Editing Longitudinal wave (section)

=== Attenuation in polycrystalline materials ===
Polycrystalline materials are made up of various crystal [[Grain boundary#:~:text=In materials science, a grain,thermal conductivity of the material.|grains]] which form the bulk material. Due to the difference in crystal structure and properties of these grains, when a wave propagating through a poly-crystal crosses a grain boundary, a [[scattering]] event occurs causing scattering based attenuation of the wave.<ref name=":1">{{Cite journal |last1=Kube |first1=Christopher M. |last2=Norris |first2=Andrew N. |date=2017-04-01 |title=Bounds on the longitudinal and shear wave attenuation ratio of polycrystalline materials |url=https://pubs.aip.org/jasa/article/141/4/2633/1059148/Bounds-on-the-longitudinal-and-shear-wave |journal=The Journal of the Acoustical Society of America |language=en |volume=141 |issue=4 |pages=2633–2636 |doi=10.1121/1.4979980 |pmid=28464650 |bibcode=2017ASAJ..141.2633K |issn=0001-4966}}</ref> Additionally it has been shown that the ratio rule for viscoelastic materials,
:<math>\frac{~\ \alpha_\ell\ }{~\ \alpha_T\ } ~\geq~ \frac{~ 4\ c_T^3\ }{~ 3\ c_\ell^3\ }
</math>
applies equally successfully to polycrystalline materials.<ref name=":1" />

A current prediction for modeling attenuation of waves in polycrystalline materials with elongated grains is the second-order approximation (SOA) model which accounts the second order of inhomogeneity allowing for the consideration multiple scattering in the crystal system.<ref name=":2">{{Cite journal |last1=Huang |first1=M. |last2=Sha |first2=G. |last3=Huthwaite |first3=P. |last4=Rokhlin |first4=S. I. |last5=Lowe |first5=M. J. S. |date=2021-04-01 |title=Longitudinal wave attenuation in polycrystals with elongated grains: 3D numerical and analytical modeling  |journal=The Journal of the Acoustical Society of America |language=en |volume=149 |issue=4 |pages=2377–2394 |doi=10.1121/10.0003955 |pmid=33940885 |bibcode=2021ASAJ..149.2377H |issn=0001-4966|doi-access=free }}</ref><ref>{{Cite journal |last1=Huang |first1=M. |last2=Sha |first2=G. |last3=Huthwaite |first3=P. |last4=Rokhlin |first4=S. I. |last5=Lowe |first5=M. J. S. |date=2020-12-01 |title=Elastic wave velocity dispersion in polycrystals with elongated grains: Theoretical and numerical analysis |url=https://pubs.aip.org/jasa/article/148/6/3645/1056424/Elastic-wave-velocity-dispersion-in-polycrystals |journal=The Journal of the Acoustical Society of America |language=en |volume=148 |issue=6 |pages=3645–3662 |doi=10.1121/10.0002916 |pmid=33379920 |bibcode=2020ASAJ..148.3645H |issn=0001-4966|doi-access=free |hdl=10044/1/85906 |hdl-access=free }}</ref> This model predicts that the shape of the grains in a poly-crystal has little effect on attenuation.<ref name=":2" />