Can a Finite Degenerate ‘String’ Hear Itself The Exact Solution to a Simplified IBVP

Анотація

The separation of variables based solution to a simplified (compared to that published earlier in JODEA, 28 (1) (2020), 1 – 42) initial boundary value problem for a 1D linear degenerate wave equation, posed in a space-time rectangle, has been presented in a fully complete form. Degeneracy of the equation is due to vanishing its coefficient in an interior point of the spatial segment being the side of the rectangle. For the sake of convenience, the solution is interpreted as a vibrating ‘string’. The solution obtained in the case of weak degeneracy is smooth and bounded, whereas that in the case of strong degeneracy is piece-wise smooth, piece-wise continuous and unbounded in a neighborhood of the point of degeneracy, nevertheless being satisfied some regularity conditions, including square-integrability. In both cases the travelling waves pass through the point of degeneracy, and this phenomenon is referred to as an ability of the ‘string’ to hear itself. The total energy of the ‘string’ is shown to conserve in both cases of degeneracy, provided the ends of the ‘string’ are fixed, though the above vibrating ‘string’ analogy fails in the case of strong degeneracy. The total energy conservation implies the uniqueness of the solution to the problem in both cases of degeneracy