37:
716:
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
1135:" or homogeneous media. Maxwell's equations, as we now understand them, retain that conclusion: in free-space or other uniform isotropic dielectrics, electro-magnetic waves are strictly transverse. However electromagnetic waves can display a longitudinal component in the electric and/or magnetic fields when traversing
1043:
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. This model predicts that
715:
of a wave in a medium describes the loss of energy a wave carries as it propagates throughout the medium. This is caused by the scattering of the wave at interfaces, the loss of energy due to the friction between molecules, or geometric divergence. The study of attenuation of elastic waves in
1154:
and Bo
Lehnert of the Swedish Royal Society, have used the Proca equation in an attempt to demonstrate photon mass as a longitudinal electromagnetic component of Maxwell's equations, suggesting that longitudinal electromagnetic waves could exist in a Dirac polarized vacuum. However
274:
1038:
876:
1150:(1897–1955) was known for developing relativistic quantum field equations bearing his name (Proca's equations) which apply to the massive vector spin-1 mesons. In recent decades some other theorists, such as
138:"Longitudinal waves" and "transverse waves" have been abbreviated by some authors as "L-waves" and "T-waves", respectively, for their own convenience. While these two abbreviations have specific meanings in
1236:
1114:; due to the fact that they would need particles to vibrate upon, the electric and magnetic fields of which the wave consists are perpendicular to the direction of the wave's propagation. However
1641:"Hear the Weird Sounds of a Black Hole Singing – As part of an effort to "sonify" the cosmos, researchers have converted the pressure waves from a black hole into an audible … something"
619:
524:
400:
442:
946:
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
1941:
154:), some authors chose to use "l-waves" (lowercase 'L') and "t-waves" instead, although they are not commonly found in physics writings except for some popular science books.
786:
759:
932:
905:
696:
670:
557:
182:
639:
1288:
20:
955:
793:
2091:
1052:
The equations for sound in a fluid given above also apply to acoustic waves in an elastic solid. Although solids also support transverse waves (known as
1910:
1118:
are longitudinal since these are not electromagnetic waves but density waves of charged particles, but which can couple to the electromagnetic field.
447:
For sound waves, the amplitude of the wave is the difference between the pressure of the undisturbed air and the maximum pressure caused by the wave.
289:
950:
event occurs causing scattering based attenuation of the wave. Additionally it has been shown that the ratio rule for viscoelastic materials,
1354:
114:, in which the displacements of the medium are at right angles to the direction of propagation. Transverse waves, for instance, describe
59:
of the medium is parallel to the direction the wave travels and displacement of the medium is in the same (or opposite) direction of the
1537:
1609:
2066:
1710:
1693:
1677:
1422:
95:
toy, where the distance between coils increases and decreases, is a good visualization. Real-world examples include sound waves (
1903:
1937:
1640:
2045:
566:
477:
364:
1896:
411:
2274:
2056:
2004:
1538:"Elastic wave velocity dispersion in polycrystals with elongated grains: Theoretical and numerical analysis"
19:
1481:"Longitudinal wave attenuation in polycrystals with elongated grains: 3D numerical and analytical modeling"
2022:
1126:
1103:
2000:
1817:
Krishan, S.; Selim, A. A. (1968). "Generation of transverse waves by non-linear wave-wave interaction".
718:
76:
2029:
1306:
Thompson, R. Bruce; Margetan, F.J.; Haldipur, P.; Yu, L.; Li, A.; Panetta, P.; Wasan, H. (April 2008).
130:
waves" to differentiate them from the (longitudinal) pressure waves that these materials also support.
1756:
Lakes, Roderic (1998). "Experimental Limits on the Photon Mass and Cosmic
Magnetic Vector Potential".
1826:
1765:
1549:
1492:
1434:
1376:
1319:
1107:
100:
1139:
materials, or inhomogeneous materials especially at interfaces (surface waves for instance) such as
2248:
2071:
1958:
147:
1863:
1669:
1645:
1366:
1151:
1065:
764:
737:
292:
Representation of the propagation of an omnidirectional pulse wave on a 2d grid (empirical shape)
127:
16:
Waves in which the direction of media displacement is parallel (along) to the direction of travel
2034:
1129:, Heaviside concluded that electromagnetic waves were not to be found as longitudinal waves in "
405:
The wavelength can be calculated as the relation between a wave's speed and ordinary frequency.
288:
2136:
1928:
1795:
1706:
1689:
1673:
1575:
1567:
1518:
1510:
1458:
1450:
1400:
1392:
1335:
1251:
332:
1270:
1220:
1216:
454:
depends on the type, temperature, and composition of the medium through which it propagates.
269:{\displaystyle y(x,t)=y_{0}\cos \!{\bigg (}\omega \!\left(t-{\frac {x}{c}}\right)\!{\bigg )}}
2081:
2061:
1855:
1834:
1773:
1557:
1500:
1442:
1384:
1327:
60:
910:
883:
681:
648:
535:
40:
2086:
2012:
1888:
1423:"Bounds on the longitudinal and shear wave attenuation ratio of polycrystalline materials"
1172:
1147:
1111:
1092:
731:
560:
163:
111:
64:
1830:
1769:
1553:
1496:
1438:
1380:
1323:
2124:
2051:
1975:
1920:
1846:
Barrow, W.L. (1936). "Transmission of
Electromagnetic Waves in Hollow Tubes of Metal".
1636:
1232:
1160:
1061:
1033:{\displaystyle {\frac {\alpha _{L}}{\alpha _{T}}}\geq {\frac {4c_{T}^{3}}{3c_{L}^{3}}}}
943:
871:{\displaystyle {\frac {\alpha _{L}}{\alpha _{T}}}\geq {\frac {4c_{T}^{3}}{3c_{L}^{3}}}}
624:
451:
1878:". Acoustics Animations, Pennsylvania State University, Graduate Program in Acoustics.
1838:
1594:
2268:
2233:
2171:
2076:
1980:
1182:
642:
1867:
2253:
1806:– A.J. Devaney, H. Levine, and T. Plona. Ann Arbor, Mich., Ann Arbor Science, 1982.
1331:
1192:
1140:
1136:
1084:
1077:
673:
1804:
Attenuation due to scattering of ultrasonic compressional waves in granular media
1307:
2228:
2186:
2141:
2106:
1990:
1970:
1953:
1777:
1536:
Huang, M.; Sha, G.; Huthwaite, P.; Rokhlin, S. I.; Lowe, M. J. S. (2020-12-01).
1479:
Huang, M.; Sha, G.; Huthwaite, P.; Rokhlin, S. I.; Lowe, M. J. S. (2021-04-01).
1115:
712:
99:
in pressure, a particle of displacement, and particle velocity propagated in an
80:
1859:
2166:
2161:
2156:
1985:
1963:
1924:
1131:
1076:, the latter of which is described (as with sound in a gas) by the material's
1057:
947:
173:
139:
1571:
1514:
1454:
1396:
1355:"An inequality for longitudinal and transverse wave attenuation coefficients"
1339:
2208:
2146:
2101:
2039:
1258:
1122:
1073:
1044:
the shape of the grains in a poly-crystal has little effect on attenuation.
316:
169:
143:
96:
56:
1579:
1522:
1462:
1404:
1159:
is strongly doubted by almost all physicists and is incompatible with the
2223:
2176:
2119:
2008:
471:
solids and liquids, the speed of a
Longitudinal wave can be described by
468:
88:
2243:
2218:
2201:
2196:
2129:
2114:
1811:
Experimental
Observation of Pressure Waves in Gas-Solids Fluidized Beds
1744:
Nikola Tesla, Lightning
Observations, and stationary waves, Appendix II
1562:
1505:
1480:
1069:
1053:
699:
1705:
Gerald E. Marsh (1996), Force-free
Magnetic Fields, World Scientific,
1446:
1388:
2213:
2191:
2181:
2151:
1187:
1156:
151:
123:
104:
92:
1809:
Schaaf, John van der, Jaap C. Schouten, and Cor M. van den Bleek, "
1371:
2238:
1813:". American Institute of Chemical Engineers. New York, N.Y., 1997.
1177:
1088:
1087:(converting astronomical data associated with pressure waves into
119:
1995:
1614:
1308:"Scattering of elastic waves in simple and complex polycrystals"
52:
41:
Detailed animation of longitudinal wave motion (CC-BY-NC-ND 4.0)
1892:
934:
are the transverse and longitudinal wave speeds respectively.
286:
is the displacement of the point on the traveling sound wave;
1040:
applies equally successfully to polycrystalline materials.
1727:
Appendices: D. On compressional electric or magnetic waves
942:
Polycrystalline materials are made up of various crystal
734:
materials, the attenuation coefficients per length alpha
23:
A type of longitudinal wave: A plane pressure pulse wave.
30:
Nonfree image: detailed animation of a longitudinal wave
1608:
Watzke, Megan; Porter, Molly; Mohon, Lee (4 May 2022).
788:
for transverse waves must satisfy the following ratio:
347:
is the time that the wave takes to travel the distance
1421:
Kube, Christopher M.; Norris, Andrew N. (2017-04-01).
1060:), longitudinal sound waves in the solid exist with a
958:
913:
886:
796:
767:
740:
684:
651:
627:
569:
538:
480:
414:
367:
185:
300:
is the distance from the point to the wave's source;
2100:
29:
1032:
926:
899:
870:
780:
753:
690:
664:
633:
613:
551:
518:
436:
394:
268:
87:, because they produce increases and decreases in
261:
258:
231:
223:
220:
1610:"New NASA Black Hole Sonifications with a Remix"
1542:The Journal of the Acoustical Society of America
1485:The Journal of the Acoustical Society of America
1427:The Journal of the Acoustical Society of America
1359:The Journal of the Acoustical Society of America
1729:. Chelsea Pub Co; 3rd edition (1971) 082840237X
1904:
1213:Stone in Architecture: Properties, Durability
8:
1688:John D. Jackson, Classical Electrodynamics,
1665:
1663:
614:{\displaystyle E_{L}=K_{b}+{\frac {4G}{3}}}
519:{\displaystyle v_{l}={\sqrt {E_{l}/\rho }}}
395:{\displaystyle f={\frac {\omega }{2\pi }}.}
168:For longitudinal harmonic sound waves, the
1911:
1897:
1889:
1561:
1504:
1370:
1091:) of the black hole at the center of the
1021:
1016:
1001:
996:
986:
975:
965:
959:
957:
918:
912:
891:
885:
859:
854:
839:
834:
824:
813:
803:
797:
795:
772:
766:
745:
739:
683:
656:
650:
626:
596:
587:
574:
568:
543:
537:
506:
500:
494:
485:
479:
421:
413:
374:
366:
260:
259:
243:
222:
221:
211:
184:
107:(created by earthquakes and explosions).
91:. A wave along the length of a stretched
938:Attenuation in polycrystalline materials
437:{\displaystyle \lambda ={\frac {c}{f}}.}
287:
18:
1876:Longitudinal and Transverse Wave Motion
1800:Elastic wave scattering and propagation
1253:Abbreviations Dictionary, Tenth Edition
1223:, Springer Science & Business Media
1204:
1146:In the development of modern physics,
83:when travelling through a medium, and
26:
1881:Longitudinal Waves, with animations "
1598:". Eric Weisstein's World of Science.
726:Attenuation in viscoelastic materials
7:
1474:
1472:
1416:
1414:
1283:
1281:
1250:Dean A. Stahl, Karen Landen (2001),
1672:, Introduction to Electrodynamics,
110:The other main type of wave is the
67:longitudinal waves are also called
1241:(3rd ed.), Oxford University Press
14:
707:Attenuation of longitudinal waves
1738:Corum, K. L., and J. F. Corum, "
1353:Norris, Andrew N. (2017-01-01).
1110:in a vacuum, which are strictly
176:can be described by the formula
35:
1083:In May 2022, NASA reported the
1332:10.1016/j.wavemoti.2007.09.008
1238:A Dictionary of Earth Sciences
201:
189:
1:
325:is the speed of the wave; and
1068:dependent on the material's
1839:10.1088/0032-1028/10/10/305
1778:10.1103/PhysRevLett.80.1826
781:{\displaystyle \alpha _{T}}
761:for longitudinal waves and
754:{\displaystyle \alpha _{l}}
458:Speed of Longitudinal Waves
2291:
1860:10.1109/JRPROC.1936.227357
1125:'s attempts to generalize
1106:lead to the prediction of
358:) of the wave is given by
161:
126:); these are also called "
2046:Music On A Long Thin Wire
1949:
1935:
1269:Francine Milford (2016),
34:
1942:Hornbostel–Sachs numbers
1740:The Zenneck surface wave
354:The ordinary frequency (
1758:Physical Review Letters
1211:Erhard Winkler (1997),
75:, because they produce
2023:Long-string instrument
1848:Proceedings of the IRE
1723:Electromagnetic theory
1093:Perseus galaxy cluster
1034:
928:
901:
872:
782:
755:
692:
666:
635:
615:
553:
520:
438:
396:
293:
270:
122:materials (but not in
24:
1883:The Physics Classroom
1592:Weisstein, Eric W., "
1108:electromagnetic waves
1035:
929:
927:{\displaystyle c_{L}}
902:
900:{\displaystyle c_{T}}
873:
783:
756:
693:
691:{\displaystyle \rho }
667:
665:{\displaystyle K_{B}}
636:
616:
554:
552:{\displaystyle E_{l}}
521:
439:
397:
291:
271:
162:Further information:
22:
1796:Vasundara V. Varadan
1794:Varadan, V. K., and
1721:Heaviside, Oliver, "
956:
911:
884:
794:
765:
738:
682:
649:
625:
567:
536:
478:
412:
365:
319:of the oscillations,
306:is the time elapsed;
183:
118:bulk sound waves in
103:medium) and seismic
1831:1968PlPh...10..931K
1770:1998PhRvL..80.1826L
1554:2020ASAJ..148.3645H
1497:2021ASAJ..149.2377H
1439:2017ASAJ..141.2633K
1381:2017ASAJ..141..475N
1324:2008WaMot..45..655T
1127:Maxwell's equations
1104:Maxwell's equations
1026:
1006:
864:
844:
148:electrocardiography
2030:Melde's experiment
1670:David J. Griffiths
1646:The New York Times
1563:10.1121/10.0002916
1506:10.1121/10.0003955
1152:Jean-Pierre Vigier
1030:
1012:
992:
924:
897:
868:
850:
830:
778:
751:
688:
662:
631:
611:
549:
516:
434:
392:
294:
266:
146:or long wave) and
49:Longitudinal waves
25:
2262:
2261:
2018:Longitudinal wave
1854:(10): 1298–1328.
1447:10.1121/1.4979980
1389:10.1121/1.4974152
1028:
981:
866:
819:
634:{\displaystyle G}
609:
514:
452:propagation speed
429:
387:
333:angular frequency
251:
73:compression waves
46:
45:
2282:
2082:String vibration
1913:
1906:
1899:
1890:
1871:
1842:
1782:
1781:
1764:(9): 1826–1829.
1753:
1747:
1736:
1730:
1719:
1713:
1703:
1697:
1686:
1680:
1667:
1658:
1657:
1655:
1653:
1633:
1627:
1626:
1624:
1622:
1605:
1599:
1590:
1584:
1583:
1565:
1548:(6): 3645–3662.
1533:
1527:
1526:
1508:
1491:(4): 2377–2394.
1476:
1467:
1466:
1433:(4): 2633–2636.
1418:
1409:
1408:
1374:
1350:
1344:
1343:
1303:
1297:
1296:
1285:
1276:
1267:
1261:
1248:
1242:
1230:
1224:
1209:
1157:photon rest mass
1112:transverse waves
1099:Electromagnetics
1039:
1037:
1036:
1031:
1029:
1027:
1025:
1020:
1007:
1005:
1000:
987:
982:
980:
979:
970:
969:
960:
933:
931:
930:
925:
923:
922:
906:
904:
903:
898:
896:
895:
877:
875:
874:
869:
867:
865:
863:
858:
845:
843:
838:
825:
820:
818:
817:
808:
807:
798:
787:
785:
784:
779:
777:
776:
760:
758:
757:
752:
750:
749:
697:
695:
694:
689:
671:
669:
668:
663:
661:
660:
640:
638:
637:
632:
620:
618:
617:
612:
610:
605:
597:
592:
591:
579:
578:
558:
556:
555:
550:
548:
547:
525:
523:
522:
517:
515:
510:
505:
504:
495:
490:
489:
463:Isotropic medium
443:
441:
440:
435:
430:
422:
401:
399:
398:
393:
388:
386:
375:
275:
273:
272:
267:
265:
264:
257:
253:
252:
244:
227:
226:
216:
215:
61:wave propagation
39:
38:
27:
2290:
2289:
2285:
2284:
2283:
2281:
2280:
2279:
2265:
2264:
2263:
2258:
2167:Japanese fiddle
2105:
2096:
2087:Transverse wave
2035:Mersenne's laws
2013:String harmonic
1945:
1931:
1917:
1874:Russell, Dan, "
1845:
1825:(10): 931–937.
1816:
1791:
1789:Further reading
1786:
1785:
1755:
1754:
1750:
1737:
1733:
1720:
1716:
1704:
1700:
1687:
1683:
1668:
1661:
1651:
1649:
1637:Overbye, Dennis
1635:
1634:
1630:
1620:
1618:
1607:
1606:
1602:
1591:
1587:
1535:
1534:
1530:
1478:
1477:
1470:
1420:
1419:
1412:
1352:
1351:
1347:
1305:
1304:
1300:
1287:
1286:
1279:
1272:The Tuning Fork
1268:
1264:
1249:
1245:
1231:
1227:
1210:
1206:
1201:
1173:Transverse wave
1169:
1148:Alexandru Proca
1101:
1050:
1008:
988:
971:
961:
954:
953:
940:
914:
909:
908:
887:
882:
881:
846:
826:
809:
799:
792:
791:
768:
763:
762:
741:
736:
735:
728:
709:
680:
679:
652:
647:
646:
623:
622:
598:
583:
570:
565:
564:
561:elastic modulus
539:
534:
533:
496:
481:
476:
475:
465:
460:
410:
409:
379:
363:
362:
314:
236:
232:
207:
181:
180:
166:
164:Acoustic theory
160:
136:
112:transverse wave
36:
17:
12:
11:
5:
2288:
2286:
2278:
2277:
2275:Wave mechanics
2267:
2266:
2260:
2259:
2257:
2256:
2251:
2246:
2241:
2236:
2231:
2226:
2221:
2216:
2211:
2206:
2205:
2204:
2199:
2194:
2189:
2184:
2179:
2169:
2164:
2159:
2154:
2149:
2144:
2139:
2134:
2133:
2132:
2125:Bladder fiddle
2122:
2117:
2111:
2109:
2098:
2097:
2095:
2094:
2089:
2084:
2079:
2074:
2069:
2064:
2059:
2054:
2049:
2042:
2037:
2032:
2027:
2026:
2025:
2015:
1998:
1993:
1988:
1983:
1978:
1973:
1968:
1967:
1966:
1956:
1950:
1947:
1946:
1936:
1933:
1932:
1918:
1916:
1915:
1908:
1901:
1893:
1887:
1886:
1879:
1872:
1843:
1819:Plasma Physics
1814:
1807:
1790:
1787:
1784:
1783:
1748:
1731:
1714:
1698:
1681:
1659:
1639:(7 May 2022).
1628:
1600:
1585:
1528:
1468:
1410:
1365:(1): 475–479.
1345:
1318:(5): 655–674.
1298:
1277:
1262:
1243:
1233:Michael Allaby
1225:
1203:
1202:
1200:
1197:
1196:
1195:
1190:
1185:
1180:
1175:
1168:
1165:
1161:Standard Model
1100:
1097:
1066:wave impedance
1049:
1048:Pressure waves
1046:
1024:
1019:
1015:
1011:
1004:
999:
995:
991:
985:
978:
974:
968:
964:
939:
936:
921:
917:
894:
890:
862:
857:
853:
849:
842:
837:
833:
829:
823:
816:
812:
806:
802:
775:
771:
748:
744:
727:
724:
708:
705:
704:
703:
702:of the medium.
687:
677:
659:
655:
630:
608:
604:
601:
595:
590:
586:
582:
577:
573:
546:
542:
527:
526:
513:
509:
503:
499:
493:
488:
484:
464:
461:
459:
456:
445:
444:
433:
428:
425:
420:
417:
403:
402:
391:
385:
382:
378:
373:
370:
337:
336:
326:
320:
312:
307:
301:
295:
277:
276:
263:
256:
250:
247:
242:
239:
235:
230:
225:
219:
214:
210:
206:
203:
200:
197:
194:
191:
188:
159:
156:
135:
132:
85:pressure waves
44:
43:
32:
31:
15:
13:
10:
9:
6:
4:
3:
2:
2287:
2276:
2273:
2272:
2270:
2255:
2252:
2250:
2247:
2245:
2242:
2240:
2237:
2235:
2234:Tromba marina
2232:
2230:
2227:
2225:
2222:
2220:
2217:
2215:
2212:
2210:
2207:
2203:
2200:
2198:
2195:
2193:
2190:
2188:
2185:
2183:
2180:
2178:
2175:
2174:
2173:
2170:
2168:
2165:
2163:
2160:
2158:
2155:
2153:
2150:
2148:
2145:
2143:
2140:
2138:
2135:
2131:
2128:
2127:
2126:
2123:
2121:
2118:
2116:
2113:
2112:
2110:
2108:
2103:
2099:
2093:
2090:
2088:
2085:
2083:
2080:
2078:
2077:Standing wave
2075:
2073:
2070:
2068:
2065:
2063:
2060:
2058:
2055:
2053:
2050:
2048:
2047:
2043:
2041:
2038:
2036:
2033:
2031:
2028:
2024:
2021:
2020:
2019:
2016:
2014:
2010:
2006:
2002:
1999:
1997:
1994:
1992:
1989:
1987:
1984:
1982:
1979:
1977:
1974:
1972:
1969:
1965:
1962:
1961:
1960:
1957:
1955:
1952:
1951:
1948:
1943:
1939:
1934:
1930:
1926:
1922:
1914:
1909:
1907:
1902:
1900:
1895:
1894:
1891:
1884:
1880:
1877:
1873:
1869:
1865:
1861:
1857:
1853:
1849:
1844:
1840:
1836:
1832:
1828:
1824:
1820:
1815:
1812:
1808:
1805:
1801:
1797:
1793:
1792:
1788:
1779:
1775:
1771:
1767:
1763:
1759:
1752:
1749:
1745:
1741:
1735:
1732:
1728:
1724:
1718:
1715:
1712:
1711:981-02-2497-4
1708:
1702:
1699:
1695:
1694:0-471-30932-X
1691:
1685:
1682:
1679:
1678:0-13-805326-X
1675:
1671:
1666:
1664:
1660:
1648:
1647:
1642:
1638:
1632:
1629:
1617:
1616:
1611:
1604:
1601:
1597:
1596:
1589:
1586:
1581:
1577:
1573:
1569:
1564:
1559:
1555:
1551:
1547:
1543:
1539:
1532:
1529:
1524:
1520:
1516:
1512:
1507:
1502:
1498:
1494:
1490:
1486:
1482:
1475:
1473:
1469:
1464:
1460:
1456:
1452:
1448:
1444:
1440:
1436:
1432:
1428:
1424:
1417:
1415:
1411:
1406:
1402:
1398:
1394:
1390:
1386:
1382:
1378:
1373:
1368:
1364:
1360:
1356:
1349:
1346:
1341:
1337:
1333:
1329:
1325:
1321:
1317:
1313:
1309:
1302:
1299:
1294:
1290:
1289:"Attenuation"
1284:
1282:
1278:
1275:
1273:
1266:
1263:
1260:
1256:
1254:
1247:
1244:
1240:
1239:
1234:
1229:
1226:
1222:
1218:
1214:
1208:
1205:
1198:
1194:
1191:
1189:
1186:
1184:
1183:Acoustic wave
1181:
1179:
1176:
1174:
1171:
1170:
1166:
1164:
1162:
1158:
1153:
1149:
1144:
1142:
1141:Zenneck waves
1138:
1134:
1133:
1128:
1124:
1119:
1117:
1113:
1109:
1105:
1098:
1096:
1094:
1090:
1086:
1081:
1079:
1075:
1071:
1067:
1063:
1059:
1055:
1047:
1045:
1041:
1022:
1017:
1013:
1009:
1002:
997:
993:
989:
983:
976:
972:
966:
962:
951:
949:
945:
937:
935:
919:
915:
892:
888:
878:
860:
855:
851:
847:
840:
835:
831:
827:
821:
814:
810:
804:
800:
789:
773:
769:
746:
742:
733:
725:
723:
722:publication.
721:
720:
714:
706:
701:
685:
678:
675:
657:
653:
644:
643:shear modulus
628:
606:
602:
599:
593:
588:
584:
580:
575:
571:
562:
544:
540:
532:
531:
530:
511:
507:
501:
497:
491:
486:
482:
474:
473:
472:
470:
462:
457:
455:
453:
448:
431:
426:
423:
418:
415:
408:
407:
406:
389:
383:
380:
376:
371:
368:
361:
360:
359:
357:
352:
350:
346:
342:
339:The quantity
334:
330:
327:
324:
321:
318:
311:
308:
305:
302:
299:
296:
290:
285:
282:
281:
280:
254:
248:
245:
240:
237:
233:
228:
217:
212:
208:
204:
198:
195:
192:
186:
179:
178:
177:
175:
171:
165:
157:
155:
153:
149:
145:
141:
133:
131:
129:
125:
121:
117:
113:
108:
106:
102:
98:
94:
90:
86:
82:
78:
74:
70:
69:compressional
66:
62:
58:
55:in which the
54:
50:
42:
33:
28:
21:
2254:Washtub bass
2107:musical bows
2067:Scale length
2044:
2017:
1964:Third bridge
1882:
1875:
1851:
1847:
1822:
1818:
1810:
1803:
1799:
1761:
1757:
1751:
1743:
1739:
1734:
1726:
1722:
1717:
1701:
1684:
1650:. Retrieved
1644:
1631:
1619:. Retrieved
1613:
1603:
1593:
1588:
1545:
1541:
1531:
1488:
1484:
1430:
1426:
1362:
1358:
1348:
1315:
1311:
1301:
1292:
1271:
1265:
1252:
1246:
1237:
1228:
1212:
1207:
1193:Plasma waves
1163:of physics.
1145:
1137:birefringent
1130:
1120:
1116:plasma waves
1102:
1085:sonification
1082:
1078:bulk modulus
1051:
1042:
952:
941:
879:
790:
732:viscoelastic
729:
717:
710:
674:bulk modulus
528:
466:
449:
446:
404:
355:
353:
348:
344:
340:
338:
335:of the wave.
328:
322:
309:
303:
297:
283:
278:
167:
142:(L-wave for
137:
134:Nomenclature
115:
109:
84:
72:
68:
48:
47:
2229:Psalmodicon
2142:Diddley bow
2001:Fundamental
1991:Fingerboard
1971:Chordophone
1929:instruments
1312:Wave Motion
719:Wave Motion
713:attenuation
158:Sound waves
81:rarefaction
77:compression
2162:Ichigenkin
2157:Ground bow
2102:Monochords
2092:Tuning peg
2072:Soundboard
1986:Enharmonic
1372:1605.04326
1274:, pp.43–44
1199:References
1132:free space
1058:seismology
948:scattering
563:such that
174:wavelength
140:seismology
97:vibrations
65:Mechanical
2209:Langeleik
2147:Duxianqin
2040:Monochord
2009:Overtones
2005:Harmonics
1572:0001-4966
1515:0001-4966
1455:0001-4966
1397:0001-4966
1340:0165-2125
1259:CRC Press
1123:Heaviside
984:≥
973:α
963:α
822:≥
811:α
801:α
770:α
743:α
686:ρ
512:ρ
469:isotropic
416:λ
384:π
377:ω
317:amplitude
241:−
229:ω
170:frequency
144:Love wave
57:vibration
2269:Category
2224:Onavillu
2177:Genggong
2172:Jaw harp
2120:Berimbau
2062:Re-entry
1919:Musical
1868:32056359
1580:33379920
1523:33940885
1463:28464650
1405:28147617
1293:SEG Wiki
1235:(2008),
1167:See also
1074:rigidity
1072:and its
1062:velocity
450:Sound's
89:pressure
2244:Umuduri
2219:Masenqo
2202:Mukkuri
2197:Morsing
2137:Đàn bầu
2130:Boom-ba
2115:Ahardin
1921:strings
1827:Bibcode
1766:Bibcode
1746:. 1994.
1550:Bibcode
1493:Bibcode
1435:Bibcode
1377:Bibcode
1320:Bibcode
1255:, p.618
1070:density
1054:S-waves
700:density
698:is the
672:is the
641:is the
559:is the
331:is the
315:is the
279:where:
105:P-waves
101:elastic
2249:Unitar
2214:Lesiba
2192:Kubing
2187:Khomuz
2182:Gogona
2152:Ektara
1976:Course
1959:Bridge
1927:, and
1866:
1709:
1692:
1676:
1652:11 May
1621:11 May
1595:P-Wave
1578:
1570:
1521:
1513:
1461:
1453:
1403:
1395:
1338:
1188:P-wave
1121:After
944:grains
880:where
621:where
529:where
152:T wave
124:fluids
93:Slinky
2239:Tumbi
1981:Drone
1925:wires
1864:S2CID
1367:arXiv
1178:Sound
1089:sound
150:(see
128:shear
120:solid
53:waves
2052:Node
1996:Fret
1938:List
1707:ISBN
1690:ISBN
1674:ISBN
1654:2022
1623:2022
1615:NASA
1576:PMID
1568:ISSN
1519:PMID
1511:ISSN
1459:PMID
1451:ISSN
1401:PMID
1393:ISSN
1336:ISSN
1221:p.57
1219:and
1217:p.55
1064:and
907:and
711:The
645:and
467:For
172:and
116:some
79:and
51:are
2104:and
2057:Nut
1954:Bow
1856:doi
1835:doi
1802:".
1798:, "
1774:doi
1742:",
1725:".
1558:doi
1546:148
1501:doi
1489:149
1443:doi
1431:141
1385:doi
1363:141
1328:doi
1056:in
730:In
218:cos
71:or
2271::
1923:,
1862:.
1852:24
1850:.
1833:.
1823:10
1821:.
1772:.
1762:80
1760:.
1662:^
1643:.
1612:.
1574:.
1566:.
1556:.
1544:.
1540:.
1517:.
1509:.
1499:.
1487:.
1483:.
1471:^
1457:.
1449:.
1441:.
1429:.
1425:.
1413:^
1399:.
1391:.
1383:.
1375:.
1361:.
1357:.
1334:.
1326:.
1316:45
1314:.
1310:.
1291:.
1280:^
1257:,
1215:,
1143:.
1095:.
1080:.
351:.
63:.
2011:/
2007:/
2003:/
1944:)
1940:(
1912:e
1905:t
1898:v
1885:"
1870:.
1858::
1841:.
1837::
1829::
1780:.
1776::
1768::
1696:.
1656:.
1625:.
1582:.
1560::
1552::
1525:.
1503::
1495::
1465:.
1445::
1437::
1407:.
1387::
1379::
1369::
1342:.
1330::
1322::
1295:.
1023:3
1018:L
1014:c
1010:3
1003:3
998:T
994:c
990:4
977:T
967:L
920:L
916:c
893:T
889:c
861:3
856:L
852:c
848:3
841:3
836:T
832:c
828:4
815:T
805:L
774:T
747:l
676:;
658:B
654:K
629:G
607:3
603:G
600:4
594:+
589:b
585:K
581:=
576:L
572:E
545:l
541:E
508:/
502:l
498:E
492:=
487:l
483:v
432:.
427:f
424:c
419:=
390:.
381:2
372:=
369:f
356:f
349:x
345:c
343:/
341:x
329:ω
323:c
313:0
310:y
304:t
298:x
284:y
262:)
255:)
249:c
246:x
238:t
234:(
224:(
213:0
209:y
205:=
202:)
199:t
196:,
193:x
190:(
187:y
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
