How can we manage repairing a broken finite vibrating string? Solutions to the problem II

The current research deals with the IBVP (JODEA, 31 (2) (2023), 89 – 114) for the 1D homogeneous wave equation with discontinuous piece-wise constant coefficient posed to study three approaches to implement the transmission conditions (called ‘repairing’ the broken composite string): 1) separation of variables; 2) Laplace transformation; 3) applying the energy rate equation. The second approach (JODEA, 33 (1) (2025), 144 – 181) has been currently completed and the solution has been shown to demonstrate an evidence of a weak solution, as it was previously noticed and proved in case of the IBVP for the 1D non-homogeneous wave equation with constant coefficient (JODEA, 33 (2) (2025), 200 – 221).