2546:
a faster group velocity than low-frequency ones, so the front of the precursor should have a period corresponding to that of the highest-frequency component of the original impulse; with increasing time, components with lower and lower frequencies arrive, so the period of the precursor becomes longer and longer until the lowest-frequency component arrives. As more and more components arrive, the amplitude of the precursor also increases. The particular type of precursor characterized by increasing period and amplitude is known as the
72:
experimental lag is mainly due to the fact that in many situations, precursors have a much smaller amplitude than the signals that give rise to them (a baseline figure given by
Brillouin is six orders of magnitude smaller). As a result, experimental confirmations could only be done after technology became available to detect precursors.
1960:
2553:
In a region of anomalous dispersion, where low-frequency components have faster group velocities than high-frequency ones, the opposite of the above situation occurs: the onset of the precursor is characterized by a long period, and the period of the signal decreases with time. This type of precursor
2562:
time and distance, the precursor waveform will consist of a superposition of both low- and high-frequency
Sommerfeld precursors. Any local extrema only correspond to single frequencies, so at these points there will be a contribution from a precursor signal with a constant period; this is known as a
2545:
Therefore, one can determine the approximate period of a precursor waveform at a particular distance and time by calculating the period of the frequency component that would arrive at that distance and time based on its group velocity. In a region of normal dispersion, high-frequency components have
2561:
In certain situations of wave propagation (for instance, fluid surface waves), two or more frequency components may have the same group velocity for particular ranges of frequency; this is typically accompanied by a local extremum in the group velocity curve. This means that for certain values of
26:
of an impulse's frequency components as it propagates through a medium. Classically, precursors precede the main signal, although in certain situations they may also follow it. Precursor phenomena exist for all types of waves, as their appearance is only predicated on the prominence of dispersion
71:
to compute the integrals involved. However, it was not until 1969 that precursors were first experimentally confirmed for the case of microwaves propagating in a waveguide, and much of the experimental work observing precursors in other types of waves has only been done since the year 2000. This
2258:
700:
540:
819:
2116:
1258:
2540:
1810:
1799:
1615:
227:
2310:
can be used to analyze the form of precursor waves without solving the general-form integral given in the Basic Theory section above. The stationary phase approximation states that for any speed of wave propagation
901:
2294:
of the first kind. This solution, which is an oscillatory function with amplitude and period that both increase with increasing time, is characteristic of a particular type of precursor known as the
1475:
551:
442:
711:
275:
1308:
2147:
1390:
1027:
63:
for the case of electromagnetic radiation propagating through a neutral dielectric in a region of normal dispersion. Sommerfeld's work was expanded in the following years by
2434:
2336:
1015:
1955:{\displaystyle \omega {\sqrt {\frac {t'}{\xi }}}=e^{ik},\qquad {\frac {d\omega }{\omega }}=idk,\qquad {\frac {d\omega }{\omega ^{2}}}=i{\sqrt {\frac {t'}{\xi }}}e^{-ik}dk}
2442:
2403:
988:
389:
352:
1358:
1971:
1410:
2288:
1626:
1483:
434:
86:
2741:
Rost, Sebastian; Garnero, Edward J.; Williams, Quentin; Manga, Michael (2005). "Seismological constraints on a possible plume root at the core–mantle boundary".
2376:
2356:
2139:
1328:
961:
941:
921:
80:
As a dispersive phenomenon, the amplitude at any distance and time of a precursor wave propagating in one dimension can be expressed by the
Fourier integral
705:
For simplicity, we assume the frequencies involved are all in a range of normal dispersion for the medium, and we let the dispersion relation take the form
398:
when idealized assumptions are made about the initial impulse and the dispersion relation, as in
Sommerfeld's derivation below. In most realistic cases,
27:
effects in a given mode of wave propagation. This non-specificity has been confirmed by the observation of precursor patterns in different types of
827:
2585:
Pleshko, Peter; Palócz, István (1969-06-02). "Experimental
Observation of Sommerfeld and Brillouin Precursors in the Microwave Domain".
1415:
2307:
695:{\displaystyle f(x,t)=-{\frac {1}{\tau }}\int e^{-i(k(\omega )x-\omega t)}{\frac {d\omega }{\omega ^{2}-(2\pi /\tau )^{2}}}.}
535:{\displaystyle f(t)=\left\{{\begin{array}{rl}0&t<0\\\sin {\frac {2\pi t}{\tau }}&t\geq 0\end{array}}\right.,}
2696:
Falcon, Éric; Laroche, Claude; Fauve, Stéphan (2003-08-07). "Observation of
Sommerfeld Precursors on a Fluid Surface".
814:{\displaystyle k(\omega )={\frac {\omega }{c}}{\sqrt {1+{\frac {a^{2}\omega _{0}^{2}}{\omega _{0}^{2}-\omega ^{2}}}}}}
68:
2620:
Aaviksoo, J.; Kuhl, J.; Ploog, K. (1991-11-01). "Observation of optical precursors at pulse propagation in GaAs".
235:
28:
358:
summed in the integral. To account for the effects of dispersion, the phase of the exponential must include the
2253:{\displaystyle f(\xi ,t')={\frac {2\pi }{\tau }}{\sqrt {\frac {t'}{\xi }}}J_{1}\left(2{\sqrt {\xi t'}}\right),}
1363:
1253:{\displaystyle f(x,t)=-{\frac {1}{\tau }}\int \exp \left{\frac {d\omega }{\omega ^{2}-(2\pi /\tau )^{2}}}.}
395:
1269:
399:
1310:, which is necessary to ensure that the solution does not violate causality by propagating faster than
2535:{\displaystyle v_{g}(\omega _{D})=\left.{\frac {d\omega }{dk}}\right|_{\omega _{D}}={\frac {x}{t}}.}
2412:
2314:
1018:
993:
359:
40:
23:
2812:
2705:
2381:
2111:{\displaystyle f(\xi ,t')=-{\frac {i}{\tau }}{\sqrt {\frac {t'}{\xi }}}\int \exp \lefte^{-ik}dk,}
966:
2797:(Academic Press, New York, NY, 1950), Vol. 4, p. 88-101, for further details of this derivation.
365:
284:
1333:
2758:
2723:
2678:
2637:
2602:
278:
64:
60:
1395:
2750:
2715:
2668:
2629:
2594:
1794:{\displaystyle f(\xi ,t')=-{\frac {1}{\tau }}\int \exp \left{\frac {d\omega }{\omega ^{2}}}}
1610:{\displaystyle f(\xi ,t')=-{\frac {1}{\tau }}\int \exp \left{\frac {d\omega }{\omega ^{2}}}}
222:{\displaystyle f(x,t)={\frac {1}{2\pi }}\int {\hat {\zeta }}_{0}(\omega )\exp \leftd\omega }
2266:
2291:
413:
2406:
2361:
2341:
2124:
1313:
946:
926:
906:
2806:
1264:
410:
Assuming the initial impulse takes the form of a sinusoid turned on abruptly at time
48:
44:
2719:
2598:
466:
2633:
32:
2762:
2727:
2682:
2641:
2606:
545:
then we can write the general-form integral given in the previous section as
405:
2673:
2656:
406:
Sommerfeld's derivation for electromagnetic waves in a neutral dielectric
2754:
2710:
355:
896:{\displaystyle a^{2}={\frac {Nq^{2}}{m\epsilon _{0}\omega _{0}^{2}}}}
2657:"Brillouin precursor propagation in the THz region in Lorentz media"
391:
factor) for the particular medium in which the wave is propagating.
1263:
To solve this integral, we first express the time in terms of the
36:
2477:
526:
2749:(7042). Springer Science and Business Media LLC: 666–669.
59:
Precursors were first theoretically predicted in 1914 by
1470:{\displaystyle \xi ={\frac {a^{2}\omega _{0}^{2}}{2c}}x}
923:
being the number of atomic oscillators in the medium,
2445:
2415:
2384:
2364:
2344:
2317:
2269:
2150:
2127:
1974:
1813:
1629:
1486:
1418:
1398:
1366:
1336:
1316:
1272:
1030:
996:
969:
949:
929:
909:
830:
714:
554:
445:
416:
368:
287:
238:
89:
2302:
Stationary-Phase-Approximation-Based Period
Analysis
2628:(9). American Physical Society (APS): R5353–R5356.
281:of the initial impulse and the complex exponential
2593:(22). American Physical Society (APS): 1201–1204.
2534:
2428:
2397:
2370:
2350:
2330:
2282:
2252:
2133:
2110:
1954:
1793:
1609:
1469:
1404:
1384:
1352:
1322:
1302:
1252:
1009:
982:
955:
935:
915:
895:
813:
694:
534:
428:
383:
346:
269:
221:
2704:(6). American Physical Society (APS): 064502.
990:the natural frequency of the oscillators, and
8:
2784:(Academic Press, New York, NY, 1960), Ch. 1.
270:{\displaystyle {\hat {\zeta }}_{0}(\omega )}
2776:
2774:
2772:
1965:allows the integral to be transformed into
22:are characteristic wave patterns caused by
2709:
2672:
2580:
2578:
2519:
2508:
2503:
2479:
2463:
2450:
2444:
2416:
2414:
2389:
2383:
2363:
2343:
2318:
2316:
2274:
2268:
2227:
2213:
2192:
2177:
2149:
2141:is simply a dummy variable, and, finally
2126:
2087:
2054:
2014:
2004:
1973:
1934:
1913:
1899:
1885:
1854:
1841:
1817:
1812:
1783:
1769:
1742:
1719:
1709:
1689:
1659:
1628:
1599:
1585:
1551:
1516:
1485:
1447:
1442:
1432:
1425:
1417:
1397:
1367:
1365:
1345:
1337:
1335:
1315:
1290:
1271:
1238:
1226:
1208:
1193:
1163:
1150:
1145:
1133:
1128:
1118:
1111:
1103:
1093:
1055:
1029:
1001:
995:
974:
968:
948:
928:
908:
884:
879:
869:
854:
844:
835:
829:
800:
787:
782:
770:
765:
755:
748:
740:
730:
713:
680:
668:
650:
635:
596:
579:
553:
493:
465:
444:
415:
394:The integral above can only be solved in
367:
286:
252:
241:
240:
237:
142:
131:
130:
111:
88:
2405:of the precursor is the frequency whose
2574:
1392:term in deference to the second-order
1385:{\displaystyle {\frac {2\pi }{\tau }}}
2667:(9). The Optical Society: 4188–4194.
402:is required to compute the integral.
7:
354:represents the individual component
2782:Wave Propagation and Group Velocity
2655:Ni, Xiaohui; Alfano, R. R. (2006).
2548:high-frequency Sommerfeld precursor
1303:{\displaystyle t'=t-{\frac {x}{c}}}
2556:low-frequency Sommerfeld precursor
14:
963:the charge and mass of each one,
2795:Lectures on Theoretical Physics
1884:
1853:
2469:
2456:
2429:{\displaystyle {\frac {x}{t}}}
2331:{\displaystyle {\frac {x}{t}}}
2308:stationary phase approximation
2171:
2154:
1995:
1978:
1650:
1633:
1507:
1490:
1346:
1338:
1235:
1217:
1046:
1034:
724:
718:
677:
659:
630:
615:
609:
603:
570:
558:
455:
449:
378:
372:
319:
313:
264:
258:
246:
188:
182:
154:
148:
136:
105:
93:
1:
2720:10.1103/physrevlett.91.064502
2338:determined from any distance
1804:and making the substitutions
1010:{\displaystyle \epsilon _{0}}
16:Dual-velocity wave phenomenon
1412:term. Lastly, we substitute
2599:10.1103/physrevlett.22.1201
2398:{\displaystyle \omega _{D}}
1021:. This yields the integral
983:{\displaystyle \omega _{0}}
2829:
384:{\displaystyle k(\omega )}
347:{\displaystyle \exp \left}
69:saddle point approximation
2634:10.1103/physreva.44.r5353
2378:, the dominant frequency
1353:{\displaystyle |\omega |}
29:electromagnetic radiation
1360:as large and ignore the
2698:Physical Review Letters
2587:Physical Review Letters
1405:{\displaystyle \omega }
2536:
2430:
2399:
2372:
2352:
2332:
2284:
2254:
2135:
2112:
1956:
1795:
1611:
1471:
1406:
1386:
1354:
1324:
1304:
1254:
1011:
984:
957:
937:
917:
897:
815:
696:
536:
430:
385:
348:
271:
223:
2537:
2431:
2400:
2373:
2353:
2333:
2285:
2283:{\displaystyle J_{1}}
2255:
2136:
2113:
1957:
1796:
1612:
1472:
1407:
1387:
1355:
1325:
1305:
1255:
1012:
985:
958:
938:
918:
898:
816:
697:
537:
431:
400:numerical integration
386:
349:
272:
224:
2674:10.1364/oe.14.004188
2443:
2413:
2382:
2362:
2342:
2315:
2296:Sommerfeld precursor
2267:
2148:
2125:
1972:
1811:
1627:
1484:
1416:
1396:
1364:
1334:
1314:
1270:
1028:
994:
967:
947:
927:
907:
828:
712:
552:
443:
414:
366:
285:
236:
87:
2793:See A. Sommerfeld,
2755:10.1038/nature03620
2564:Brillouin precursor
1452:
1155:
1138:
1019:vacuum permittivity
889:
792:
775:
429:{\displaystyle t=0}
360:dispersion relation
45:fluid surface waves
41:terahertz radiation
2780:See L. Brillouin,
2532:
2426:
2395:
2368:
2348:
2328:
2280:
2250:
2131:
2108:
1952:
1791:
1620:Rewriting this as
1607:
1467:
1438:
1402:
1382:
1350:
1320:
1300:
1250:
1141:
1124:
1007:
980:
953:
933:
913:
893:
875:
811:
778:
761:
692:
532:
524:
426:
381:
344:
267:
219:
67:, who applied the
2622:Physical Review A
2527:
2497:
2424:
2371:{\displaystyle t}
2351:{\displaystyle x}
2326:
2240:
2207:
2206:
2190:
2134:{\displaystyle k}
2067:
2029:
2028:
2012:
1928:
1927:
1905:
1867:
1832:
1831:
1789:
1757:
1756:
1734:
1733:
1717:
1702:
1667:
1605:
1559:
1524:
1462:
1380:
1323:{\displaystyle c}
1298:
1245:
1172:
1170:
1101:
1063:
956:{\displaystyle m}
936:{\displaystyle q}
916:{\displaystyle N}
891:
809:
807:
738:
687:
587:
509:
279:Fourier transform
249:
139:
124:
61:Arnold Sommerfeld
2820:
2798:
2791:
2785:
2778:
2767:
2766:
2738:
2732:
2731:
2713:
2693:
2687:
2686:
2676:
2652:
2646:
2645:
2617:
2611:
2610:
2582:
2541:
2539:
2538:
2533:
2528:
2520:
2515:
2514:
2513:
2512:
2502:
2498:
2496:
2488:
2480:
2468:
2467:
2455:
2454:
2435:
2433:
2432:
2427:
2425:
2417:
2404:
2402:
2401:
2396:
2394:
2393:
2377:
2375:
2374:
2369:
2357:
2355:
2354:
2349:
2337:
2335:
2334:
2329:
2327:
2319:
2289:
2287:
2286:
2281:
2279:
2278:
2259:
2257:
2256:
2251:
2246:
2242:
2241:
2239:
2228:
2218:
2217:
2208:
2202:
2194:
2193:
2191:
2186:
2178:
2170:
2140:
2138:
2137:
2132:
2117:
2115:
2114:
2109:
2098:
2097:
2082:
2078:
2068:
2066:
2055:
2030:
2024:
2016:
2015:
2013:
2005:
1994:
1961:
1959:
1958:
1953:
1945:
1944:
1929:
1923:
1915:
1914:
1906:
1904:
1903:
1894:
1886:
1868:
1863:
1855:
1849:
1848:
1833:
1827:
1819:
1818:
1800:
1798:
1797:
1792:
1790:
1788:
1787:
1778:
1770:
1768:
1764:
1763:
1759:
1758:
1752:
1744:
1743:
1735:
1732:
1721:
1720:
1718:
1710:
1703:
1701:
1690:
1668:
1660:
1649:
1616:
1614:
1613:
1608:
1606:
1604:
1603:
1594:
1586:
1584:
1580:
1579:
1575:
1574:
1560:
1552:
1525:
1517:
1506:
1476:
1474:
1473:
1468:
1463:
1461:
1453:
1451:
1446:
1437:
1436:
1426:
1411:
1409:
1408:
1403:
1391:
1389:
1388:
1383:
1381:
1376:
1368:
1359:
1357:
1356:
1351:
1349:
1341:
1330:. We also treat
1329:
1327:
1326:
1321:
1309:
1307:
1306:
1301:
1299:
1291:
1280:
1259:
1257:
1256:
1251:
1246:
1244:
1243:
1242:
1230:
1213:
1212:
1202:
1194:
1192:
1188:
1187:
1183:
1173:
1171:
1169:
1168:
1167:
1154:
1149:
1139:
1137:
1132:
1123:
1122:
1112:
1104:
1102:
1094:
1064:
1056:
1016:
1014:
1013:
1008:
1006:
1005:
989:
987:
986:
981:
979:
978:
962:
960:
959:
954:
942:
940:
939:
934:
922:
920:
919:
914:
902:
900:
899:
894:
892:
890:
888:
883:
874:
873:
860:
859:
858:
845:
840:
839:
820:
818:
817:
812:
810:
808:
806:
805:
804:
791:
786:
776:
774:
769:
760:
759:
749:
741:
739:
731:
701:
699:
698:
693:
688:
686:
685:
684:
672:
655:
654:
644:
636:
634:
633:
588:
580:
541:
539:
538:
533:
528:
525:
510:
505:
494:
435:
433:
432:
427:
390:
388:
387:
382:
353:
351:
350:
345:
343:
339:
338:
334:
276:
274:
273:
268:
257:
256:
251:
250:
242:
228:
226:
225:
220:
212:
208:
207:
203:
147:
146:
141:
140:
132:
125:
123:
112:
43:) as well as in
2828:
2827:
2823:
2822:
2821:
2819:
2818:
2817:
2803:
2802:
2801:
2792:
2788:
2779:
2770:
2740:
2739:
2735:
2711:physics/0307032
2695:
2694:
2690:
2654:
2653:
2649:
2619:
2618:
2614:
2584:
2583:
2576:
2572:
2504:
2489:
2481:
2476:
2475:
2459:
2446:
2441:
2440:
2411:
2410:
2385:
2380:
2379:
2360:
2359:
2340:
2339:
2313:
2312:
2304:
2292:Bessel function
2270:
2265:
2264:
2232:
2223:
2219:
2209:
2195:
2179:
2163:
2146:
2145:
2123:
2122:
2083:
2059:
2044:
2040:
2017:
1987:
1970:
1969:
1930:
1916:
1895:
1887:
1856:
1837:
1820:
1809:
1808:
1779:
1771:
1745:
1725:
1708:
1704:
1694:
1682:
1678:
1642:
1625:
1624:
1595:
1587:
1567:
1550:
1546:
1539:
1535:
1499:
1482:
1481:
1454:
1428:
1427:
1414:
1413:
1394:
1393:
1369:
1362:
1361:
1332:
1331:
1312:
1311:
1273:
1268:
1267:
1234:
1204:
1203:
1195:
1159:
1140:
1114:
1113:
1089:
1085:
1078:
1074:
1026:
1025:
997:
992:
991:
970:
965:
964:
945:
944:
925:
924:
905:
904:
865:
861:
850:
846:
831:
826:
825:
796:
777:
751:
750:
710:
709:
676:
646:
645:
637:
592:
550:
549:
523:
522:
511:
495:
484:
483:
472:
461:
441:
440:
412:
411:
408:
364:
363:
309:
305:
298:
294:
283:
282:
239:
234:
233:
178:
174:
167:
163:
129:
116:
85:
84:
78:
57:
17:
12:
11:
5:
2826:
2824:
2816:
2815:
2805:
2804:
2800:
2799:
2786:
2768:
2733:
2688:
2661:Optics Express
2647:
2612:
2573:
2571:
2568:
2543:
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2523:
2518:
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2507:
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2492:
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2484:
2478:
2474:
2471:
2466:
2462:
2458:
2453:
2449:
2423:
2420:
2407:group velocity
2392:
2388:
2367:
2347:
2325:
2322:
2303:
2300:
2277:
2273:
2261:
2260:
2249:
2245:
2238:
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2226:
2222:
2216:
2212:
2205:
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2198:
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2182:
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2173:
2169:
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2162:
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2119:
2118:
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2065:
2062:
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2053:
2050:
2047:
2043:
2039:
2036:
2033:
2027:
2023:
2020:
2011:
2008:
2003:
2000:
1997:
1993:
1990:
1986:
1983:
1980:
1977:
1963:
1962:
1951:
1948:
1943:
1940:
1937:
1933:
1926:
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1909:
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977:
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872:
868:
864:
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843:
838:
834:
822:
821:
803:
799:
795:
790:
785:
781:
773:
768:
764:
758:
754:
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737:
734:
729:
726:
723:
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703:
702:
691:
683:
679:
675:
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664:
661:
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632:
629:
626:
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614:
611:
608:
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602:
599:
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591:
586:
583:
578:
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569:
566:
563:
560:
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543:
542:
531:
527:
521:
518:
515:
512:
508:
504:
501:
498:
492:
489:
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482:
479:
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468:
467:
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460:
457:
454:
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425:
422:
419:
407:
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380:
377:
374:
371:
342:
337:
333:
330:
327:
324:
321:
318:
315:
312:
308:
304:
301:
297:
293:
290:
266:
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245:
230:
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206:
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199:
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177:
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159:
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101:
98:
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92:
77:
74:
65:Léon Brillouin
56:
53:
15:
13:
10:
9:
6:
4:
3:
2:
2825:
2814:
2811:
2810:
2808:
2796:
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2783:
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2769:
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2220:
2214:
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2203:
2199:
2196:
2187:
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2167:
2164:
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2144:
2143:
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2128:
2105:
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2056:
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2048:
2045:
2041:
2037:
2034:
2031:
2025:
2021:
2018:
2009:
2006:
2001:
1998:
1991:
1988:
1984:
1981:
1975:
1968:
1967:
1966:
1949:
1946:
1941:
1938:
1935:
1931:
1924:
1920:
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1910:
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1600:
1596:
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1464:
1458:
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1448:
1443:
1439:
1433:
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1422:
1419:
1399:
1377:
1373:
1370:
1342:
1317:
1295:
1292:
1287:
1284:
1281:
1277:
1274:
1266:
1265:retarded time
1247:
1239:
1231:
1227:
1223:
1220:
1214:
1209:
1205:
1199:
1196:
1189:
1184:
1180:
1177:
1174:
1164:
1160:
1156:
1151:
1146:
1142:
1134:
1129:
1125:
1119:
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1108:
1105:
1098:
1095:
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1086:
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1079:
1075:
1071:
1068:
1065:
1060:
1057:
1052:
1049:
1043:
1040:
1037:
1031:
1024:
1023:
1022:
1020:
1002:
998:
975:
971:
950:
930:
910:
885:
880:
876:
870:
866:
862:
855:
851:
847:
841:
836:
832:
801:
797:
793:
788:
783:
779:
771:
766:
762:
756:
752:
745:
742:
735:
732:
727:
721:
715:
708:
707:
706:
689:
681:
673:
669:
665:
662:
656:
651:
647:
641:
638:
627:
624:
621:
618:
612:
606:
600:
597:
593:
589:
584:
581:
576:
573:
567:
564:
561:
555:
548:
547:
546:
529:
519:
516:
513:
506:
502:
499:
496:
490:
487:
480:
477:
474:
469:
462:
458:
452:
446:
439:
438:
437:
423:
420:
417:
403:
401:
397:
392:
375:
369:
361:
357:
340:
335:
331:
328:
325:
322:
316:
310:
306:
302:
299:
295:
291:
288:
280:
261:
253:
243:
216:
213:
209:
204:
200:
197:
194:
191:
185:
179:
175:
171:
168:
164:
160:
157:
151:
143:
133:
126:
120:
117:
113:
108:
102:
99:
96:
90:
83:
82:
81:
75:
73:
70:
66:
62:
54:
52:
50:
49:seismic waves
46:
42:
38:
37:visible light
34:
30:
25:
21:
2794:
2789:
2781:
2746:
2742:
2736:
2701:
2697:
2691:
2664:
2660:
2650:
2625:
2621:
2615:
2590:
2586:
2563:
2560:
2555:
2554:is called a
2552:
2547:
2544:
2305:
2295:
2262:
2120:
1964:
1803:
1619:
1262:
823:
704:
544:
409:
393:
231:
79:
76:Basic theory
58:
19:
18:
396:closed form
362:(here, the
2570:References
1477:, getting
33:microwaves
24:dispersion
20:Precursors
2813:Radiation
2763:0028-0836
2728:0031-9007
2683:1094-4087
2642:1050-2947
2607:0031-9007
2506:ω
2486:ω
2461:ω
2387:ω
2358:and time
2230:ξ
2204:ξ
2188:τ
2184:π
2158:ξ
2089:−
2073:
2057:ξ
2046:−
2038:
2032:∫
2026:ξ
2010:τ
2002:−
1982:ξ
1936:−
1925:ξ
1897:ω
1892:ω
1865:ω
1861:ω
1829:ξ
1815:ω
1781:ω
1776:ω
1754:ξ
1740:ω
1723:ξ
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1684:−
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1670:∫
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1541:−
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1527:∫
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1288:−
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1200:ω
1178:ω
1175:−
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1126:ω
1096:ω
1080:−
1072:
1066:∫
1061:τ
1053:−
999:ϵ
972:ω
877:ω
867:ϵ
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780:ω
763:ω
733:ω
722:ω
674:τ
666:π
657:−
648:ω
642:ω
625:ω
622:−
613:ω
598:−
590:∫
585:τ
577:−
517:≥
507:τ
500:π
491:
376:ω
329:ω
326:−
317:ω
300:−
292:
262:ω
247:^
244:ζ
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134:ζ
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2807:Category
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356:wavelets
2409:equals
277:is the
55:History
2761:
2743:Nature
2726:
2681:
2640:
2605:
2263:where
2121:where
824:where
232:where
39:, and
2706:arXiv
2290:is a
2759:ISSN
2724:ISSN
2679:ISSN
2638:ISSN
2603:ISSN
2306:The
1017:the
943:and
478:<
47:and
2751:doi
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2716:doi
2669:doi
2630:doi
2595:doi
2070:cos
2035:exp
1673:exp
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488:sin
289:exp
158:exp
2809::
2771:^
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2722:.
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