Abstract
We targeted a new kind of inverse problem for the wave-diffusion equation. Namely, using the given data we have found not only a solution of the wave-diffusion equation satisfying certain boundary and initial conditions but also an unknown time-moment starting from which the wave process transmitted into the diffusion process. In the practical point of view, even finding the first such time-moment seems very interesting. Our main tool of investigation is a spectral expansion method. In addition, we will use the solution of the corresponding Cauchy problem for differential equation in time-variable.