Existence and uniqueness for wave propagation in inhomogeneous elastic solids
The problem of wave reflection and transmission in a layer sandwiched between two half-spaces is not a standard initial-boundary value problem because we do not known a priori a surface where the solution is known. For this problem, in one spatial dimension and when the inhomogeneity is taken into account, the authors are able to prove uniqueness of regular solutions by energy method, and to investigate the existence using Fourier analysis. The key idea is to write the boundary conditions in a way which directly accounts for the unknown waves propagating away from the scatterer, and to require the finiteness of the speed of propagation. In the particular homogeneous case this method provides also an exact solution of the problem in closed form. The authors note an analogy of this problem with some existence and uniqueness problems investigated by \textit{P. Bassanini} in electromagnetism and seismology [Wave Motion 8, 311--319 (1986; Zbl 0585.73178); Atti Semin. Mat. Fis. Univ. Modena 35, 335--356 (1987; Zbl 0673.35071)]. We point out that very important studies on the subject have been done by \textit{C. S. Morawetz} and coworkers [Proc. R. Ir. Acad., Sect. A 72, 113--120 (1972; Zbl 0239.35057); Comput. Math. Appl. 7, 319--331 (1981; Zbl 0465.35083); Math. Comput. 52, No. 186, 321-338 (1989; Zbl 0692.65068)].
scientific article; zbMATH DE number 2099238
scientific article