An Effective Fractional Paraxial Wave Equation for Wave-Fronts in Randomly Layered Media with Long-Range Correlations

IF 16.4 1区 化学 Q1 CHEMISTRY, MULTIDISCIPLINARY
Christophe Gomez
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引用次数: 1

Abstract

This work concerns the asymptotic analysis of high-frequency wave propagation in randomly layered media with fast variations and long-range correlations. The analysis takes place in the three-dimensional physical space and weak-coupling regime. The role played by the slow decay of the correlations on a propagating pulse is twofold. First we observe a random travel time characterized by a fractional Brownian motion that appears to have a standard deviation larger than the pulse width, which is in contrast with the standard O’Doherty–Anstey theory for random propagation media with mixing properties. Second, a deterministic pulse deformation is described as the solution of a paraxial wave equation involving a pseudodifferential operator. This operator is characterized by the autocorrelation function of the medium fluctuations. In case of fluctuations with long-range correlations this operator is close to a fractional Weyl derivative whose order, between 2 and 3, depends on the power decay of the autocorrelation function. In the frequency domain, the pseudodifferential operator exhibits a frequency-dependent power-law attenuation with exponent corresponding to the order of the fractional derivative, and a frequency-dependent phase modulation, both ensuring the causality of the limiting paraxial wave equation as well as the Kramers–Kronig relations. The mathematical analysis is based on an approximation-diffusion theorem for random ordinary differential equations with long-range correlations.
具有长程相关的随机分层介质中波前的有效分数阶傍轴波动方程
这项工作涉及高频波在随机分层介质中传播的渐近分析,具有快速变化和远程相关性。分析发生在三维物理空间和弱耦合状态下。在传播脉冲中,相关性的缓慢衰减所起的作用是双重的。首先,我们观察到一个以分数布朗运动为特征的随机旅行时间,其标准差似乎大于脉冲宽度,这与具有混合特性的随机传播介质的标准O 'Doherty-Anstey理论相反。其次,将确定性脉冲变形描述为包含伪微分算子的近轴波动方程的解。该算子的特征是介质波动的自相关函数。在具有长期相关性的波动情况下,该算子接近于分数阶Weyl导数,其阶数在2和3之间,取决于自相关函数的幂衰减。在频域,伪微分算子表现出频率相关的幂律衰减,其指数对应于分数阶导数的阶数,以及频率相关的相位调制,两者都确保了极限傍轴波方程的因果关系以及Kramers-Kronig关系。对具有长程相关的随机常微分方程,利用近似扩散定理进行了数学分析。
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来源期刊
Accounts of Chemical Research
Accounts of Chemical Research 化学-化学综合
CiteScore
31.40
自引率
1.10%
发文量
312
审稿时长
2 months
期刊介绍: Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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