Representation of Wave Energy of a Rotating Flow in Terms of the Dispersion Relation

This chapter provides an outline of the link of the wave energy, in the context of Kelvin waves confined in a circular cylinder, with a derivative of the dispersion relation. The chapter begins with a concise description of derivation of the wave energy using the Lagrangian displacement field. The authors briefly recall the Kelvin waves, a family of neutrally stable linear oscillations, in a confined geometry. They take, as a basic flow, the rigid-body rotation of an inviscid incompressible fluid confined in a cylinder of circular cross-section and of unit radius. The chapter establishes the relation of the formula of the wave energy with a derivative of the dispersion relation with respect to the frequency. Finally, it discusses the derivation of the mean flow induced by the nonlinear interactions of Kelvin waves and their utility for deriving the amplitude equations to third order.