The multiple-scale wave equation

An analytic and numerical study of the behavior of the linear nonhomogeneou s wave. equation of the form epsilon(2)u(tt) = Delta u + f with high wave s peed (epsilon much less than 1) is carried out. This study was initially mo tivated by meteorological observations which have indicated the presence of large spatial scab gravity waves in the neighborhood of a number of summer and winter storms, mainly from visible images of ripples in clouds in sate llite photos. There is a question as to whether the presence of these waxes is caused by the nearby storms. Since the linear wave equation is an appro ximation to the full system describing pressure waves in the atmosphere, ye t is considerably more tractable, we have chosen to analyse the behavior of the linear nonhomogeneous wave equation with high wave speed. The analysis is shown to be valid in one, two, and three space dimensions. Partly becau se of the high wave speed, the solution is known to consist of behavior whi ch changes on two different time scales, one rapid and one slow. Additional ly, because of the presence of the nonhomogeneous forcing term f, we show t hat there is a component of the solution which will vary only on a very lar ge spatial scale. Since even the linearized wave equation can give rise to persistent large spatial scale waves under the right conditions, the implic ation is that certain storms could be responsible for the generation of lar ge-scale waves. Numerical simulations in one and two dimensions confirm ana lytic results. (C) 1998 Elsevier Science Ltd. All rights reserved.