The Combinatorics of Scattering in Layered Media

Abstract

Reflection and transmission of waves in piecewise constant layered media are important in various imaging modalities and have been studied extensively. Despite this, no exact time domain formulas for the Green's functions have been established. Indeed, there is an underlying combinatorial obstacle: the analysis of scattering sequences. In the present paper we exploit a representation of scattering sequences in terms of trees to solve completely the inherent combinatorial problem, and thereby derive new, explicit formulas for the reflection and transmission Green's functions.

A scattering‐based algorithm for wave propagation in one dimension

“…To match the scale of the ramp, the amplitude of the waveforms has been scaled up by a factor of 32. Right: a log-log plot of relative rms error E(n) of the plotted waveform for n = 2 p , p = 7, 8,9,10,11,12,13. The slope is very close to −1, corresponding to E(n) ∼ = 7.4/n.…”

“…If the jump points of ζ are equally spaced, one can choose P such that ζ P = ζ , whereby the algorithm is exact up to roundoff error (by Theorem 2). Comparison with the (much slower) exact methods of [9] shows that a good approximation is also obtained if the jump points of ζ are not equally spaced, provided the underlying grid is sufficiently fine. Indeed, Algorithm 1 reveals a remarkable qualitative phenomenon whereby smooth initial data can lead to discontinuities, as illustrated in Figure 3.…”

“…9]). Recent theoretical insights into the case where ζ is piecewise constant offer a fresh perspective from which to consider the numerical analysis of (1.1) for general ζ [9,10].…”

A geometry where everything is better than nice

“…The tensor products n j=1 u (k j ,k j+1 ) (R j ) thus have a combinatorial interpretation in that they count weighted Dyck paths having 2k j edges at height j. See [2]. Figure 1.…”

A Path-Integral Method for Solution of the Wave Equation with Continuously Varying Coefficients

“…Among the vast literature on solutions of the wave equation in heterogeneous media there are other approaches that bear some relation to ours. In most cases, a piecewise-constant approximation is used; a combinatorial solution for scattering from arbitrary piecewise-constant media was developed in [9,10]. For an approach similar to ours but in the setting of time-harmonic solutions, see [2,14,17].…”