Abstract
We review discuss derivation and then write down equations of weak nonlinear wavefront and shock front.
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Notes
- 1.
Generally, in any book on wave propagation we use an intuitive idea of a wave and wavefront. A mathematical definition is available in Sect. 3.2.1 of [42], according to which a wave is an approximate concept in high-frequency approximation—see the next section. This precise definition was first given in 1977 [36].
- 2.
Note that only \(d{-1}\) components of unit normal \(\varvec{n}\) are independent.
- 3.
Unlike a wavefront, which is an approximate concept in high-frequency approximation, a shock in a solution of a system of conservation laws is an exact phenomenon and high-frequency approximation is exactly satisfied.
- 4.
We shall explain later the reason of using the word ‘sufficient’ for just one more unknown.
- 5.
In this section, for an element the arc length \(d\ell \) of a linear ray, we have \(d\ell = a_0 dt\). A proof of the result \(\frac{1}{\mathcal {A}} \frac{d \mathcal {A}}{d \ell } = - \frac{1}{2}{\varOmega }\) for a general ray system is given in Sect. 7.1.
- 6.
\(\tilde{w}\) in [32, 42] is non-dimensional.
- 7.
- 8.
Note that the symbols \(\Omega _{t}\) and \({\varOmega }\) represent respectively a surface in \(I\!\!R^{d}\) (here d =3) and mean curvature of \(\Omega _t\).
- 9.
This theorem is reproduced from Sect. 3 of [45] with some changes. The author thanks Indian Academy Sciences for kindly giving permission for reproduction.
- 10.
See (5.50) below, \(\mu _1\) has dimension of \(\frac{a_0}{L}\) and \(\mu _2\) has dimension of \(\frac{a^2_0}{L^2}\).
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Prasad, P. (2017). Equations of Nonlinear Wavefront and Shock Front. In: Propagation of Multidimensional Nonlinear Waves and Kinematical Conservation Laws. Infosys Science Foundation Series(). Springer, Singapore. https://doi.org/10.1007/978-981-10-7581-0_5
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DOI: https://doi.org/10.1007/978-981-10-7581-0_5
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