Waves are common in astrophysical fluids. They define the morphology of outflows, which are related to accretion disks surrounding compact objects. Waves often appear spontaneously, in response to instabilities commonly associated with shear flows. The canonical example of a shear-driven instability is the Kelvin–Helmholtz instability. Even in the absence of shear, stratified flows with different densities can become unstable in the presence of acceleration and/or gravity – the Rayleigh–Taylor instability. Such instabilities do not fundamentally depend on compressibility, and hence they are appropriately discussed in the approximation of incompressible flows. In rotating fluids, instabilities represent a tendency to redistribute angular momentum leading towards a lower energy state. These, likewise, can be studied in the limit of incompressible flows.

A torus around a black hole is a fluid bound to a central potential well. The fluid in the torus is a rotational shear flow, which is generally more rapidly rotating on the inner face than on the outer one. In particular, when driven by a spin-connection to the black hole, the inner face develops a super-Keplerian state, while the outer one develops a sub-Keplerian state by angular momentum loss in winds. The induced effective gravity – centrifugal on the inner face and centripetal on the outer face – allows surface waves to appear very similar to water waves in channels of finite depth.