518:
1211:
A true plane wave cannot physically exist, because it would have to fill all space. Nevertheless, the plane wave model is important and widely used in physics. The waves emitted by any source with finite extent into a large homogeneous region of space can be well approximated by plane waves when
2216:
1138:
1964:
658:
1326:
179:
2090:
1640:
1396:
798:
324:
2257:
1020:
1774:
1577:
1498:
1994:
2286:
2085:
1850:
1814:
1706:
1669:
944:
882:
827:
498:
411:
382:
353:
208:
67:
1202:
2056:
722:
247:
1855:
1011:
911:
687:
566:
748:
2306:
2014:
1538:
1518:
1459:
1439:
1416:
1346:
1182:
1158:
964:
853:
561:
465:
434:
271:
87:
1229:
1235:
92:
1780:
1212:
viewed over any part of that region that is sufficiently small compared to its distance from the source. That is the case, for example, of the
37:: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space.
1228:
is a field whose value can be expressed as the product of two functions, one depending only on position, the other only on time. A
1582:
1787:
or not, applied to a plane wave yields a plane wave. Any linear combination of plane waves with the same normal vector
2211:{\displaystyle \nabla \cdot {\vec {F}}({\vec {x}},t)\;=\;{\vec {n}}\cdot \partial _{1}G({\vec {x}}\cdot {\vec {n}},t)}
441:
2402:
2368:
2389:
1351:
753:
279:
2384:
2221:
1711:
2437:
1543:
1464:
1969:
831:
2432:
2322:
1133:{\displaystyle F({\vec {x}},t)=A\sin \left(2\pi f({\vec {x}}\cdot {\vec {n}}-ct)+\varphi \right)}
966:; and the value of the field is then the same, and constant in time, at every one of its points.
274:
2262:
2061:
1826:
1790:
1682:
1645:
920:
858:
803:
474:
387:
358:
329:
184:
43:
1500:
is bounded in the time interval of interest (which is usually the case in physical contexts),
1187:
1959:{\displaystyle \nabla F({\vec {x}},t)={\vec {n}}\partial _{1}G({\vec {x}}\cdot {\vec {n}},t)}
1679:
A plane wave can be studied by ignoring the directions perpendicular to the direction vector
981:
975:
2026:
692:
538:, whose evolution in time can be described as simple translation of the field at a constant
217:
2317:
534:
987:
887:
663:
8:
2352:
727:
2291:
1999:
1523:
1503:
1444:
1424:
1401:
1331:
1167:
1143:
949:
838:
563:
along the direction perpendicular to the wavefronts. Such a field can be written as
546:
450:
419:
256:
72:
436:
may be scalars, vectors, or any other physical or mathematical quantity. They can be
2398:
2394:
2364:
468:
724:, that describes the "profile" of the wave, namely the value of the field at time
1784:
501:
437:
34:
2360:
2332:
1421:
This representation is not unique, since the same field values are obtained if
540:
30:
2426:
2327:
1225:
653:{\displaystyle F({\vec {x}},t)=G\left({\vec {x}}\cdot {\vec {n}}-ct\right)\,}
980:
The term is also used, even more specifically, to mean a "monochromatic" or
2023:
of a vector-valued plane wave depends only on the projection of the vector
517:
1205:
250:
211:
2020:
1213:
1014:
1161:
914:
522:
249:
is a function that gives the field's value as dependent on only two
1820:
1321:{\displaystyle F({\vec {x}},t)=G({\vec {x}}\cdot {\vec {n}})\,S(t)}
526:
22:
384:. The displacement is constant over each plane perpendicular to
174:{\displaystyle F({\vec {x}},t)=G({\vec {x}}\cdot {\vec {n}},t),}
1635:{\displaystyle \left|G({\vec {x}}\cdot {\vec {n}})\right|}
917:". This plane travels along the direction of propagation
1819:
For a scalar plane wave in two or three dimensions, the
1348:
is a function of one scalar parameter (the displacement
1642:
will be the maximum field magnitude seen at the point
2294:
2265:
2224:
2093:
2064:
2029:
2002:
1972:
1858:
1829:
1793:
1714:
1685:
1648:
1585:
1546:
1526:
1506:
1467:
1447:
1427:
1404:
1354:
1334:
1238:
1190:
1170:
1146:
1023:
990:
952:
923:
890:
861:
841:
806:
756:
730:
695:
666:
569:
549:
532:
Often the term "plane wave" refers specifically to a
504:
if they are always orthogonal (perpendicular) to it.
477:
471:
if the vectors are always collinear with the vector
453:
422:
390:
361:
332:
282:
259:
220:
187:
95:
75:
46:
1823:
of the field is always collinear with the direction
1160:, which may be a scalar or a vector, is called the
2300:
2280:
2251:
2218:In particular, a transverse planar wave satisfies
2210:
2079:
2050:
2008:
1988:
1958:
1844:
1808:
1768:
1700:
1663:
1634:
1571:
1532:
1512:
1492:
1453:
1433:
1410:
1390:
1340:
1320:
1196:
1176:
1152:
1132:
1005:
958:
938:
905:
876:
847:
821:
792:
742:
716:
681:
652:
555:
492:
459:
428:
405:
376:
347:
318:
265:
241:
202:
173:
81:
61:
2424:
1216:from a distant star that arrive at a telescope.
89:, the value of such a field can be written as
689:is now a function of a single real parameter
2351:
1391:{\displaystyle d={\vec {x}}\cdot {\vec {n}}}
793:{\displaystyle d={\vec {x}}\cdot {\vec {n}}}
319:{\displaystyle d={\vec {x}}\cdot {\vec {n}}}
1540:can be scaled so that the maximum value of
1184:is its "spatial frequency"; and the scalar
2140:
2136:
2252:{\displaystyle \nabla \cdot {\vec {F}}=0}
1305:
649:
1776:as a wave in a one-dimensional medium.
984:: a travelling plane wave whose profile
969:
516:
18:Type of wave propagating in 3 dimensions
2383:
1769:{\displaystyle G(z,t)=F(z{\vec {n}},t)}
1708:; that is, by considering the function
512:
2425:
1219:
467:are vectors, the wave is said to be a
2016:with respect to the first argument.
1461:are scaled by reciprocal factors. If
1232:, in particular, can be expressed as
1398:) with scalar or vector values, and
1164:of the wave; the scalar coefficient
855:, the moving plane perpendicular to
13:
2225:
2157:
2094:
1974:
1905:
1859:
14:
2449:
1572:{\displaystyle \left|S(t)\right|}
1493:{\displaystyle \left|S(t)\right|}
1418:is a scalar function of time.
507:
2377:
2345:
2272:
2237:
2205:
2193:
2178:
2169:
2147:
2133:
2121:
2112:
2106:
2071:
2045:
2033:
1989:{\displaystyle \partial _{1}G}
1953:
1941:
1926:
1917:
1898:
1886:
1874:
1865:
1836:
1800:
1763:
1751:
1739:
1730:
1718:
1692:
1655:
1624:
1618:
1603:
1594:
1561:
1555:
1482:
1476:
1382:
1367:
1315:
1309:
1302:
1296:
1281:
1272:
1263:
1251:
1242:
1116:
1101:
1086:
1077:
1048:
1036:
1027:
1000:
994:
930:
913:from the origin is called a "
868:
813:
784:
769:
676:
670:
629:
614:
594:
582:
573:
484:
442:complex exponential plane wave
397:
368:
339:
310:
295:
236:
224:
194:
165:
153:
138:
129:
120:
108:
99:
53:
1:
2338:
1996:is the partial derivative of
1674:
525:of a plane wave traveling in
7:
2311:
10:
2454:
2281:{\displaystyle {\vec {x}}}
2080:{\displaystyle {\vec {n}}}
1845:{\displaystyle {\vec {n}}}
1809:{\displaystyle {\vec {n}}}
1701:{\displaystyle {\vec {n}}}
1664:{\displaystyle {\vec {x}}}
973:
939:{\displaystyle {\vec {n}}}
877:{\displaystyle {\vec {n}}}
822:{\displaystyle {\vec {n}}}
493:{\displaystyle {\vec {n}}}
406:{\displaystyle {\vec {n}}}
377:{\displaystyle {\vec {n}}}
348:{\displaystyle {\vec {x}}}
203:{\displaystyle {\vec {n}}}
62:{\displaystyle {\vec {x}}}
2390:Classical Electrodynamics
2393:(3 ed.). New York:
2359:(2 ed.). New York:
1197:{\displaystyle \varphi }
835:. For each displacement
832:direction of propagation
750:, for each displacement
416:The values of the field
273:, and the scalar-valued
2323:Rectilinear propagation
29:is a special case of a
2357:Waves in Layered Media
2302:
2282:
2253:
2212:
2081:
2052:
2051:{\displaystyle G(d,t)}
2010:
1990:
1960:
1846:
1816:is also a plane wave.
1810:
1770:
1702:
1665:
1636:
1573:
1534:
1514:
1494:
1455:
1435:
1412:
1392:
1342:
1322:
1198:
1178:
1154:
1134:
1007:
960:
940:
907:
878:
849:
823:
794:
744:
718:
717:{\displaystyle u=d-ct}
683:
654:
557:
529:
494:
461:
430:
407:
378:
349:
320:
267:
243:
242:{\displaystyle G(d,t)}
204:
175:
83:
69:in space and any time
63:
2303:
2283:
2254:
2213:
2082:
2053:
2011:
1991:
1961:
1847:
1811:
1771:
1703:
1666:
1637:
1574:
1535:
1515:
1495:
1456:
1436:
1413:
1393:
1343:
1323:
1199:
1179:
1155:
1135:
1008:
982:sinusoidal plane wave
976:Sinusoidal plane wave
970:Sinusoidal plane wave
961:
941:
908:
879:
850:
824:
795:
745:
719:
684:
655:
558:
520:
495:
462:
431:
408:
379:
350:
321:
268:
253:parameters: the time
244:
205:
176:
84:
64:
2318:Plane-wave expansion
2292:
2263:
2222:
2091:
2062:
2027:
2000:
1970:
1856:
1827:
1791:
1712:
1683:
1646:
1583:
1544:
1524:
1504:
1465:
1445:
1425:
1402:
1352:
1332:
1236:
1188:
1168:
1144:
1021:
1017:function. That is,
1006:{\displaystyle G(u)}
988:
950:
921:
906:{\displaystyle d+ct}
888:
859:
839:
804:
754:
728:
693:
682:{\displaystyle G(u)}
664:
567:
547:
535:traveling plane wave
513:Traveling plane wave
475:
451:
420:
388:
359:
355:along the direction
330:
280:
257:
218:
185:
93:
73:
44:
2385:Jackson, John David
1230:plane standing wave
1220:Plane standing wave
743:{\displaystyle t=0}
447:When the values of
2298:
2278:
2249:
2208:
2077:
2048:
2006:
1986:
1956:
1842:
1806:
1766:
1698:
1661:
1632:
1569:
1530:
1510:
1490:
1451:
1431:
1408:
1388:
1338:
1318:
1194:
1174:
1150:
1130:
1003:
956:
936:
903:
874:
845:
819:
790:
740:
714:
679:
650:
553:
530:
490:
457:
426:
403:
374:
345:
316:
263:
239:
212:unit-length vector
200:
171:
79:
59:
2438:Planes (geometry)
2301:{\displaystyle t}
2275:
2240:
2196:
2181:
2150:
2124:
2109:
2074:
2058:in the direction
2009:{\displaystyle G}
1944:
1929:
1901:
1877:
1839:
1803:
1754:
1695:
1658:
1621:
1606:
1533:{\displaystyle G}
1513:{\displaystyle S}
1454:{\displaystyle G}
1434:{\displaystyle S}
1411:{\displaystyle S}
1385:
1370:
1341:{\displaystyle G}
1299:
1284:
1254:
1177:{\displaystyle f}
1153:{\displaystyle A}
1104:
1089:
1039:
959:{\displaystyle c}
933:
871:
848:{\displaystyle d}
816:
800:. In that case,
787:
772:
632:
617:
585:
556:{\displaystyle c}
487:
469:longitudinal wave
460:{\displaystyle F}
429:{\displaystyle F}
400:
371:
342:
313:
298:
266:{\displaystyle t}
197:
156:
141:
111:
82:{\displaystyle t}
56:
40:For any position
2445:
2409:
2408:
2381:
2375:
2374:
2363:. pp. 1–3.
2353:Brekhovskikh, L.
2349:
2307:
2305:
2304:
2299:
2287:
2285:
2284:
2279:
2277:
2276:
2268:
2258:
2256:
2255:
2250:
2242:
2241:
2233:
2217:
2215:
2214:
2209:
2198:
2197:
2189:
2183:
2182:
2174:
2165:
2164:
2152:
2151:
2143:
2126:
2125:
2117:
2111:
2110:
2102:
2087:. Specifically,
2086:
2084:
2083:
2078:
2076:
2075:
2067:
2057:
2055:
2054:
2049:
2015:
2013:
2012:
2007:
1995:
1993:
1992:
1987:
1982:
1981:
1965:
1963:
1962:
1957:
1946:
1945:
1937:
1931:
1930:
1922:
1913:
1912:
1903:
1902:
1894:
1879:
1878:
1870:
1852:; specifically,
1851:
1849:
1848:
1843:
1841:
1840:
1832:
1815:
1813:
1812:
1807:
1805:
1804:
1796:
1775:
1773:
1772:
1767:
1756:
1755:
1747:
1707:
1705:
1704:
1699:
1697:
1696:
1688:
1670:
1668:
1667:
1662:
1660:
1659:
1651:
1641:
1639:
1638:
1633:
1631:
1627:
1623:
1622:
1614:
1608:
1607:
1599:
1578:
1576:
1575:
1570:
1568:
1564:
1539:
1537:
1536:
1531:
1519:
1517:
1516:
1511:
1499:
1497:
1496:
1491:
1489:
1485:
1460:
1458:
1457:
1452:
1440:
1438:
1437:
1432:
1417:
1415:
1414:
1409:
1397:
1395:
1394:
1389:
1387:
1386:
1378:
1372:
1371:
1363:
1347:
1345:
1344:
1339:
1327:
1325:
1324:
1319:
1301:
1300:
1292:
1286:
1285:
1277:
1256:
1255:
1247:
1203:
1201:
1200:
1195:
1183:
1181:
1180:
1175:
1159:
1157:
1156:
1151:
1139:
1137:
1136:
1131:
1129:
1125:
1106:
1105:
1097:
1091:
1090:
1082:
1041:
1040:
1032:
1012:
1010:
1009:
1004:
965:
963:
962:
957:
945:
943:
942:
937:
935:
934:
926:
912:
910:
909:
904:
883:
881:
880:
875:
873:
872:
864:
854:
852:
851:
846:
828:
826:
825:
820:
818:
817:
809:
799:
797:
796:
791:
789:
788:
780:
774:
773:
765:
749:
747:
746:
741:
723:
721:
720:
715:
688:
686:
685:
680:
659:
657:
656:
651:
648:
644:
634:
633:
625:
619:
618:
610:
587:
586:
578:
562:
560:
559:
554:
499:
497:
496:
491:
489:
488:
480:
466:
464:
463:
458:
435:
433:
432:
427:
412:
410:
409:
404:
402:
401:
393:
383:
381:
380:
375:
373:
372:
364:
354:
352:
351:
346:
344:
343:
335:
325:
323:
322:
317:
315:
314:
306:
300:
299:
291:
272:
270:
269:
264:
248:
246:
245:
240:
209:
207:
206:
201:
199:
198:
190:
180:
178:
177:
172:
158:
157:
149:
143:
142:
134:
113:
112:
104:
88:
86:
85:
80:
68:
66:
65:
60:
58:
57:
49:
2453:
2452:
2448:
2447:
2446:
2444:
2443:
2442:
2423:
2422:
2421:
2413:
2412:
2405:
2397:. p. 296.
2382:
2378:
2371:
2350:
2346:
2341:
2314:
2293:
2290:
2289:
2267:
2266:
2264:
2261:
2260:
2232:
2231:
2223:
2220:
2219:
2188:
2187:
2173:
2172:
2160:
2156:
2142:
2141:
2116:
2115:
2101:
2100:
2092:
2089:
2088:
2066:
2065:
2063:
2060:
2059:
2028:
2025:
2024:
2001:
1998:
1997:
1977:
1973:
1971:
1968:
1967:
1936:
1935:
1921:
1920:
1908:
1904:
1893:
1892:
1869:
1868:
1857:
1854:
1853:
1831:
1830:
1828:
1825:
1824:
1795:
1794:
1792:
1789:
1788:
1746:
1745:
1713:
1710:
1709:
1687:
1686:
1684:
1681:
1680:
1677:
1650:
1649:
1647:
1644:
1643:
1613:
1612:
1598:
1597:
1590:
1586:
1584:
1581:
1580:
1551:
1547:
1545:
1542:
1541:
1525:
1522:
1521:
1505:
1502:
1501:
1472:
1468:
1466:
1463:
1462:
1446:
1443:
1442:
1426:
1423:
1422:
1403:
1400:
1399:
1377:
1376:
1362:
1361:
1353:
1350:
1349:
1333:
1330:
1329:
1291:
1290:
1276:
1275:
1246:
1245:
1237:
1234:
1233:
1222:
1189:
1186:
1185:
1169:
1166:
1165:
1145:
1142:
1141:
1096:
1095:
1081:
1080:
1067:
1063:
1031:
1030:
1022:
1019:
1018:
989:
986:
985:
978:
972:
951:
948:
947:
925:
924:
922:
919:
918:
889:
886:
885:
863:
862:
860:
857:
856:
840:
837:
836:
808:
807:
805:
802:
801:
779:
778:
764:
763:
755:
752:
751:
729:
726:
725:
694:
691:
690:
665:
662:
661:
624:
623:
609:
608:
607:
603:
577:
576:
568:
565:
564:
548:
545:
544:
515:
510:
502:transverse wave
479:
478:
476:
473:
472:
452:
449:
448:
438:complex numbers
421:
418:
417:
392:
391:
389:
386:
385:
363:
362:
360:
357:
356:
334:
333:
331:
328:
327:
305:
304:
290:
289:
281:
278:
277:
258:
255:
254:
219:
216:
215:
189:
188:
186:
183:
182:
148:
147:
133:
132:
103:
102:
94:
91:
90:
74:
71:
70:
48:
47:
45:
42:
41:
19:
12:
11:
5:
2451:
2441:
2440:
2435:
2433:Wave mechanics
2420:
2419:
2417:
2414:
2411:
2410:
2403:
2376:
2369:
2361:Academic Press
2343:
2342:
2340:
2337:
2336:
2335:
2333:Weyl expansion
2330:
2325:
2320:
2313:
2310:
2297:
2274:
2271:
2248:
2245:
2239:
2236:
2230:
2227:
2207:
2204:
2201:
2195:
2192:
2186:
2180:
2177:
2171:
2168:
2163:
2159:
2155:
2149:
2146:
2139:
2135:
2132:
2129:
2123:
2120:
2114:
2108:
2105:
2099:
2096:
2073:
2070:
2047:
2044:
2041:
2038:
2035:
2032:
2005:
1985:
1980:
1976:
1955:
1952:
1949:
1943:
1940:
1934:
1928:
1925:
1919:
1916:
1911:
1907:
1900:
1897:
1891:
1888:
1885:
1882:
1876:
1873:
1867:
1864:
1861:
1838:
1835:
1802:
1799:
1781:local operator
1765:
1762:
1759:
1753:
1750:
1744:
1741:
1738:
1735:
1732:
1729:
1726:
1723:
1720:
1717:
1694:
1691:
1676:
1673:
1657:
1654:
1630:
1626:
1620:
1617:
1611:
1605:
1602:
1596:
1593:
1589:
1567:
1563:
1560:
1557:
1554:
1550:
1529:
1509:
1488:
1484:
1481:
1478:
1475:
1471:
1450:
1430:
1407:
1384:
1381:
1375:
1369:
1366:
1360:
1357:
1337:
1317:
1314:
1311:
1308:
1304:
1298:
1295:
1289:
1283:
1280:
1274:
1271:
1268:
1265:
1262:
1259:
1253:
1250:
1244:
1241:
1221:
1218:
1193:
1173:
1149:
1140:The parameter
1128:
1124:
1121:
1118:
1115:
1112:
1109:
1103:
1100:
1094:
1088:
1085:
1079:
1076:
1073:
1070:
1066:
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1038:
1035:
1029:
1026:
1002:
999:
996:
993:
974:Main article:
971:
968:
955:
946:with velocity
932:
929:
902:
899:
896:
893:
870:
867:
844:
829:is called the
815:
812:
786:
783:
777:
771:
768:
762:
759:
739:
736:
733:
713:
710:
707:
704:
701:
698:
678:
675:
672:
669:
647:
643:
640:
637:
631:
628:
622:
616:
613:
606:
602:
599:
596:
593:
590:
584:
581:
575:
572:
552:
514:
511:
509:
506:
486:
483:
456:
425:
399:
396:
370:
367:
341:
338:
312:
309:
303:
297:
294:
288:
285:
262:
238:
235:
232:
229:
226:
223:
196:
193:
170:
167:
164:
161:
155:
152:
146:
140:
137:
131:
128:
125:
122:
119:
116:
110:
107:
101:
98:
78:
55:
52:
17:
9:
6:
4:
3:
2:
2450:
2439:
2436:
2434:
2431:
2430:
2428:
2418:
2416:
2415:
2406:
2404:9780471309321
2400:
2396:
2392:
2391:
2386:
2380:
2372:
2370:9780323161626
2366:
2362:
2358:
2354:
2348:
2344:
2334:
2331:
2329:
2328:Wave equation
2326:
2324:
2321:
2319:
2316:
2315:
2309:
2295:
2269:
2246:
2243:
2234:
2228:
2202:
2199:
2190:
2184:
2175:
2166:
2161:
2153:
2144:
2137:
2130:
2127:
2118:
2103:
2097:
2068:
2042:
2039:
2036:
2030:
2022:
2017:
2003:
1983:
1978:
1950:
1947:
1938:
1932:
1923:
1914:
1909:
1895:
1889:
1883:
1880:
1871:
1862:
1833:
1822:
1817:
1797:
1786:
1782:
1777:
1760:
1757:
1748:
1742:
1736:
1733:
1727:
1724:
1721:
1715:
1689:
1672:
1652:
1628:
1615:
1609:
1600:
1591:
1587:
1565:
1558:
1552:
1548:
1527:
1507:
1486:
1479:
1473:
1469:
1448:
1428:
1419:
1405:
1379:
1373:
1364:
1358:
1355:
1335:
1312:
1306:
1293:
1287:
1278:
1269:
1266:
1260:
1257:
1248:
1239:
1231:
1227:
1226:standing wave
1217:
1215:
1209:
1207:
1191:
1171:
1163:
1147:
1126:
1122:
1119:
1113:
1110:
1107:
1098:
1092:
1083:
1074:
1071:
1068:
1064:
1060:
1057:
1054:
1051:
1045:
1042:
1033:
1024:
1016:
997:
991:
983:
977:
967:
953:
927:
916:
900:
897:
894:
891:
865:
842:
834:
833:
810:
781:
775:
766:
760:
757:
737:
734:
731:
711:
708:
705:
702:
699:
696:
673:
667:
645:
641:
638:
635:
626:
620:
611:
604:
600:
597:
591:
588:
579:
570:
550:
543:
542:
537:
536:
528:
524:
519:
508:Special types
505:
503:
481:
470:
454:
445:
443:
439:
423:
414:
394:
365:
336:
326:of the point
307:
301:
292:
286:
283:
276:
260:
252:
233:
230:
227:
221:
213:
191:
168:
162:
159:
150:
144:
135:
126:
123:
117:
114:
105:
96:
76:
50:
38:
36:
32:
28:
24:
16:
2388:
2379:
2356:
2347:
2018:
1818:
1778:
1678:
1579:is 1. Then
1420:
1223:
1210:
979:
884:at distance
830:
539:
533:
531:
446:
415:
275:displacement
39:
26:
20:
15:
1214:light waves
1206:phase shift
2427:Categories
2339:References
2021:divergence
1675:Properties
1015:sinusoidal
541:wave speed
523:wavefronts
440:, as in a
27:plane wave
2273:→
2238:→
2229:⋅
2226:∇
2194:→
2185:⋅
2179:→
2158:∂
2154:⋅
2148:→
2122:→
2107:→
2098:⋅
2095:∇
2072:→
1975:∂
1942:→
1933:⋅
1927:→
1906:∂
1899:→
1875:→
1860:∇
1837:→
1801:→
1752:→
1693:→
1656:→
1619:→
1610:⋅
1604:→
1383:→
1374:⋅
1368:→
1297:→
1288:⋅
1282:→
1252:→
1192:φ
1162:amplitude
1123:φ
1108:−
1102:→
1093:⋅
1087:→
1072:π
1061:
1037:→
931:→
915:wavefront
869:→
814:→
785:→
776:⋅
770:→
706:−
636:−
630:→
621:⋅
615:→
583:→
485:→
398:→
369:→
340:→
311:→
302:⋅
296:→
195:→
154:→
145:⋅
139:→
109:→
54:→
2387:(1998).
2355:(1980).
2312:See also
2259:for all
1966:, where
1821:gradient
1204:is its "
500:, and a
527:3-space
23:physics
2401:
2367:
1785:linear
1328:where
660:where
214:, and
181:where
2395:Wiley
1013:is a
210:is a
35:field
2399:ISBN
2365:ISBN
2288:and
2019:The
1779:Any
1520:and
1441:and
521:The
444:.
251:real
31:wave
25:, a
1208:".
1058:sin
33:or
21:In
2429::
2308:.
1783:,
1671:.
1224:A
413:.
2407:.
2373:.
2296:t
2270:x
2247:0
2244:=
2235:F
2206:)
2203:t
2200:,
2191:n
2176:x
2170:(
2167:G
2162:1
2145:n
2138:=
2134:)
2131:t
2128:,
2119:x
2113:(
2104:F
2069:n
2046:)
2043:t
2040:,
2037:d
2034:(
2031:G
2004:G
1984:G
1979:1
1954:)
1951:t
1948:,
1939:n
1924:x
1918:(
1915:G
1910:1
1896:n
1890:=
1887:)
1884:t
1881:,
1872:x
1866:(
1863:F
1834:n
1798:n
1764:)
1761:t
1758:,
1749:n
1743:z
1740:(
1737:F
1734:=
1731:)
1728:t
1725:,
1722:z
1719:(
1716:G
1690:n
1653:x
1629:|
1625:)
1616:n
1601:x
1595:(
1592:G
1588:|
1566:|
1562:)
1559:t
1556:(
1553:S
1549:|
1528:G
1508:S
1487:|
1483:)
1480:t
1477:(
1474:S
1470:|
1449:G
1429:S
1406:S
1380:n
1365:x
1359:=
1356:d
1336:G
1316:)
1313:t
1310:(
1307:S
1303:)
1294:n
1279:x
1273:(
1270:G
1267:=
1264:)
1261:t
1258:,
1249:x
1243:(
1240:F
1172:f
1148:A
1127:)
1120:+
1117:)
1114:t
1111:c
1099:n
1084:x
1078:(
1075:f
1069:2
1065:(
1055:A
1052:=
1049:)
1046:t
1043:,
1034:x
1028:(
1025:F
1001:)
998:u
995:(
992:G
954:c
928:n
901:t
898:c
895:+
892:d
866:n
843:d
811:n
782:n
767:x
761:=
758:d
738:0
735:=
732:t
712:t
709:c
703:d
700:=
697:u
677:)
674:u
671:(
668:G
646:)
642:t
639:c
627:n
612:x
605:(
601:G
598:=
595:)
592:t
589:,
580:x
574:(
571:F
551:c
482:n
455:F
424:F
395:n
366:n
337:x
308:n
293:x
287:=
284:d
261:t
237:)
234:t
231:,
228:d
225:(
222:G
192:n
169:,
166:)
163:t
160:,
151:n
136:x
130:(
127:G
124:=
121:)
118:t
115:,
106:x
100:(
97:F
77:t
51:x
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
