Abstract
In this paper we consider an acoustic problem of wave propagation through a discontinuous medium. The problem is reduced to the dissipative wave equation with distributional dissipation. We show that this problem has a so-called very weak solution, we analyse its properties and illustrate the theoretical results through some numerical simulations by approximating the solutions to the full dissipative model for a particular synthetic piecewise continuous medium. In particular, we discover numerically a very interesting phenomenon of the appearance of a new wave at the singular point. For the acoustic problem this can be interpreted as an echo effect at the discontinuity interface of the medium.
Highlights
This work is devoted to the investigation of the 1D wave propagation through a medium with positive piecewise regular density and wave speed functions
For these non-smooth data we show that the problem has a so-called ‘very weak’ solution. This notion has been introduced in [9] in the analysis of second order hyperbolic equations, and in [24,26] it was applied to show the well-posedness of the wave equations for the Landau Hamiltonian with irregular electro-magnetic fields
If the Cauchy data (u0, u1) is in the Sobolev spaces Hs+1 × Hs, s ≥ 0, the Cauchy problem (1.4) has a very weak solution of order s; the very weak solution is unique in an appropriate sense;
Summary
This work is devoted to the investigation of the 1D wave propagation through a medium with positive piecewise regular density and wave speed functions For these non-smooth data we show that the problem has a so-called ‘very weak’ solution. In this paper we are interested in the problem of existence of solutions of the model equation (1.3) in the situation when the density ρ and the wave speed c of the medium are irregular. These will be the assumptions for our analysis, namely, we assume that the product b = ζ = ρc of the density and the wave speed of the medium is an increasing piecewise continuous function If they are smooth (or at least C1), this is a natural physical assumption (see [2]).
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