Abstract
Wave motion in a 3-D elastic medium can be expressed in a very elegant way by the use of tensor notation and differential operators. Because of this the resulting form of the equations of motion becomes independent of the selection of the coordinate system. This makes such a form of the equations of motion universal and more suitable for further studies.
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Notes
- 1.
These conditions concern the smoothness of the vector field under consideration as well as its fast decay at infinity.
- 2.
In a coordinate system different from the Cartesian coordinate system, i.e. cylindrical, spherical or others, different formulae must be used to calculate the components of the displacement vector \(\textbf{u}\).
- 3.
The Helmholtz equation, which describes the process of diffusion or wave propagation, presents the eigenvalue problem for the Laplace operator, i.e. \(\nabla ^{2}\phi = - k^{2}\phi \), and is an important equation in many fields of science.
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Żak, A. (2024). Waves in a 3-D Elastic Space. In: A Finite Element Approach for Wave Propagation in Elastic Solids. Lecture Notes on Numerical Methods in Engineering and Sciences. Springer, Cham. https://doi.org/10.1007/978-3-031-56836-7_3
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DOI: https://doi.org/10.1007/978-3-031-56836-7_3
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