A schematic diagram of the base state of the system. The flow under investigation represents a small perturbation away from this state. While the base state is parallel, the perturbation velocity has components in both directions.
The equation is derived by solving a linearized version of the Navier–Stokes equation, in presence of gravity and a mean density gradient (with gradient-length ), for the perturbation velocity field
where is the unperturbed or basic flow. The perturbation velocity has the wave-like solution (real part understood). Using this knowledge, and the streamfunction representation for the flow, the following dimensional form of the Taylor–Goldstein equation is obtained:
where denotes the Brunt–Väisälä frequency. The eigenvalue parameter of the problem is . If the imaginary part of the wave speed is positive, then the flow is unstable, and the small perturbation introduced to the system is amplified in time.