An Asymptotic Theory for a Class of Initial-Boundary Value Problems for Weakly Nonlinear Wave Equations with an Application to a Model of the Galloping Oscillations of Overhead Transmission Lines

Asymptotics for a system of nonlinearly coupled wave equations with an application to the galloping oscillations of overhead transmission lines

“…Moreover, the asymptotic results presented in this paper can be seen as a generalization of the asymptotic results obtained in [7], This paper, being an attempt to contribute to the foundations of the asymptotic methods for (systems of) weakly nonlinear hyperbolic partial differential equations, is organized as follows. In Sec.…”

“…4 it follows from an aero-elastic analysis that this initial-boundary value problem may be regarded as a model which describes the growth of wind-induced oscillations of overhead transmission lines in the vertical and in the horizontal direction. In fact this initial-boundary value problem is an extension of a model (describing only the vertical displacements of the transmission lines) which has beenfpostulated in the early seventies in [11,12] and which recently has been derived in [7], From a practical point of view it is interesting to investigate the vertical as well as the horizontal displacements of the transmission line, since one or both of the displacements may give rise to conductor damage due to impact of conductor lines or due to flashover as a result of a phase difference between conductor lines, for which the mutual distance has become too small.…”

“…The asymptotic theory in [6,7] and the asymptotic theory presented in this paper can be regarded as an extension of the asymptotic theory for ordinary differential equations as for instance described in [1,2,4,14]. Moreover, the asymptotic results presented in this paper can be seen as a generalization of the asymptotic results obtained in [7], This paper, being an attempt to contribute to the foundations of the asymptotic methods for (systems of) weakly nonlinear hyperbolic partial differential equations, is organized as follows.…”

“…The asymptotic theory presented here implies the well-posedness in the classical sense of the initial-boundary value problem (1.1)-(1.3) as well as the asymptotic validity of formal approximations. In this paper formal approximations are defined to be vectorvalued functions satisfying the differential equations and the initial conditions up to some order depending on the small parameter s. For scalar-valued functions similar asymptotic theories have been developed in [6] for an initial-boundary value problem for the weakly semi-linear telegraph equation *Received January 22, 1988. ©1989 Brown University 197 m« -uxx + u + ef{x,t,u\e) -0, and in [7] for an initial-boundary value problem for the weakly nonlinear wave equation u" -uxx + ef(x,t,u,ut,ux\e) = 0. Both types of equations were considered subject to the initial values u(x, 0;e) = Mo(x;e) and ut(x, 0;e) = u\{x\e) and the boundary values u(0,t\e) = u(ji,t\e) = 0.…”

“…Using a two-time-scale perturbation method, as for instance successfully used in [3,6,7,9,10], an asymptotic approximation of the solution of the aforementioned initial-boundary value problem will be constructed. In some sense it is remarkable that the two-time-scale perturbation method as developed in [3,10] and applied in [3,6,7,9,10] to an initial-boundary value problem for a single perturbed wave equation may also be used for an initial-boundary value problem for the aforementioned system of perturbed wave equations.…”