{\displaystyle L_{0}} is the wavelength in deep water: L0=gT2/(2π),{\displaystyle L_{0}=gT^{2}/(2\pi ),} with T{\displaystyle T} the wave period and g{\displaystyle g} the gravity of Earth. The blue line is the shoaling coefficient according to Green's law for waves in shallow water, i.e. valid when the water depth is less than 1/20 times the local wavelength L=Tgh.{\displaystyle L=T\,{\sqrt {gh}}.}

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.  For waves affected by refraction and shoaling (i.e. within the geometric optics approximation), the rate of change of the wave energy transport is: 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:

1. Rate of energy transfer per unit area.
2. 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.

${\frac {d}{ds}}(bc_{g}E)=0,$ where $s$ is the co-ordinate along the wave ray and $bc_{g}E$ 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 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:

$H\,{\sqrt[{4}]{h}}={\text{constant}},$ with $h$ the mean water depth, $H$ the wave height and ${\sqrt[{4}]{h}}$ the fourth root of $h.$ Water wave refraction

Following Phillips (1977) and Mei (1989),   denote the phase of a wave ray as

$S=S(\mathbf {x} ,t),\qquad 0\leq S<2\pi$ .

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

$\mathbf {k} =\nabla S$ ,

and the angular frequency 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;

${\frac {\partial k}{\partial t}}+{\frac {\partial \omega }{\partial x}}=0$ .

Assuming stationary conditions ($\partial /\partial t=0$ ), this implies that wave crests are conserved and the frequency must remain constant along a wave ray as $\partial \omega /\partial x=0$ . As waves enter shallower waters, the decrease in group velocity caused by the reduction in water depth leads to a reduction in wave length $\lambda =2\pi /k$ because the nondispersive shallow water limit of the dispersion relation for the wave phase speed,

$\omega /k\equiv c={\sqrt {gh}}$ dictates that

$k=\omega /{\sqrt {gh}}$ ,

i.e., a steady increase in k (decrease in $\lambda$ ) as the phase speed decreases under constant $\omega$ .