The Maslov integral representation of slowly varying dispersive wavetrainsin inhomogeneous moving media

The Maslov integral representation of slowly varying surface gravity wavetr ains is developed, allowing for smooth but otherwise arbitrary variations o f both bathymetry and horizontal currents. Although we focus on the surface gravity wave problem, the results presented can be applied - and are prese nted in a form which facilitates their application - to any type of scalar small amplitude dispersive wave motion in a slowly varying environment, wit h or without background flow. The Maslov integral provides an asymptoticall y valid solution, on the wavelength scale, to the initial value problem und er conditions in which exact solutions are unavailable owing to nonseparabi lity of the equations of motion. Caustics of arbitrary complexity are prope rly treated. Away from caustics, stationary phase evaluation of the Maslov integral reduces it to a superposition of locally plane, slowly varying dis persive wavetrains that conserve wave action. An important step in the deve lopment of the Maslov integral is the geometric construction of exact solut ions to the time-dependent wave action conservation equation. It is shown t hat in an unbounded homogeneous environment the Maslov integral reduces as a special case to the usual Fourier integral solution to the initial value problem. (C) 2000 Elsevier Science B.V. All rights reserved.