Internal waves are gravity wave

Gravity wave

In fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media which has the restoring force of gravity or buoyancy....

s that oscillate

Oscillation

Oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. Familiar examples include a swinging pendulum and AC power. The term vibration is sometimes used more narrowly to mean a mechanical oscillation but sometimes...

 within, rather than on the surface of, a fluid medium. They are one of many types of wave motion in stratified fluids (another example being Lee waves

Lee waves

In meteorology, lee waves are atmospheric standing waves. The most common form is mountain waves, which are atmospheric internal gravity waves...

). A simple example is a wave propagating on the interface between two fluids of different densities

Density

The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

, such as oil and water. Internal wave motions are ubiquitous in both the ocean and atmosphere, where they create wave cloud

Wave cloud

A wave cloud is a cloud form created by atmospheric internal waves.-Formation:The atmospheric internal waves that form wave clouds are created as stable air flows over a raised land feature such as a mountain range, and can form either directly above or in the lee of the feature. As an air mass...

s. Nonlinear solitary internal waves are called soliton

Soliton

In mathematics and physics, a soliton is a self-reinforcing solitary wave that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium...

s.
Suppose a water column is in hydrostatic equilibrium and displace a small packet of fluid with density vertically by a distance .