In fluid dynamics, gravity waves are waves in a fluid medium or at the interface between two media when the force of gravity or buoyancy tries to restore equilibrium.

An example of such an interface is that between the atmosphere and the ocean, which gives rise to wind waves.

A gravity wave results when fluid is displaced from a position of equilibrium.

The period of wind-generated gravity waves on the free surface of the Earth's ponds, lakes, seas and oceans are predominantly between 0.3 and 30 seconds (corresponding to frequencies between 3 Hz and .03 Hz).

[2] In the Earth's atmosphere, gravity waves are a mechanism that produce the transfer of momentum from the troposphere to the stratosphere and mesosphere.

Gravity waves are generated in the troposphere by frontal systems or by airflow over mountains.

But as the waves reach more rarefied (thin) air at higher altitudes, their amplitude increases, and nonlinear effects cause the waves to break, transferring their momentum to the mean flow.

This transfer of momentum is responsible for the forcing of the many large-scale dynamical features of the atmosphere.

For example, this momentum transfer is partly responsible for the driving of the Quasi-Biennial Oscillation, and in the mesosphere, it is thought to be the major driving force of the Semi-Annual Oscillation.

Thus, this process plays a key role in the dynamics of the middle atmosphere.

The gravity wave represents a perturbation around a stationary state, in which there is no velocity.

Thus, the perturbation introduced to the system is described by a velocity field of infinitesimally small amplitude,

Because the fluid is assumed incompressible, this velocity field has the streamfunction representation where the subscripts indicate partial derivatives.

Next, because of the translational invariance of the system in the x-direction, it is possible to make the ansatz where k is a spatial wavenumber.

Thus, the problem reduces to solving the equation We work in a sea of infinite depth, so the boundary condition is at

is given by the Young–Laplace equation: where σ is the surface tension and κ is the curvature of the interface, which in a linear approximation is Thus, However, this condition refers to the total pressure (base+perturbed), thus (As usual, The perturbed quantities can be linearized onto the surface z=0.)

A wave in which the group and phase velocities differ is called dispersive.

When the water depth is h, 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.

As with other resonance effects, the amplitude of this wave grows linearly with time.

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.

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.

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.

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.

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.

[7] 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.