Editing Longitudinal wave (section)

== Attenuation of longitudinal waves ==
The [[attenuation]] of a wave in a medium describes the loss of energy a wave carries as it propagates throughout the medium.<ref name=":0">{{Cite web |title=Attenuation |url=https://wiki.seg.org/wiki/Attenuation#:~:text=Attenuation%20%E2%80%94%20the%20falloff%20of%20a,which%20is%20the%20conversion%20of |website=SEG Wiki}}</ref> This is caused by the scattering of the wave at interfaces, the loss of energy due to the friction between molecules, or geometric divergence.<ref name=":0" /> The study of attenuation of elastic waves in materials has increased in recent years, particularly within the study of polycrystalline materials where researchers aim to "nondestructively evaluate the degree of damage of engineering components" and to "develop improved procedures for characterizing microstructures" according to a research team led by R. Bruce Thompson in a ''[[Wave Motion (journal)|Wave Motion]]'' publication.<ref>{{Cite journal |last1=Thompson |first1=R. Bruce |last2=Margetan |first2=F.J. |last3=Haldipur |first3=P. |last4=Yu |first4=L. |last5=Li |first5=A. |last6=Panetta |first6=P. |last7=Wasan |first7=H. |date=April 2008 |title=Scattering of elastic waves in simple and complex polycrystals |url=https://doi.org/10.1016/j.wavemoti.2007.09.008 |journal=Wave Motion |volume=45 |issue=5 |pages=655–674 |doi=10.1016/j.wavemoti.2007.09.008 |bibcode=2008WaMot..45..655T |issn=0165-2125|url-access=subscription }}</ref>

=== Attenuation in viscoelastic materials ===
In [[viscoelastic]] materials, the attenuation coefficients per length  <math>\ \alpha_\ell\ </math> for longitudinal waves and <math>\ \alpha_T\ </math> for transverse waves must satisfy the following ratio:

:<math>\ \frac{~\ \alpha_\ell\ }{~\ \alpha_T\ } ~\geq~ \frac{~ 4\ c_T^3\ }{~ 3\ c_\ell^3\ }\ </math>

where <math>\ c_T\ </math> and <math>\ c_\ell\ </math> are the transverse and longitudinal wave speeds respectively.<ref>
{{cite journal
 |last=Norris |first=Andrew N.
 |year=2017
 |title=An inequality for longitudinal and transverse wave attenuation coefficients
 |journal=The Journal of the Acoustical Society of America
 |volume=141  |issue=1  |pages=475–479
 |doi=10.1121/1.4974152  |pmid=28147617  |issn=0001-4966
 |arxiv=1605.04326   |bibcode=2017ASAJ..141..475N
 |url=https://pubs.aip.org/jasa/article/141/1/475/1058243/An-inequality-for-longitudinal-and-transverse-wave
 |via=pubs.aip.org/jasa  |lang=en
}}
</ref>

=== 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|url-access=subscription }}</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" />