1/f spectral trend and frequency power law of lossy media

Abstract

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.