2023/05/06 by Demanet, Laurent, Lafitte, Olivier
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.2305.04071
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(x1, x2, x3) = c0 (1 + ( (x1)/(ℓ))+α)-1/2, for α> 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 ℓ ω/ c0 regime as R = \fracΓ(α+ 1)(2 i)α+ 2 ( (c0)/(ℓ ω) )α + lower order terms. The amplitude is proportional to ω-α, and the phase rotation behavior is obtained from the i-(α+2) factor. The proof method does not rely on representing the solution by special functions, since α> 0 is general. In the multi-dimensional layered case, we obtain a similar result where the nondimensional variable ℓ ω/ c0 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.