Gravity wave

Quantitative description

Deep water

The phase velocity ${\displaystyle c}$ of a linear gravity wave with wavenumber ${\displaystyle k}$ is given by the formula

${\displaystyle c=\sqrt{\frac{g}{k}},}$

where g is the acceleration due to gravity. When surface tension is important, this is modified to

${\displaystyle c=\sqrt{\frac{g}{k}+\frac{\sigma k}{\rho }},}$

where σ is the surface tension coefficient and ρ is the density.

Since ${\displaystyle c=\omega /k}$ is the phase speed in terms of the angular frequency ${\displaystyle \omega }$ and the wavenumber, the gravity wave angular frequency can be expressed as

${\displaystyle \omega =\sqrt{gk}.}$

The group velocity of a wave (that is, the speed at which a wave packet travels) is given by

${\displaystyle c_g=\frac{d\omega }{dk},}$

and thus for a gravity wave,

${\displaystyle c_g=\frac{1}{2}\sqrt{\frac{g}{k}}=\frac{1}{2}c.}$

The group velocity is one half the phase velocity. A wave in which the group and phase velocities differ is called dispersive.

Shallow water

Gravity waves traveling in shallow water (where the depth is much less than the wavelength), are nondispersive: the phase and group velocities are identical and independent of wavelength and frequency. When the water depth is h,

${\displaystyle c_p=c_g=\sqrt{gh}.}$

Generation of ocean waves by wind

Wind waves, as their name suggests, are generated by wind transferring energy from the atmosphere to the ocean's surface, and capillary-gravity waves play an essential role in this effect. There are two distinct mechanisms involved, called after their proponents, Phillips and Miles.

In the work of Phillips, the ocean surface is imagined to be initially flat (glassy), and a turbulent wind blows over the surface. When a flow is turbulent, one observes a randomly fluctuating velocity field superimposed on a mean flow (contrast with a laminar flow, in which the fluid motion is ordered and smooth). The fluctuating velocity field gives rise to fluctuating stresses (both tangential and normal) that act on the air-water interface. The normal stress, or fluctuating pressure acts as a forcing term (much like pushing a swing introduces a forcing term). If the frequency and wavenumber ${\displaystyle \left(\omega ,k\right)}$ of this forcing term match a mode of vibration of the capillary-gravity wave (as derived above), then there is a resonance, and the wave grows in amplitude. As with other resonance effects, the amplitude of this wave grows linearly with time.

The air-water interface is now endowed with a surface roughness due to the capillary-gravity waves, and a second phase of wave growth takes place. A wave established on the surface either spontaneously as described above, or in laboratory conditions, interacts with the turbulent mean flow in a manner described by Miles. This is the so-called critical-layer mechanism. A critical layer forms at a height where the wave speed c equals the mean turbulent flow U. As the flow is turbulent, its mean profile is logarithmic, and its second derivative is thus negative. This is precisely the condition for the mean flow to impart its energy to the interface through the critical layer. This supply of energy to the interface is destabilizing and causes the amplitude of the wave on the interface to grow in time. As in other examples of linear instability, the growth rate of the disturbance in this phase is exponential in time.

This Miles–Phillips Mechanism process can continue until an equilibrium is reached, or until the wind stops transferring energy to the waves (i.e., blowing them along) or when they run out of ocean distance, also known as fetch length.

Analog gravity models and surface gravity waves

Surface gravity waves have been recognized as a powerful tool for studying analog gravity models, providing experimental platforms for phenomena typically found in black hole physics. In an experiment, surface gravity waves were utilized to simulate phase space horizons, akin to event horizons of black holes. This experiment observed logarithmic phase singularities, which are central to phenomena like Hawking radiation, and the emergence of Fermi-Dirac distributions, which parallel quantum mechanical systems.

By propagating surface gravity water waves, researchers were able to recreate the energy wave functions of an inverted harmonic oscillator, a system that serves as an analog for black hole physics. The experiment demonstrated how the free evolution of these classical waves in a controlled laboratory environment can reveal the formation of horizons and singularities, shedding light on fundamental aspects of gravitational theories and quantum mechanics.

Clear-air turbulence

Gravity waves are generated as air is forced to rise and sink over smaller mountain ranges, creating oscillations in buoyancy and vertical wind. These waves typically remain stable in gently stratified conditions like those found over ranges such as the Appalachians. However, as terrain height increases, the resulting vertical displacement amplifies these perturbations and enhances vertical wind shear. This destabilization can be quantified using the Richardson number, the ratio of buoyancy and flow shear. If the shear becomes strong enough, the stretching forces from wind gradients are more positive than the buoyant resistance which causes air layers to roll over each other. This can be further analyzed with the Kelvin-Helmholtz instability equation. This breakdown of wave structure results in clear-air turbulence, which is smaller scale atmospheric movement typically felt on aircraft.