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$c(x_1,x_2,x_3)=c_0{(1+{\left(\frac{x_1}{\ell }\right)}_{+}^{\alpha })}^{-1/2},$ 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$R=\frac{\Gamma (\alpha +1)}{{\left(2i\right)}^{\alpha +2}}{\left(\frac{c_0}{\ell \omega }\right)}^{\alpha }+\text{lower order terms.}$ 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 −α.

In practice, the knowledge of the dependency of both the amplitude and the phase of R on ω and α might be able to inform the kind of signal processing needed to characterize the fractional nature of reflectors, for instance in geophysics.