Abstract
The contribution A c(z, t) to the asymptotic behavior of the propagated plane wave field A(z, t) that is due to the presence of any simple pole singularities of the spectral function \(\tilde {u}(\omega -\omega _c)\), where.
“A disturbance traveling with the velocity of light arrives first followed by transients which are eventually overwhelmed by the main signal.” D. S. Jones, The Theory of Electromagnetism (1964).
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Notes
- 1.
The value of θ s depends upon which Olver-type path is chosen for P(θ). If that path is taken to lie along the path of steepest descent that passes through the saddle point nearest the pole, then the value of θ s is specified by the relation Υ(ω SP, θ s) = Υ(ω p, θ s).
- 2.
The asymptotic dominance of the saddle point contribution over the pole contribution at θ = θ s is guaranteed by the fact that P(θ s) is an Olver-type path.
- 3.
The numerically determined wave field A(z, t) in Fig. 14.2, as well as that used in obtaining the results presented in Figs. 14.3, 14.4 and 14.5, has been slightly shifted in space-time by a fixed amount (Δθ = 0.060) that is determined by the requirement that the peak amplitude point in A(z, t) for a sufficiently large propagation distance occurs at the same space-time point (θ = θ 0) as that described by the asymptotic description of the Brillouin precursor.
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- 5.
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Oughstun, K.E. (2019). Evolution of the Signal. In: Electromagnetic and Optical Pulse Propagation . Springer Series in Optical Sciences, vol 225. Springer, Cham. https://doi.org/10.1007/978-3-030-20692-5_5
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