多层介质中归一化能量密度的守恒：一般多层二维SH问题的证明

中文摘要 AI 辅助

归一化能量密度（NED）由Goto等人（2011）提出，与常规能量不同，它在分层弹性介质的材料界面间是守恒的。其守恒性仅在两层结构中被解析证明；对于三层或更多层，仅通过蒙特卡洛模拟支持。我补充了二维SH问题中一般多层（n层）结构的缺失证明。关键步骤是不再将角频率视为单一变量，而是将各层相位提升为n维环面上的独立坐标。原始论文中识别的障碍——即走时组合之比通常是无理数，使得用于两层的周期性论证失效——消失了，证明简化为Weyl等分布定理与两层情形中相同基本积分的n次重复。我在两个独立的设定中证明了守恒性，它们共同覆盖了所有实际感兴趣的情况：走时在有理数上线性无关，以及走时可公度的情况，包括等走时（Goupillaud）离散化。我还表明，任意分层结构可以通过无反射界面细分为一个走时仅取d个不同的、有理数上独立值的结构，其中d是原始走时的有理数张成的维数；这将两个已证明的情形置于端点d=1和d=n处，并将剩余的开放情形1<d<n以规范形式隔离出来。

英文摘要

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.

发表机构

• Disaster Prevention Research Institute, Kyoto University（京都大学防灾研究所）

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