Uniformly Valid Asymptotics for Carrier's Mathematical Model of String Oscillations

Abstract

In the paper, an asymptotic analysis of G.F. Carrier's mathematical model of string oscillation is presented. The model consists of a system of two nonlinear second order partial differential equations and periodic initial conditions. The longitudinal and transversal string oscillations are analyzed together when at the initial moment of time the system's solutions have amplitudes proportional to a small parameter. The problem is reduced to a system of two weakly nonlinear wave equations. The resonant interaction of periodic waves is analyzed. An uniformly valid asymptotic approximation in the long time interval, which is inversely proportional to the small parameter, is constructed. This asymptotic approximation is a solution of averaged along characteristics integro-differential system. Conditions of appearance of combinatoric resonances in the system have been established. The results of numerical experiments are presented.

Keywords:

• asymptotic analysis
• averaging
• nonlinear waves
• resonances

AMS Subject Classification:

• 35L60
• 41A60
• 74J30