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Preprints, Working Papers, ... Year : 2023

The reflection coefficient of a fractional reflector


This paper considers the question of characterizing the behavior of waves reflected by a fractional singularity of the wave speed profile $c(x_1)=c_0(1+((\frac{x_1}{l_0})_+)^{\alpha})^{-\frac12}$. We first focus on the case of one spatial dimension and a harmonic time dependence. We define the reflection coefficient $R$ from a limiting absorption principle. We provide an exact formula for R in terms of the solution to a Volterra equation. We obtain the asymptotic limit of this coefficient in the large $\frac{l_0\omega ω}{c_0}$ regime as $R= \frac{\Gamma(\alpha+1)}{(2i)^{\alpha+2}}\theta+o(1)$ The amplitude is proportional to $\omega^{-\alpha}$, and the phase rotation behavior is obtained from the $i^{−{\alpha+2}}$ factor. The proof method does not rely on representing the solution by special functions, since $\alpha > 0$ is general. In the multi-dimensional layered case, we obtain a similar result where the nondimen- sional variable lω/c0 is modified to account for the angle of incidence. The asymptotic analysis requires the waves to be non-glancing. The resulting reflection coefficient can now be interpreted as a Fourier multiplier of order $-\alpha$ In practice, the knowledge of the dependency of both the amplitude and the phase of $R$ on $\omega$ and $\alpha$ might be able to inform the kind of signal processing needed to characterize the fractional nature of reflectors, for instance in geophysics.
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Dates and versions

hal-04093250 , version 1 (09-05-2023)


Attribution - CC BY 4.0


  • HAL Id : hal-04093250 , version 1


Laurent Demanet, Olivier Lafitte. The reflection coefficient of a fractional reflector. 2023. ⟨hal-04093250⟩
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