The reflection coefficient of a fractional reflector

Abstract

This paper considers the question of characterizing the behavior of waves reflected by a fractional singularity of the wave speed profile, i.e., of the form \[ c(x_1, x_2, x_3) = c_0 \left(1 + \left( \frac{x_1}{\ell}\right)_{+}^\alpha \right)^{-1/2}, \] for $\alpha > 0$ not necessarily integer. 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 $\ell \omega / c_0$ regime as \[ R = \frac{\Gamma(\alpha + 1)}{(2 i)^{\alpha + 2}} \left( \frac{c_0}{\ell \omega} \right)^{\alpha} + \mbox{lower order terms.} \] 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 nondimensional variable $\ell \omega / c_0$ is modified to account for the angle of incidence. The asymptotic analysis now 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.