Internal waves in a Lagrangian reference frame

Several recent studies of internal gravity waves have been expressed in a Lagrangian reference frame, motivated by the observation that in this frame the dispersion relation then excludes the Eulerian Doppler shifting term due to a background flow. Here the dispersion relation in a Lagrangian reference frame is explicitly derived for a background flow and background density that are slowly varying with respect to the waves, but are otherwise arbitrary functions of space and time. Two derivations are given, both yielding the same result. The first derivation involves a transformation of the dispersion relation from Eulerian to Lagrangian coordinates, while the second derivation involves a wave-packet analysis of the equations of motion directly in Lagrangian coordinates. The authors show that, although the Eulerian Doppler shifting term is removed from the dispersion relation by the transformation of the frequency when passing from an Eulerian to a Lagrangian reference frame, a dependence on the background shear is then introduced by the transformation of the wavenumber. This dependence on the background shear is the term that accounts for wave refraction in the Lagrangian frame, and its role has apparently not been fully appreciated in the aforementioned previous studies.