Theory and application of equivalent transformation relationships between plane wave and spherical wave
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Based on the governing equations and the equivalent models, we propose an equivalent transformation relationships between a plane wave in a one-dimensional medium and a spherical wave in globular geometry with radially inhomogeneous properties. These equivalent relationships can help us to obtain the analytical solutions of the elastodynamic issues in an inhomogeneous medium. The physical essence of the presented equivalent transformations is the equivalent relationships between the geometry and the material properties. It indicates that the spherical wave problem in globular geometry can be transformed into the plane wave problem in the bar with variable property fields, and its inverse transformation is valid as well. Four different examples of wave motion problems in the inhomogeneous media are solved based on the presented equivalent relationships. We obtain two basic analytical solution forms in Examples I and II, investigate the reflection behavior of inhomogeneous half-space in Example III, and exhibit a special inhomogeneity in Example IV, which can keep the traveling spherical wave in constant amplitude. This study implies that our idea makes solving the associated problem easier.
Keywordselastodynamic issue equivalent transformation relationship governing equation inhomogeneous medium
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This work is supported by the Scientific Research Fund of Institute of Engineering Mechanics, China Earthquake Administration (Grant No. 2017QJGJ06), the National Science and Technology Pillar Program (Grant No. 2015BAK17B06), the Earthquake Industry Special Science Research Foundation Project (Grant No. 201508026-02), the Fundamental Research Funds for the Central Universities (Grant No. HEUCF170202) and the program for Innovative Research Team in China Earthquake Administration.
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