A wave solution for energy dissipation and exchange at nonclassical boundaries of a traveling string

•The governing equation and reflection equations for a travelling string with classical and nonclassical boundaries are given.•A novel reflective wave superposition method is proposed to obtain analytical solutions of vibration for travelling string.•A new calculation method for vibration energy of travelling string is put forward.•The laws of energy change in boundaries under different motion conditions and boundary conditions are revealed.