Editing Internal wave (section)

==Buoyancy, reduced gravity and buoyancy frequency==
According to [[Archimedes' principle]], the weight of an immersed object is reduced by the weight of fluid it displaces. This holds for a fluid parcel of density <math>\rho</math> surrounded by an ambient fluid of density <math>\rho_0</math>. Its weight per unit volume is <math>g(\rho-\rho_0)</math>, in which <math>g</math> is the acceleration of gravity. Dividing by a characteristic density, <math>\rho_{00}</math>, gives the definition of the reduced gravity:

:<math>g^\prime \equiv g \frac{\rho-\rho_0}{\rho_{00}}</math>

If <math>\rho>\rho_0</math>, <math>g^\prime</math> is positive though generally much smaller than <math>g</math>. Because water is much more dense than air, the displacement of water by air from a surface [[gravity wave]] feels nearly the full force of gravity (<math>g^\prime \sim g</math>). The displacement of the [[thermocline]] of a lake, which separates warmer surface from cooler deep water, feels the buoyancy force expressed through the reduced gravity. For example, the density difference between ice water and room temperature water is 0.002 the characteristic density of water. So the reduced gravity is 0.2% that of gravity. It is for this reason that internal waves move in slow-motion relative to surface waves.

Whereas the reduced gravity is the key variable describing buoyancy for interfacial internal waves, a different quantity is used to describe buoyancy in continuously stratified fluid whose density varies with height as <math>\rho_0(z)</math>. Suppose a water column is in [[hydrostatic balance|hydrostatic equilibrium]] and a small parcel of fluid with density <math>\rho_0(z_0)</math> is displaced vertically by a small distance <math>\Delta z</math>. The [[buoyancy|buoyant]] restoring force results in a vertical acceleration, given by<ref name=Tritton>{{harv|Tritton|1990|pages = 208–214}}</ref><ref name=Sutherland>(Sutherland 2010, pp 141-151)</ref>

:<math>\frac{d^2 \Delta z}{dt^2} = - g^\prime = - g (\rho_0(z_0)-\rho_0(z_0+\Delta z))/\rho_0(z_0) \simeq - g \left(-\frac{d\rho_0}{dz} \Delta z\right)/\rho_0(z_0)</math>

This is the spring equation whose solution predicts oscillatory vertical displacement about <math>z_0</math> in time about with frequency given by the [[Brunt–Väisälä frequency|buoyancy frequency]]:

:<math> N = \left(-\frac{g}{\rho_0} \frac{d\rho_0}{dz}\right)^{1/2}.</math>

The above argument can be generalized to predict the frequency, <math>\omega</math>, of a fluid parcel that oscillates along a line at an angle <math>\Theta</math> to the vertical:

:<math>\omega = N \cos\Theta</math>.

This is one way to write the dispersion relation for internal waves whose lines of constant phase lie at an angle <math>\Theta</math> to the vertical. In particular, this shows that the [[Brunt–Väisälä frequency|buoyancy frequency]] is an upper limit of allowed internal wave frequencies.