31:
2564:
825:
2485:
1237:
1357:
1938:
1569:
380:
1729:
1034:
692:
1822:
as differences between energy levels, energy being proportional to wavenumber, or frequency. However, spectroscopic data kept being tabulated in terms of spectroscopic wavenumber rather than frequency or energy.
1797:
614:
1460:
224:
2098:
1128:
The sign convention is chosen for consistency with propagation in lossy media. If the attenuation constant is positive, then the wave amplitude decreases as the wave propagates in the x-direction.
1086:
296:
1272:
1840:
1734:
The historical reason for using this spectroscopic wavenumber rather than frequency is that it is a convenient unit when studying atomic spectra by counting fringes per cm with an
1247:
Here we assume that the wave is regular in the sense that the different quantities describing the wave such as the wavelength, frequency and thus the wavenumber are constants. See
2042:
1148:
501:
1652:
1123:
891:
626:
is the free-space wavenumber, as above. The imaginary part of the wavenumber expresses attenuation per unit distance and is useful in the study of exponentially decaying
1502:
681:
2331:
977:
923:
854:
531:
316:
948:
2115:. Note that the wavelength of light changes as it passes through different media, however, the spectroscopic wavenumber (i.e., frequency) remains constant.
1496:, for example an electron wave, in the non-relativistic approximation (in the case of a free particle, that is, the particle has no potential energy):
255:
In theoretical physics, a wave number, defined as the number of radians per unit distance, sometimes called "angular wavenumber", is more often used:
2483:, Ray, James C. & Asari, Logan R., "Method and apparatus for spectroscopic comparison of compositions", published 1991-09-10
2118:
Often spatial frequencies are stated by some authors "in wavenumbers", incorrectly transferring the name of the quantity to the CGS unit cm itself.
1668:
979:
2451:
1815:
820:{\displaystyle k=k'-jk''={\sqrt {-\left(\omega \mu ''+j\omega \mu '\right)\left(\sigma +\omega \varepsilon ''+j\omega \varepsilon '\right)}}\;}
2258:
2225:
2192:
252:, in centimeters per nanosecond); conversely, an electromagnetic wave at 29.9792458 GHz has a wavelength of 1 cm in free space.
1802:
which remains essentially the same in air, and so the spectroscopic wavenumber is directly related to the angles of light scattered from
2589:
2543:
1744:
546:
2427:
240:
For example, a wavenumber in inverse centimeters can be converted to a frequency expressed in the unit gigahertz by multiplying by
1414:
991:
1393:
of the wave. The dependence of the wavenumber on the frequency (or more commonly the frequency on the wavenumber) is known as a
182:
1088:
504:
313:
is still being represented, albeit indirectly. As described in the spectroscopy section, this is done through the relationship
2053:
638:
The propagation factor of a sinusoidal plane wave propagating in the positive x direction in a linear material is given by
407:
30:
2594:
1352:{\displaystyle k={\frac {2\pi }{\lambda }}={\frac {2\pi \nu }{v_{\mathrm {p} }}}={\frac {\omega }{v_{\mathrm {p} }}}}
261:
2350:
454:
1810:, when those instruments are operated in air or vacuum. Such wavenumbers were first used in the calculations of
1966:
1827:
1659:
96:
1933:{\displaystyle {\tilde {\nu }}=R\left({\frac {1}{{n_{\text{f}}}^{2}}}-{\frac {1}{{n_{\text{i}}}^{2}}}\right),}
1818:
of 1908 was also formulated in terms of wavenumbers. A few years later spectral lines could be understood in
1232:{\displaystyle \lambda ={\frac {2\pi }{k'}}\qquad v_{p}={\frac {\omega }{k'}}\qquad \delta ={\frac {1}{k''}}}
2584:
2372:
Murthy, V. L. R.; Lakshman, S. V. J. (1981). "Electronic absorption spectrum of cobalt antipyrine complex".
1603:
1478:
1259:
140:
2003:
2568:
1043:
34:
Diagram illustrating the relationship between the wavenumber and the other properties of harmonic waves.
2480:
479:
2460:
2381:
1994:
1628:
1401:
893:
399:
153:
128:
65:
1095:
1803:
1564:{\displaystyle k\equiv {\frac {2\pi }{\lambda }}={\frac {p}{\hbar }}={\frac {\sqrt {2mE}}{\hbar }}}
1394:
132:
124:
2599:
2512:
2127:
136:
2280:
644:
2345:
2220:. Remote Sensing and Digital Image Processing. Springer International Publishing. p. 679.
2539:
2495:
2423:
2325:
2254:
2221:
2215:
2188:
2132:
1819:
1379:
959:
905:
509:
403:
375:{\textstyle {\frac {\nu _{\text{s}}}{c}}\;=\;{\frac {1}{\lambda }}\;\equiv \;{\tilde {\nu }},}
234:
145:
120:
56:
2533:
2182:
1738: : the spectroscopic wavenumber is the reciprocal of the wavelength of light in vacuum:
2504:
2468:
2389:
2248:
2142:
2137:
2108:
1948:
1811:
1655:
627:
534:
418:(i.e., reciprocal centimeters; cm); in this context, the wavenumber was formerly called the
107:
39:
152:
Wavenumber can be used to specify quantities other than spatial frequency. For example, in
2306:
1831:
868:
423:
831:
2464:
2385:
930:
233:
is the wavelength. It is sometimes called the "spectroscopic wavenumber". It equals the
2112:
1807:
1735:
1610:
1595:
1486:
1390:
1135:
983:
856:
462:
249:
157:
2472:
2578:
2516:
2393:
1987:
473:
A complex-valued wavenumber can be defined for a medium with complex-valued relative
447:
2406:
2508:
1622:
1036:
474:
411:
169:
2250:
Digital
Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
2217:
Understanding Earth
Observation: The Electromagnetic Foundation of Remote Sensing
1493:
1263:
1139:
303:
101:
1658:, cm) refers to a temporal frequency (in hertz) which has been divided by the
1248:
1131:
173:
116:
2447:"Absorption and the dimensionless overlap integral for two-photon excitation"
17:
1724:{\displaystyle {\tilde {\nu }}={\frac {\nu }{c}}={\frac {\omega }{2\pi c}}.}
310:
81:
2253:. Geophysical references. Society of Exploration Geophysicists. p. 9.
2563:
2165:
1579:
415:
156:, it is often used as a unit of temporal frequency assuming a certain
1470:
860:
458:
443:
439:
395:. This is done for convenience as frequencies tend to be very large.
112:
73:
2446:
1251:
for discussion of the case when these quantities are not constant.
457:
in vacuum, wavenumber is directly proportional to frequency and to
897:
392:
1404:
in a vacuum, in which the wave propagates at the speed of light,
1792:{\displaystyle \lambda _{\rm {vac}}={\frac {1}{\tilde {\nu }}},}
1587:
609:{\displaystyle k=k_{0}{\sqrt {\varepsilon _{r}\mu _{r}}}=k_{0}n}
61:
1142:
have simple relationships to the components of the wavenumber:
84:, which is defined as the number of wave cycles per unit time (
2167:
ISO 80000-3:2019 Quantities and units – Part 3: Space and time
461:
energy. Because of this, wavenumbers are used as a convenient
426:(some older scientific papers used this unit, abbreviated as
2452:
1455:{\displaystyle k={\frac {E}{\hbar c}}={\frac {\omega }{c}}}
1029:{\displaystyle \varepsilon =\varepsilon '-j\varepsilon ''=}
219:{\displaystyle {\tilde {\nu }}\;=\;{\frac {1}{\lambda }},}
111:. Wave numbers and wave vectors play an essential role in
438:
cm). The angular wavenumber may be expressed in the unit
2093:{\displaystyle \lambda ={\frac {1}{n{\tilde {\nu }}}},}
172:
and most chemistry fields, is defined as the number of
2307:"Wavenumber -- from Eric Weisstein's World of Physics"
319:
2056:
2006:
1843:
1747:
1671:
1631:
1505:
1417:
1275:
1151:
1098:
1046:
994:
962:
933:
908:
871:
834:
695:
647:
549:
512:
482:
264:
185:
2281:"NIST: Wavenumber Calibration Tables - Description"
2047:It can also be converted into wavelength of light:
2092:
2036:
1932:
1826:For example, the spectroscopic wavenumbers of the
1791:
1723:
1646:
1563:
1454:
1351:
1231:
1117:
1080:
1028:
971:
942:
917:
885:
848:
819:
675:
608:
525:
495:
374:
290:
218:
2187:. Springer International Publishing. p. 73.
1986:A spectroscopic wavenumber can be converted into
2181:Rodrigues, A.; Sardinha, R.A.; Pita, G. (2021).
442:per meter (rad⋅m), or as above, since the
176:per unit distance, typically centimeters (cm):
2184:Fundamental Principles of Environmental Physics
291:{\displaystyle k\;=\;{\frac {2\pi }{\lambda }}}
1969:of the initial and final levels respectively (
301:When wavenumber is represented by the symbol
8:
2330:: CS1 maint: multiple names: authors list (
2160:
2158:
1662:(usually in centimeters per second, cm⋅s):
816:
356:
352:
341:
337:
272:
268:
202:
198:
2073:
2072:
2063:
2055:
2020:
2019:
2005:
1914:
1907:
1902:
1896:
1885:
1878:
1873:
1867:
1845:
1844:
1842:
1774:
1769:
1753:
1752:
1746:
1700:
1687:
1673:
1672:
1670:
1633:
1632:
1630:
1543:
1530:
1512:
1504:
1442:
1424:
1416:
1340:
1339:
1330:
1318:
1317:
1300:
1282:
1274:
1214:
1192:
1183:
1158:
1150:
1105:
1097:
1045:
993:
961:
932:
907:
870:
833:
727:
694:
658:
646:
597:
582:
572:
566:
560:
548:
517:
511:
487:
481:
358:
357:
342:
326:
320:
318:
273:
263:
203:
187:
186:
184:
99:, the wavenumber is the magnitude of the
139:waves, the wavenumber multiplied by the
29:
2538:. John Wiley & Sons. p. xxii.
2154:
1556:
1535:
1430:
2323:
1609:Wavenumber is also used to define the
105:. The space of wave vectors is called
391:is a frequency expressed in the unit
7:
2420:Time-Harmonic Electromagnetic Fields
2247:Robinson, E.A.; Treitel, S. (2008).
2037:{\displaystyle E=hc{\tilde {\nu }}.}
1828:emission spectrum of atomic hydrogen
1806:and the distance between fringes in
410:is the reciprocal of meters (m). In
2493:"Boson Peaks and Glass Formation".
1254:In general, the angular wavenumber
1081:{\displaystyle \mu =\mu '-j\mu ''=}
414:it is usual to give wavenumbers in
1816:Rydberg–Ritz combination principle
1760:
1757:
1754:
1341:
1319:
25:
2562:
496:{\displaystyle \varepsilon _{r}}
1647:{\displaystyle {\tilde {\nu }}}
1207:
1178:
80:). It is analogous to temporal
2509:10.1126/science.308.5726.1221a
2078:
2025:
1850:
1779:
1678:
1638:
1366:is the frequency of the wave,
1118:{\displaystyle j={\sqrt {-1}}}
363:
192:
1:
2473:10.1016/S0022-4073(00)00044-3
2422:(1st ed.), McGraw-Hill,
2418:Harrington, Roger F. (1961),
2394:10.1016/0038-1098(81)90960-1
88:) or radians per unit time (
2532:Hollas, J. Michael (2004).
1400:For the special case of an
634:Plane waves in linear media
27:Spatial frequency of a wave
2616:
2590:Scalar physical quantities
2374:Solid State Communications
1492:For the special case of a
676:{\displaystyle P=e^{-jkx}}
1967:principal quantum numbers
950:distance traveled in the
455:electromagnetic radiation
2311:scienceworld.wolfram.com
1660:speed of light in vacuum
972:{\displaystyle \sigma =}
918:{\displaystyle \omega =}
526:{\displaystyle \mu _{r}}
115:and the physics of wave
97:multidimensional systems
2351:Encyclopædia Britannica
1604:reduced Planck constant
1479:reduced Planck constant
168:Wavenumber, as used in
141:reduced Planck constant
2445:Fiechtner, G. (2001).
2094:
2038:
1934:
1793:
1725:
1656:reciprocal centimeters
1648:
1565:
1456:
1353:
1233:
1119:
1082:
1030:
973:
944:
919:
887:
850:
821:
677:
610:
527:
497:
376:
292:
220:
35:
2305:W., Weisstein, Eric.
2214:Solimini, D. (2016).
2095:
2039:
1935:
1794:
1726:
1649:
1598:of the particle, and
1566:
1457:
1354:
1234:
1120:
1083:
1031:
974:
945:
920:
888:
851:
822:
678:
611:
528:
498:
377:
293:
221:
33:
2571:at Wikimedia Commons
2503:(5726): 1221. 2005.
2054:
2004:
1841:
1804:diffraction gratings
1745:
1669:
1629:
1503:
1415:
1402:electromagnetic wave
1273:
1149:
1096:
1089:complex permeability
1044:
1037:complex permittivity
992:
960:
931:
906:
894:attenuation constant
886:{\displaystyle k''=}
869:
832:
693:
645:
547:
510:
480:
317:
262:
183:
154:optical spectroscopy
129:electron diffraction
2535:Modern spectroscopy
2465:2001JQSRT..68..543F
2386:1981SSCom..38..651M
1395:dispersion relation
1370:is the wavelength,
849:{\displaystyle k'=}
133:elementary particle
125:neutron diffraction
76:per unit distance (
70:ordinary wavenumber
68:per unit distance (
2595:Units of frequency
2128:Angular wavelength
2090:
2034:
1930:
1814:in the 1880s. The
1789:
1721:
1644:
1561:
1452:
1349:
1229:
1115:
1078:
1026:
969:
943:{\displaystyle x=}
940:
915:
883:
846:
817:
673:
606:
523:
493:
372:
288:
216:
146:canonical momentum
137:quantum mechanical
86:ordinary frequency
78:angular wavenumber
36:
2567:Media related to
2441:See for example,
2260:978-1-56080-148-1
2227:978-3-319-25633-7
2194:978-3-030-69025-0
2133:Spatial frequency
2085:
2081:
2028:
1995:Planck's relation
1988:energy per photon
1920:
1910:
1891:
1881:
1853:
1830:are given by the
1784:
1782:
1716:
1695:
1681:
1641:
1590:of the particle,
1582:of the particle,
1559:
1555:
1538:
1525:
1450:
1437:
1382:of the wave, and
1380:angular frequency
1347:
1325:
1295:
1243:In wave equations
1227:
1205:
1176:
1113:
925:angular frequency
814:
628:evanescent fields
588:
465:in spectroscopy.
404:reciprocal length
366:
350:
335:
329:
286:
235:spatial frequency
211:
195:
121:X-ray diffraction
90:angular frequency
57:spatial frequency
50:), also known as
40:physical sciences
16:(Redirected from
2607:
2566:
2550:
2549:
2529:
2523:
2520:
2489:
2488:
2484:
2476:
2439:
2433:
2432:
2415:
2409:
2404:
2398:
2397:
2369:
2363:
2362:
2360:
2358:
2342:
2336:
2335:
2329:
2321:
2319:
2317:
2302:
2296:
2295:
2293:
2291:
2285:physics.nist.gov
2277:
2271:
2270:
2268:
2267:
2244:
2238:
2237:
2235:
2234:
2211:
2205:
2204:
2202:
2201:
2178:
2172:
2171:
2162:
2143:Zonal wavenumber
2138:Refractive index
2109:refractive index
2099:
2097:
2096:
2091:
2086:
2084:
2083:
2082:
2074:
2064:
2043:
2041:
2040:
2035:
2030:
2029:
2021:
1976:is greater than
1949:Rydberg constant
1939:
1937:
1936:
1931:
1926:
1922:
1921:
1919:
1918:
1913:
1912:
1911:
1908:
1897:
1892:
1890:
1889:
1884:
1883:
1882:
1879:
1868:
1855:
1854:
1846:
1812:Johannes Rydberg
1798:
1796:
1795:
1790:
1785:
1783:
1775:
1770:
1765:
1764:
1763:
1730:
1728:
1727:
1722:
1717:
1715:
1701:
1696:
1688:
1683:
1682:
1674:
1653:
1651:
1650:
1645:
1643:
1642:
1634:
1570:
1568:
1567:
1562:
1560:
1545:
1544:
1539:
1531:
1526:
1521:
1513:
1461:
1459:
1458:
1453:
1451:
1443:
1438:
1436:
1425:
1358:
1356:
1355:
1350:
1348:
1346:
1345:
1344:
1331:
1326:
1324:
1323:
1322:
1312:
1301:
1296:
1291:
1283:
1238:
1236:
1235:
1230:
1228:
1226:
1215:
1206:
1204:
1193:
1188:
1187:
1177:
1175:
1167:
1159:
1124:
1122:
1121:
1116:
1114:
1106:
1087:
1085:
1084:
1079:
1074:
1060:
1035:
1033:
1032:
1027:
1022:
1008:
978:
976:
975:
970:
949:
947:
946:
941:
924:
922:
921:
916:
896:in the units of
892:
890:
889:
884:
879:
859:in the units of
855:
853:
852:
847:
842:
826:
824:
823:
818:
815:
813:
809:
808:
791:
769:
765:
764:
747:
728:
723:
709:
682:
680:
679:
674:
672:
671:
615:
613:
612:
607:
602:
601:
589:
587:
586:
577:
576:
567:
565:
564:
535:refraction index
532:
530:
529:
524:
522:
521:
502:
500:
499:
494:
492:
491:
437:
433:
387:
381:
379:
378:
373:
368:
367:
359:
351:
343:
336:
331:
330:
327:
321:
308:
297:
295:
294:
289:
287:
282:
274:
247:
245:
225:
223:
222:
217:
212:
204:
197:
196:
188:
108:reciprocal space
21:
2615:
2614:
2610:
2609:
2608:
2606:
2605:
2604:
2575:
2574:
2559:
2554:
2553:
2546:
2531:
2530:
2526:
2492:
2486:
2479:
2444:
2440:
2436:
2430:
2417:
2416:
2412:
2405:
2401:
2371:
2370:
2366:
2356:
2354:
2344:
2343:
2339:
2322:
2315:
2313:
2304:
2303:
2299:
2289:
2287:
2279:
2278:
2274:
2265:
2263:
2261:
2246:
2245:
2241:
2232:
2230:
2228:
2213:
2212:
2208:
2199:
2197:
2195:
2180:
2179:
2175:
2164:
2163:
2156:
2151:
2124:
2068:
2052:
2051:
2002:
2001:
1983:for emission).
1982:
1975:
1964:
1957:
1903:
1901:
1874:
1872:
1866:
1862:
1839:
1838:
1832:Rydberg formula
1808:interferometers
1748:
1743:
1742:
1705:
1667:
1666:
1627:
1626:
1625:, "wavenumber"
1619:
1617:In spectroscopy
1514:
1501:
1500:
1429:
1413:
1412:
1388:
1335:
1313:
1302:
1284:
1271:
1270:
1245:
1219:
1197:
1179:
1168:
1160:
1147:
1146:
1094:
1093:
1067:
1053:
1042:
1041:
1015:
1001:
990:
989:
958:
957:
929:
928:
904:
903:
872:
867:
866:
835:
830:
829:
801:
784:
774:
770:
757:
740:
736:
732:
716:
702:
691:
690:
654:
643:
642:
636:
625:
593:
578:
568:
556:
545:
544:
513:
508:
507:
483:
478:
477:
471:
435:
431:
424:Heinrich Kayser
398:Wavenumber has
390:
386:
383:
322:
315:
314:
306:
302:
275:
260:
259:
243:
241:
181:
180:
166:
28:
23:
22:
15:
12:
11:
5:
2613:
2611:
2603:
2602:
2597:
2592:
2587:
2585:Wave mechanics
2577:
2576:
2573:
2572:
2558:
2557:External links
2555:
2552:
2551:
2545:978-0470844151
2544:
2524:
2522:
2521:
2490:
2477:
2459:(5): 543–557.
2434:
2428:
2410:
2399:
2380:(7): 651–652.
2364:
2337:
2297:
2272:
2259:
2239:
2226:
2206:
2193:
2173:
2153:
2152:
2150:
2147:
2146:
2145:
2140:
2135:
2130:
2123:
2120:
2101:
2100:
2089:
2080:
2077:
2071:
2067:
2062:
2059:
2045:
2044:
2033:
2027:
2024:
2018:
2015:
2012:
2009:
1980:
1973:
1962:
1955:
1941:
1940:
1929:
1925:
1917:
1906:
1900:
1895:
1888:
1877:
1871:
1865:
1861:
1858:
1852:
1849:
1820:quantum theory
1800:
1799:
1788:
1781:
1778:
1773:
1768:
1762:
1759:
1756:
1751:
1736:interferometer
1732:
1731:
1720:
1714:
1711:
1708:
1704:
1699:
1694:
1691:
1686:
1680:
1677:
1640:
1637:
1618:
1615:
1611:group velocity
1596:kinetic energy
1572:
1571:
1558:
1554:
1551:
1548:
1542:
1537:
1534:
1529:
1524:
1520:
1517:
1511:
1508:
1487:speed of light
1463:
1462:
1449:
1446:
1441:
1435:
1432:
1428:
1423:
1420:
1391:phase velocity
1386:
1360:
1359:
1343:
1338:
1334:
1329:
1321:
1316:
1311:
1308:
1305:
1299:
1294:
1290:
1287:
1281:
1278:
1266:) is given by
1244:
1241:
1240:
1239:
1225:
1222:
1218:
1213:
1210:
1203:
1200:
1196:
1191:
1186:
1182:
1174:
1171:
1166:
1163:
1157:
1154:
1136:phase velocity
1126:
1125:
1112:
1109:
1104:
1101:
1091:
1077:
1073:
1070:
1066:
1063:
1059:
1056:
1052:
1049:
1039:
1025:
1021:
1018:
1014:
1011:
1007:
1004:
1000:
997:
987:
968:
965:
955:
939:
936:
926:
914:
911:
901:
882:
878:
875:
864:
857:phase constant
845:
841:
838:
827:
812:
807:
804:
800:
797:
794:
790:
787:
783:
780:
777:
773:
768:
763:
760:
756:
753:
750:
746:
743:
739:
735:
731:
726:
722:
719:
715:
712:
708:
705:
701:
698:
684:
683:
670:
667:
664:
661:
657:
653:
650:
635:
632:
623:
617:
616:
605:
600:
596:
592:
585:
581:
575:
571:
563:
559:
555:
552:
520:
516:
490:
486:
470:
467:
463:unit of energy
388:
384:
371:
365:
362:
355:
349:
346:
340:
334:
325:
304:
299:
298:
285:
281:
278:
271:
267:
250:speed of light
227:
226:
215:
210:
207:
201:
194:
191:
165:
162:
158:speed of light
64:, measured in
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2612:
2601:
2598:
2596:
2593:
2591:
2588:
2586:
2583:
2582:
2580:
2570:
2565:
2561:
2560:
2556:
2547:
2541:
2537:
2536:
2528:
2525:
2518:
2514:
2510:
2506:
2502:
2498:
2497:
2491:
2482:
2478:
2474:
2470:
2466:
2462:
2458:
2454:
2453:
2448:
2443:
2442:
2438:
2435:
2431:
2429:0-07-026745-6
2425:
2421:
2414:
2411:
2408:, eq.(2.13.3)
2407:
2403:
2400:
2395:
2391:
2387:
2383:
2379:
2375:
2368:
2365:
2353:
2352:
2347:
2346:"Wave number"
2341:
2338:
2333:
2327:
2312:
2308:
2301:
2298:
2286:
2282:
2276:
2273:
2262:
2256:
2252:
2251:
2243:
2240:
2229:
2223:
2219:
2218:
2210:
2207:
2196:
2190:
2186:
2185:
2177:
2174:
2169:
2168:
2161:
2159:
2155:
2148:
2144:
2141:
2139:
2136:
2134:
2131:
2129:
2126:
2125:
2121:
2119:
2116:
2114:
2110:
2106:
2087:
2075:
2069:
2065:
2060:
2057:
2050:
2049:
2048:
2031:
2022:
2016:
2013:
2010:
2007:
2000:
1999:
1998:
1996:
1992:
1989:
1984:
1979:
1972:
1968:
1961:
1954:
1950:
1946:
1927:
1923:
1915:
1904:
1898:
1893:
1886:
1875:
1869:
1863:
1859:
1856:
1847:
1837:
1836:
1835:
1833:
1829:
1824:
1821:
1817:
1813:
1809:
1805:
1786:
1776:
1771:
1766:
1749:
1741:
1740:
1739:
1737:
1718:
1712:
1709:
1706:
1702:
1697:
1692:
1689:
1684:
1675:
1665:
1664:
1663:
1661:
1657:
1635:
1624:
1616:
1614:
1612:
1607:
1605:
1601:
1597:
1593:
1589:
1585:
1581:
1577:
1552:
1549:
1546:
1540:
1532:
1527:
1522:
1518:
1515:
1509:
1506:
1499:
1498:
1497:
1495:
1490:
1489:in a vacuum.
1488:
1484:
1480:
1476:
1473:of the wave,
1472:
1468:
1447:
1444:
1439:
1433:
1426:
1421:
1418:
1411:
1410:
1409:
1408:is given by:
1407:
1403:
1398:
1396:
1392:
1385:
1381:
1377:
1373:
1369:
1365:
1336:
1332:
1327:
1314:
1309:
1306:
1303:
1297:
1292:
1288:
1285:
1279:
1276:
1269:
1268:
1267:
1265:
1261:
1257:
1252:
1250:
1242:
1223:
1220:
1216:
1211:
1208:
1201:
1198:
1194:
1189:
1184:
1180:
1172:
1169:
1164:
1161:
1155:
1152:
1145:
1144:
1143:
1141:
1137:
1133:
1129:
1110:
1107:
1102:
1099:
1092:
1090:
1075:
1071:
1068:
1064:
1061:
1057:
1054:
1050:
1047:
1040:
1038:
1023:
1019:
1016:
1012:
1009:
1005:
1002:
998:
995:
988:
985:
981:
966:
963:
956:
953:
937:
934:
927:
912:
909:
902:
899:
895:
880:
876:
873:
865:
862:
858:
843:
839:
836:
828:
810:
805:
802:
798:
795:
792:
788:
785:
781:
778:
775:
771:
766:
761:
758:
754:
751:
748:
744:
741:
737:
733:
729:
724:
720:
717:
713:
710:
706:
703:
699:
696:
689:
688:
687:
668:
665:
662:
659:
655:
651:
648:
641:
640:
639:
633:
631:
629:
622:
603:
598:
594:
590:
583:
579:
573:
569:
561:
557:
553:
550:
543:
542:
541:
539:
536:
518:
514:
506:
488:
484:
476:
468:
466:
464:
460:
456:
451:
449:
448:dimensionless
445:
441:
429:
425:
421:
417:
413:
409:
405:
401:
396:
394:
369:
360:
353:
347:
344:
338:
332:
323:
312:
307:
283:
279:
276:
269:
265:
258:
257:
256:
253:
251:
238:
236:
232:
213:
208:
205:
199:
189:
179:
178:
177:
175:
171:
163:
161:
159:
155:
150:
148:
147:
142:
138:
135:physics. For
134:
130:
126:
122:
118:
114:
110:
109:
104:
103:
98:
93:
91:
87:
83:
79:
75:
71:
67:
63:
59:
58:
53:
49:
45:
41:
32:
19:
18:Kayser (unit)
2534:
2527:
2500:
2494:
2456:
2450:
2437:
2419:
2413:
2402:
2377:
2373:
2367:
2355:. Retrieved
2349:
2340:
2314:. Retrieved
2310:
2300:
2288:. Retrieved
2284:
2275:
2264:. Retrieved
2249:
2242:
2231:. Retrieved
2216:
2209:
2198:. Retrieved
2183:
2176:
2166:
2117:
2104:
2102:
2046:
1990:
1985:
1977:
1970:
1959:
1952:
1944:
1942:
1825:
1801:
1733:
1623:spectroscopy
1620:
1608:
1599:
1591:
1583:
1575:
1573:
1491:
1482:
1474:
1466:
1464:
1405:
1399:
1383:
1375:
1371:
1367:
1363:
1361:
1255:
1253:
1246:
1130:
1127:
980:conductivity
951:
685:
637:
620:
618:
537:
505:permeability
475:permittivity
472:
452:
427:
419:
412:spectroscopy
397:
300:
254:
239:
230:
228:
170:spectroscopy
167:
151:
144:
106:
100:
94:
89:
85:
77:
69:
55:
51:
47:
43:
37:
1494:matter wave
1264:wave vector
503:, relative
246: cm/ns
174:wavelengths
102:wave vector
48:wave number
2579:Categories
2569:Wavenumber
2481:US 5046846
2266:2022-12-04
2233:2022-12-04
2200:2022-12-04
2149:References
1258:(i.e. the
1249:wavepacket
1140:skin depth
1132:Wavelength
400:dimensions
164:Definition
119:, such as
117:scattering
44:wavenumber
2600:Quotients
2517:220096687
2079:~
2076:ν
2058:λ
2026:~
2023:ν
1894:−
1851:~
1848:ν
1780:~
1777:ν
1750:λ
1710:π
1703:ω
1690:ν
1679:~
1676:ν
1639:~
1636:ν
1557:ℏ
1536:ℏ
1523:λ
1519:π
1510:≡
1445:ω
1431:ℏ
1333:ω
1310:ν
1307:π
1293:λ
1289:π
1260:magnitude
1209:δ
1195:ω
1165:π
1153:λ
1108:−
1069:μ
1062:−
1055:μ
1048:μ
1017:ε
1010:−
1003:ε
996:ε
964:σ
954:direction
910:ω
803:ε
799:ω
786:ε
782:ω
776:σ
759:μ
755:ω
742:μ
738:ω
730:−
711:−
660:−
580:μ
570:ε
515:μ
485:ε
430:, where 1
406:, so its
364:~
361:ν
354:≡
348:λ
324:ν
311:frequency
284:λ
280:π
209:λ
193:~
190:ν
82:frequency
54:, is the
52:repetency
2357:19 April
2326:cite web
2316:19 March
2290:19 March
2122:See also
1965:are the
1580:momentum
1224:″
1202:′
1173:′
1072:″
1058:′
1020:″
1006:′
877:″
840:′
806:′
789:″
762:′
745:″
721:″
707:′
422:, after
416:cgs unit
2496:Science
2461:Bibcode
2382:Bibcode
2111:of the
2107:is the
1947:is the
1602:is the
1594:is the
1586:is the
1578:is the
1485:is the
1477:is the
1469:is the
1389:is the
1378:is the
1262:of the
984:Siemens
861:radians
469:Complex
408:SI unit
143:is the
74:radians
38:In the
2542:
2515:
2487:
2426:
2257:
2224:
2191:
2113:medium
2103:where
1951:, and
1943:where
1481:, and
1471:energy
1465:where
1362:where
1138:, and
986:/meter
900:/meter
898:nepers
863:/meter
686:where
619:where
459:photon
444:radian
440:radian
436:
432:
420:kayser
385:ν
382:where
305:ν
242:29.979
229:where
131:, and
113:optics
66:cycles
42:, the
2513:S2CID
1574:Here
434:K = 1
393:hertz
248:(the
72:) or
60:of a
2540:ISBN
2424:ISBN
2359:2015
2332:link
2318:2018
2292:2018
2255:ISBN
2222:ISBN
2189:ISBN
1958:and
1654:(in
1588:mass
540:as:
533:and
453:For
309:, a
244:2458
62:wave
46:(or
2505:doi
2501:308
2469:doi
2390:doi
1993:by
1621:In
1374:= 2
982:in
446:is
402:of
237:.
95:In
92:).
2581::
2511:.
2499:.
2467:.
2457:68
2455:.
2449:.
2388:.
2378:38
2376:.
2348:.
2328:}}
2324:{{
2309:.
2283:.
2157:^
1997::
1834::
1613:.
1606:.
1397:.
1376:πν
1134:,
630:.
450:.
160:.
149:.
127:,
123:,
2548:.
2519:.
2507::
2475:.
2471::
2463::
2396:.
2392::
2384::
2361:.
2334:)
2320:.
2294:.
2269:.
2236:.
2203:.
2170:.
2105:n
2088:,
2070:n
2066:1
2061:=
2032:.
2017:c
2014:h
2011:=
2008:E
1991:E
1981:f
1978:n
1974:i
1971:n
1963:f
1960:n
1956:i
1953:n
1945:R
1928:,
1924:)
1916:2
1909:i
1905:n
1899:1
1887:2
1880:f
1876:n
1870:1
1864:(
1860:R
1857:=
1787:,
1772:1
1767:=
1761:c
1758:a
1755:v
1719:.
1713:c
1707:2
1698:=
1693:c
1685:=
1600:ħ
1592:E
1584:m
1576:p
1553:E
1550:m
1547:2
1541:=
1533:p
1528:=
1516:2
1507:k
1483:c
1475:ħ
1467:E
1448:c
1440:=
1434:c
1427:E
1422:=
1419:k
1406:k
1387:p
1384:v
1372:ω
1368:λ
1364:ν
1342:p
1337:v
1328:=
1320:p
1315:v
1304:2
1298:=
1286:2
1280:=
1277:k
1256:k
1221:k
1217:1
1212:=
1199:k
1190:=
1185:p
1181:v
1170:k
1162:2
1156:=
1111:1
1103:=
1100:j
1076:=
1065:j
1051:=
1024:=
1013:j
999:=
967:=
952:x
938:=
935:x
913:=
881:=
874:k
844:=
837:k
811:)
796:j
793:+
779:+
772:(
767:)
752:j
749:+
734:(
725:=
718:k
714:j
704:k
700:=
697:k
669:x
666:k
663:j
656:e
652:=
649:P
624:0
621:k
604:n
599:0
595:k
591:=
584:r
574:r
562:0
558:k
554:=
551:k
538:n
519:r
489:r
428:K
389:s
370:,
345:1
339:=
333:c
328:s
277:2
270:=
266:k
231:λ
214:,
206:1
200:=
20:)
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