TY - JOUR

T1 - The Maslov integral representation of slowly varying dispersive wavetrains in inhomogeneous moving media

AU - Brown, Michael G.

N1 - Funding Information:
Support for this work was provided by the Division of Ocean Sciences of the US National Science Foundation, Code 321OA of the US Office of Naval Research, and the Norwegian Research Council.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2000/9

Y1 - 2000/9

N2 - The Maslov integral representation of slowly varying surface gravity wavetrains is developed, allowing for smooth but otherwise arbitrary variations of both bathymetry and horizontal currents. Although we focus on the surface gravity wave problem, the results presented can be applied - and are presented in a form which facilitates their application - to any type of scalar small amplitude dispersive wave motion in a slowly varying environment, with or without background flow. The Maslov integral provides an asymptotically valid solution, on the wavelength scale, to the initial value problem under conditions in which exact solutions are unavailable owing to nonseparability of the equations of motion. Caustics of arbitrary complexity are properly treated. Away from caustics, stationary phase evaluation of the Maslov integral reduces it to a superposition of locally plane, slowly varying dispersive wavetrains that conserve wave action. An important step in the development of the Maslov integral is the geometric construction of exact solutions to the time-dependent wave action conservation equation. It is shown that in an unbounded homogeneous environment the Maslov integral reduces as a special case to the usual Fourier integral solution to the initial value problem.

AB - The Maslov integral representation of slowly varying surface gravity wavetrains is developed, allowing for smooth but otherwise arbitrary variations of both bathymetry and horizontal currents. Although we focus on the surface gravity wave problem, the results presented can be applied - and are presented in a form which facilitates their application - to any type of scalar small amplitude dispersive wave motion in a slowly varying environment, with or without background flow. The Maslov integral provides an asymptotically valid solution, on the wavelength scale, to the initial value problem under conditions in which exact solutions are unavailable owing to nonseparability of the equations of motion. Caustics of arbitrary complexity are properly treated. Away from caustics, stationary phase evaluation of the Maslov integral reduces it to a superposition of locally plane, slowly varying dispersive wavetrains that conserve wave action. An important step in the development of the Maslov integral is the geometric construction of exact solutions to the time-dependent wave action conservation equation. It is shown that in an unbounded homogeneous environment the Maslov integral reduces as a special case to the usual Fourier integral solution to the initial value problem.

UR - http://www.scopus.com/inward/record.url?scp=0009833643&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0009833643&partnerID=8YFLogxK

U2 - 10.1016/s0165-2125(00)00046-9

DO - 10.1016/s0165-2125(00)00046-9

M3 - Article

AN - SCOPUS:0009833643

VL - 32

SP - 247

EP - 266

JO - Wave Motion

JF - Wave Motion

SN - 0165-2125

IS - 3

ER -