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Mathematical and physical interpretation of the solution to the initial–boundary value problem in linear hyperbolic thermoelasticity theory
Author(s) -
Gawinecki J.A.,
Sikorska B.,
Nakamura G.,
Rafa J.
Publication year - 2007
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.200610347
Subject(s) - thermoelastic damping , mathematical analysis , boundary value problem , mathematics , gravitational singularity , boundary (topology) , type (biology) , domain (mathematical analysis) , fourier transform , plane (geometry) , classical mechanics , physics , geometry , thermal , ecology , meteorology , biology
A new system of equations describing a thermoelastic medium in three‐dimensional space is considered. Such a system also describes the propagation of heat with finite speed. We consider the initial‐boundary value problem (with the boundary condition of Lamb's type) for this system of equations. Using the modified Cagniard‐de Hoop method we construct an explicit solution of this problem. Based on the constructed solution of the above mentioned problem, we describe the type of waves which propagate in the thermoelastic medium and the domain of their propagation. The main role in our considerations is played by the physical interpetation of the singular and branch points of the integrands in the corresponding Fourier integrals. The values of the singular and branch points and their localization on the complex plane are responsible for the velocities of the thermoelastic waves and for the type of such waves. The analysis of these singularities has allowed for the physical intepretation of the solution. Namely, new kind of waves in a thermoelastic medium have been discovered: frontal waves of the Schmidt's type surface waves of the Rayleigh's type We have determined the exact geometric description of the wave surfaces and the character of their origin. We provided the quantitative characterization of the field of temperature, as well.