Abstract
In this chapter, we start from the Cauchy problems for second-order partial differential equations and first-order systems of partial differential equations, classifying them in elliptic, parabolic, and hyperbolic. Then, we identify the characteristic surfaces of Cauchy’s problem with the singular surfaces of ordinary waves of Hadamard. The acoustic tensor is defined and it is proved that the normal speeds of ordinary waves are equal to the square of the eigenvalues of the acoustic tensor. Further, it is proved that the evolution of the singular surfaces is given by the eikonal equation. Finally, after defining shock waves, we determine the Rankine–Hugoniot conditions. After a section of exercises, the programs PdeEqClass, PdeSysClass, WavesI, and WavesII, written with MathematicaⓇ, are described. These programs allow to classify equations and systems and determine the normal speeds of ordinary waves.
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- 1.
It is clear that this procedure is applicable when a solution is known, for instance when u 0(x) = const.
- 2.
The equation is semi-linear if it has the form
$$\displaystyle{a_{ij}(\mathbf{x}) \frac{\partial ^{2}u} {\partial x_{i}\partial x_{j}} = h(\mathbf{x},u,\nabla u),\qquad i,j = 1,\cdots \,,n,\quad \mathbf{x} \in \Omega,}$$and linear if h(x, u, ∇u) is a linear function of u and ∇u.
- 3.
Remember that the time occupies the first place, i.e., x 1 ≡ t, in the list of the independent variables.
- 4.
We remark that in MathematicaⓇ [69] the functional dependence is expressed by brackets. Therefore, instead of p = p ρ we have to write p = p ρ.
- 5.
We recall that the time always occupies the first position in the list of the variables, i.e., x 1 ≡ t.
References
A. Jeffrey, Quasilinear Hyperbolic Systems and Waves (Pitman Publishing, London–San Francisco–Melbourne, 1976)
P.D. Lax, Hyperbolic systems of conservation laws. Comm. Pure Appl. Math., vol. 10 (Wiley, New York, 1957)
S. Wolfram, Mathematica ®;. A System for Doing Mathematics by Computer (Addison-Wesley Redwood City, California, 1991)
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© 2014 Springer Science+Business Media New York
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Romano, A., Marasco, A. (2014). Wave Propagation. In: Continuum Mechanics using Mathematica®. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-1604-7_8
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DOI: https://doi.org/10.1007/978-1-4939-1604-7_8
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