| Summary: | Transverse vibrations of a semi-bounded string consisting of different materials are considered. The mathematical model is a homogeneous wave equation with piecewise constant coefficients. As a first step, we investigate the solution of this equation with zero Cauchy data. The existence and uniqueness of the generalized solution of the problem is proved and its properties are analyzed. In the first quadrant of the plane, two domains are distinguished, in each of which the solution is given by a separate formula. In general, the solution is a continuous function, and the conditions for matching its first partial derivatives, which are a consequence of Hooke's laws and conservation of momentum, are satisfied at the boundary between the domains. The specificity of the obtained conclusions is noted, in particular, the zone of propagation of oscillations and the zone of their absence are indicated. The results are constructive and can serve as a basis for the creation of a numerical algorithm. The importance of such problems is caused by their use in the theory of sensing inhomogeneous media by physical signals.
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