有耗介质的1/f谱趋势和频率幂律
1/f spectral trend and frequency power law of lossy media
摘要：
长期以来，人们一直认为声波传播的耗散遵循频率的经验幂函数，其指数参数在不同的介质中变化。本文旨在揭示耗散频率幂律与1/f谱趋势之间的内在关系。因此，1/f光谱趋势可以通过介质耗散机制进行物理解释，1/f光谱趋势的所谓红外灾难也是如此。另一方面，耗散频率幂律最近已成功地在时间-空间域中通过分数阶微积分建模，也被发现是基于介质的Levy分布，而1/f谱趋势已知与分形有简单的关系。因此，可以直接将1/f谱趋势、分形、Levy统计、分数阶微积分和耗散幂律联系起来。所有这些数学方法只是以不同的方式反映了复杂现象的本质。我们还讨论了一些令人困惑的问题，从这项研究中产生的。
英文摘要：
The dissipation of acoustic wave propagation has long been found to obey an empirical power function of frequency, whose exponent parameter varies through different media. This note aims to unveil the inherent relationship between this dissipative frequency power law and 1/f spectral trend. Accordingly, the 1/f spectral trend can physically be interpreted via the media dissipation mechanism, so does the so-called infrared catastrophe of 1/f spectral trend4. On the other hand, the dissipative frequency power law has recently been modeled in time-space domain successfully via the fractional calculus and is also found to underlie the Levy distribution of media, while the 1/f spectral trend is known to have simple relationship with the fractal. As a result, it is straightforward to correlate 1/f spectral trend, fractal, Levy statistics, fractional calculus, and dissipative power law. All these mathematical methodologies simply reflect the essence of complex phenomena in different fashion. We also discuss some perplexing issues arising from this study.
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