Conservation of Normalized Energy Density in multi-layered media: a proof for the general multi-layer 2D SH problem
Abstract
The Normalized Energy Density (NED) was introduced in Goto et al.(2011) as a quantity that, unlike the conventional energy, is conserved across material interfaces in a layered elastic medium. Its conservation was proved analytically only for a two-layer structure; for three or more layers it was supported by Monte Carlo simulation. I supply the missing proof for a general multi-layer (n-layer) structure in the 2D SH problem. The key step is to stop treating the angular frequency as a single variable and instead lift the layer phases to independent coordinates on the n-torus. The obstruction identified in the original paper---that the ratio of travel-time combinations is generically irrational, so that the periodicity argument used for two layers fails and disappears, and the proof reduces to Weyl's equidistribution theorem combined with n repetitions of the same elementary integral used in the two-layer case. I prove conservation in two independent settings that together cover all cases of practical interest: travel times that are linearly independent over Q, and commensurable travel times, including the equal-travel-time (Goupillaud) discretization. I also show that an arbitrary layered structure can be subdivided by reflectionless interfaces into one whose travel times take only d distinct, Q-independent values, where d is the dimension of the Q-span of the original travel times; this places the two proved cases at the endpoints d=1 and d=n and isolates the remaining open case 1<d<n in a canonical form.
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BibTeX
@misc{goto2026conservation,
title = {{Conservation of Normalized Energy Density in multi-layered media: a proof for the general multi-layer 2D SH problem}},
author = {Hiroyuki Goto},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05924},
eprint = {2609.05924},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05924},
}
RIS
TY - UNPB
TI - Conservation of Normalized Energy Density in multi-layered media: a proof for the general multi-layer 2D SH problem
AU - Hiroyuki Goto
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.05924
UR - https://arxiv.org/abs/2609.05924
AB - The Normalized Energy Density (NED) was introduced in Goto et al.(2011) as a quantity that, unlike the conventional energy, is conserved across material interfaces in a layered elastic medium. Its conservation was proved analytically only for a two-layer structure; for three or more layers it was supported by Monte Carlo simulation. I supply the missing proof for a general multi-layer (n-layer) structure in the 2D SH problem. The key step is to stop treating the angular frequency as a single variable and instead lift the layer phases to independent coordinates on the n-torus. The obstruction identified in the original paper---that the ratio of travel-time combinations is generically irrational, so that the periodicity argument used for two layers fails and disappears, and the proof reduces to Weyl's equidistribution theorem combined with n repetitions of the same elementary integral used in the two-layer case. I prove conservation in two independent settings that together cover all cases of practical interest: travel times that are linearly independent over Q, and commensurable travel times, including the equal-travel-time (Goupillaud) discretization. I also show that an arbitrary layered structure can be subdivided by reflectionless interfaces into one whose travel times take only d distinct, Q-independent values, where d is the dimension of the Q-span of the original travel times; this places the two proved cases at the endpoints d=1 and d=n and isolates the remaining open case 1<d<n in a canonical form.
ER -