Editing Fractional calculus (section)

===Acoustic wave equations for complex media===
The propagation of acoustical waves in complex media, such as in biological tissue, commonly implies attenuation obeying a frequency power-law. This kind of phenomenon may be described using a causal wave equation which incorporates fractional time derivatives:
<math display="block">\nabla^2 u -\dfrac 1{c_0^2} \frac{\partial^2 u}{\partial t^2} + \tau_\sigma^\alpha \dfrac{\partial^\alpha}{\partial t^\alpha}\nabla^2 u - \dfrac {\tau_\epsilon^\beta}{c_0^2} \dfrac{\partial^{\beta+2} u}{\partial t^{\beta+2}} = 0\,.</math>

See also Holm & Näsholm (2011)<ref>{{cite journal |last1=Holm |first1=S. |last2=Näsholm |first2=S. P. |s2cid=7804006 |year=2011 |title=A causal and fractional all-frequency wave equation for lossy media |journal=Journal of the Acoustical Society of America |volume=130 |issue=4 |pages=2195–2201 |bibcode=2011ASAJ..130.2195H |doi=10.1121/1.3631626 |pmid=21973374|hdl=10852/103311 |hdl-access=free }}</ref> and the references therein. Such models are linked to the commonly recognized hypothesis that multiple relaxation phenomena give rise to the attenuation measured in complex media. This link is further described in Näsholm & Holm (2011b)<ref>{{cite journal |last1=Näsholm |first1=S. P. |last2=Holm |first2=S. |s2cid=10376751 |year=2011 |title=Linking multiple relaxation, power-law attenuation, and fractional wave equations |journal=Journal of the Acoustical Society of America |volume=130 |issue=5 |pages=3038–3045 |bibcode=2011ASAJ..130.3038N |doi=10.1121/1.3641457 |pmid=22087931|hdl=10852/103312 |hdl-access=free }}</ref> and in the survey paper,<ref name=Nasholm2>{{cite journal |last1=Näsholm |first1=S. P. |last2=Holm |first2=S. |year=2012 |title=On a Fractional Zener Elastic Wave Equation |journal=Fract. Calc. Appl. Anal. |volume=16 |pages=26–50 |arxiv=1212.4024 |doi=10.2478/s13540-013-0003-1 |s2cid=120348311}}</ref> as well as the ''[[Acoustic attenuation]]'' article. See Holm & Nasholm (2013)<ref name=HolmNasholm2014>{{cite journal |last1=Holm |first1=S. |last2=Näsholm |first2=S. P. |year=2013 |title=Comparison of fractional wave equations for power law attenuation in ultrasound and elastography |journal=Ultrasound in Medicine & Biology |volume=40 |issue=4 |pages=695–703 |arxiv=1306.6507 |citeseerx=10.1.1.765.120 |doi=10.1016/j.ultrasmedbio.2013.09.033 |pmid=24433745  |s2cid=11983716}}</ref> for a paper which compares fractional wave equations which model power-law attenuation. This book on power-law attenuation also covers the topic in more detail.<ref name=Holm2019>{{cite book |last=Holm |first=S. |url=https://link.springer.com/book/10.1007/978-3-030-14927-7 |title=Waves with Power-Law Attenuation |publisher=Springer and Acoustical Society of America Press |year=2019 |doi=10.1007/978-3-030-14927-7 |bibcode=2019wpla.book.....H |isbn=978-3-030-14926-0|s2cid=145880744 }}</ref>

Pandey and Holm gave a physical meaning to fractional differential equations by deriving them from physical principles and interpreting the fractional-order in terms of the parameters of the acoustical media, example in fluid-saturated granular unconsolidated marine sediments.<ref name=Pandey2016>{{cite journal |last1=Pandey |first1=Vikash |last2=Holm |first2=Sverre |date=2016-12-01 |title=Connecting the grain-shearing mechanism of wave propagation in marine sediments to fractional order wave equations |journal=The Journal of the Acoustical Society of America |volume=140 |issue=6 |pages=4225–4236 |doi=10.1121/1.4971289 |pmid=28039990 |issn=0001-4966 |arxiv=1612.05557 |bibcode=2016ASAJ..140.4225P |s2cid=29552742}}</ref> Interestingly, Pandey and Holm derived [[Cinna Lomnitz|Lomnitz's law]] in [[seismology]] and Nutting's law in [[non-Newtonian fluid|non-Newtonian rheology]] using the framework of fractional calculus.<ref>{{cite journal |last1=Pandey |first1=Vikash |last2=Holm |first2=Sverre |date=2016-09-23 |title=Linking the fractional derivative and the Lomnitz creep law to non-Newtonian time-varying viscosity |journal=Physical Review E |volume=94 |issue=3 |pages=032606 |doi=10.1103/PhysRevE.94.032606 |pmid=27739858 |bibcode=2016PhRvE..94c2606P |doi-access=free|hdl=10852/53091 |hdl-access=free }}</ref> Nutting's law was used to model the wave propagation in marine sediments using fractional derivatives.<ref name=Pandey2016/>