On reflected waves in a semi-infinite string due to a boundary condition with a cubic nonlinearity

Abstract

In this paper, initial-boundary value problems (IBVPs) for a semi-infinite string with a tuned-mass-damper (TMD) system attached at one end are studied. While previous studies have focused primarily on the linear behavior of springs, we extend the analysis to include cubic nonlinearity. Four types of TMD system are considered, that is, a dashpot-linear spring, a mass-dashpot-linear spring, a dashpot-nonlinear spring, and a mass-dashpot-nonlinear spring, to assess wave reflections under these configurations. A key contribution of this research is that, rather than assuming predefined forms for the reflected wave, we derive the reflection shapes directly from calculations, offering new insights into wave dynamics. The D’Alembert formula is used to describe the general solution of the wave equation, accounting for the string’s initial velocity and displacement. For the nonlinear cases, the Multiple Scales Perturbation (MSP) method is used to approximate solutions. Our results demonstrate that the mass, spring, and damper coefficients strongly influence the wave reflections. Additionally, adjusting the damping coefficient to small values reveals a completely different behavior of the reflected waves compared to large values. Numerical simulations using a fourth-order Runge-Kutta (RK4) method and a central finite difference scheme support the analytical results. Energy dissipation is studied for all scenarios, confirming that the solutions remain bounded over large timescales.