Exact closed-form solution of the hyperbolic equation of string vibrations with material relaxation properties taken into account

摘要

The differential equation of damped string vibrations was obtained with the finite speed of extension and strain propagation in the Hooke's law formula taken into account. In contrast to the well-known equations, the obtained equation contains the first and third time derivatives of the displacement and the mixed derivative with respect to the space and time variables. Separation of variables was used to obtain its exact closed-form solution, whose analysis showed that, for large values of the relaxation coefficient, the string return to the initial state after its escape from equilibrium is accompanied by high-frequency low-amplitude damped vibrations, which occur on the initial time interval only in the region of positive displacements. And in the limit, for some large values of the relaxation coefficient, the string return to the initial state occurs practically without any oscillatory process.