On modelling of long waves in the Lagrangian and Eulerian description

Long wave equations derived by means of the Lagrangian and Eulerian methods are discussed herein. For both approaches differences are pointed out as well as a transition from one to another description is presented. First, selected examples of the small-amplitude wave theory are given. They concern cases of a shallow-water area and a swash zone. Some analogies between the linear Lagrangian wave and the 2nd Stokes' wave are shown as well. Finally, a simple finite-amplitude model elaborated in the Lagrangian manner is presented and tentatively compared with its Eulerian counterpart. Some discrepancies in the solution for both methods are also noticed in this case.