Abstract
This chapter presents some important background concerning the transient wave problems. In the first part, the classical models, i.e. acoustics, linear elastodynamics and electromagnetism, are recalled. Some elements about their analysis are next given with a particular focus on their variational formulations and their well-posedness based on the Hille-Yosida theorem. Finally, their plane wave solutions, which are an important tool for understanding and analyzing wave phenomena, are derived.
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Notes
- 1.
Underlined characters indicate a vector of \({\mathbb {R}}^{d}\).
- 2.
T in upperscript denotes a transposed matrix.
- 3.
A more appropriate term would be “celerity” but “velocity” is widely used in literature.
- 4.
Double underline indicates a matrix or a tensor.
- 5.
- 6.
Similar but very complex forms can also be obtained for the anisotropic case.
- 7.
Also called primary or longitudinal wave.
- 8.
Also called secondary or transverse wave.
- 9.
Or derived from a polynomial.
- 10.
We recall that \(C^{0}(\varOmega )\) is the space of continuous functions on \(\varOmega \) and \(C^{n}(\varOmega )\), the space of functions whose derivatives are continuous to the nth-order.
- 11.
With a right-hand side, the correct equation would actually be
$$\displaystyle {\frac{1}{c^{2}}} \displaystyle {\frac{\partial ^{2} u}{\partial t^{2}}} - \varDelta u = f $$but the formulation given below is equivalent and easier to manipulate when \(f = 0\).
- 12.
The positivity of the expression under the square root comes from the positive character of the matrix \(\underline{\underline{\varepsilon _{0}}}^{-1}\).
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Cohen, G., Pernet, S. (2017). Classical Continuous Models and Their Analysis. In: Finite Element and Discontinuous Galerkin Methods for Transient Wave Equations. Scientific Computation. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-7761-2_1
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DOI: https://doi.org/10.1007/978-94-017-7761-2_1
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