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Traveling plane wave

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For a scalar traveling plane wave in two or three dimensions, the gradient of the field is always collinear with the direction
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A traveling plane wave can be studied by ignoring the dimensions of space perpendicular to the vector
1149: 513:; and the value of the field is then the same, and constant in time, at every one of its points. 758: 635: 467: 403: 84: 1091: 1083: 248: 51:
whose evolution in time can be described as simple translation of its values at a constant
574: 432: 219: 48: 19: 1087: 307: 899: 1182: 1133: 1049: 944: 924: 735: 607: 547: 519: 496: 383: 283: 59: 53: 1176: 1054: 1037: 541: 537: 889:{\displaystyle \nabla F({\vec {x}},t)={\vec {n}}G'({\vec {x}}\cdot {\vec {n}}-ct)} 1129: 32: 1026:{\displaystyle \nabla F=-{\frac {\vec {n}}{c}}{\frac {\partial F}{\partial t}}} 1075: 601: 44: 206:{\displaystyle F({\vec {x}},t)=G\left({\vec {x}}\cdot {\vec {n}}-ct\right)\,} 460: 24: 1121: 36: 18: 732:
on a one-dimensional medium, with a single position coordinate
464:. This plane too travels along the direction of propagation 1036:
Plane traveling waves are also special solutions of the
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of a traveling plane wave in three-dimensional space.
304:of the wave, namely the value of the field at time 1025: 953: 933: 913: 888: 776: 744: 724: 653: 616: 592: 556: 528: 505: 485: 450: 421: 392: 372: 322: 292: 272: 237: 205: 102: 68: 1082:, Berlin, Heidelberg: Springer, pp. 89–102, 1157: 1074:Tohyama, Mikio (2011), Tohyama, Mikio (ed.), 8: 373:{\displaystyle d={\vec {x}}\cdot {\vec {n}}} 1164: 1150: 1003: 988: 986: 972: 946: 926: 901: 863: 862: 848: 847: 825: 824: 801: 800: 789: 763: 762: 760: 737: 678: 677: 666: 640: 639: 637: 609: 576: 549: 521: 498: 472: 471: 469: 434: 408: 407: 405: 385: 359: 358: 344: 343: 335: 309: 285: 250: 245:is a function of a single real parameter 221: 202: 177: 176: 162: 161: 130: 129: 121: 89: 88: 86: 61: 725:{\displaystyle F(z{\vec {n}},t)=G(z-ct)} 1066: 1076:"Waves and Speed of Sound in the Air" 941:. Moreover, a traveling plane wave 7: 1118: 1116: 400:, the moving plane perpendicular to 661:; that is, by considering the wave 1088:10.1007/978-3-642-20122-6_6#citeas 1014: 1006: 974: 791: 14: 113:Such a field can be written as 1120: 544:; its values are the values of 993: 883: 868: 853: 844: 830: 818: 806: 797: 768: 719: 704: 695: 683: 671: 645: 587: 581: 477: 413: 364: 349: 232: 226: 182: 167: 147: 135: 126: 94: 1: 963:partial differential equation 1136:. You can help Knowledge by 458:from the origin is called a 961:of any shape satisfies the 1204: 1115: 1040:in an homogeneous medium. 777:{\displaystyle {\vec {n}}} 654:{\displaystyle {\vec {n}}} 486:{\displaystyle {\vec {n}}} 422:{\displaystyle {\vec {n}}} 103:{\displaystyle {\vec {n}}} 380:. For each displacement 571:is a special case, when 330:, for each displacement 79:direction of propagation 1132:-related article is a 1027: 955: 935: 915: 890: 778: 746: 726: 655: 618: 594: 558: 530: 507: 487: 452: 423: 394: 374: 324: 294: 274: 273:{\displaystyle u=d-ct} 239: 207: 104: 70: 28: 1028: 956: 936: 921:is the derivative of 916: 891: 779: 747: 727: 656: 619: 595: 569:sinusoidal plane wave 559: 531: 508: 488: 453: 424: 395: 375: 325: 295: 275: 240: 208: 105: 71: 43:is a special case of 22: 971: 945: 925: 900: 788: 759: 736: 665: 636: 608: 593:{\displaystyle G(u)} 575: 548: 520: 497: 468: 451:{\displaystyle d+ct} 433: 404: 384: 334: 308: 284: 249: 238:{\displaystyle G(u)} 220: 120: 85: 60: 41:traveling plane wave 323:{\displaystyle t=0} 1023: 951: 931: 914:{\displaystyle G'} 911: 886: 774: 742: 722: 651: 614: 590: 554: 526: 503: 483: 448: 419: 390: 370: 320: 290: 270: 235: 203: 100: 66: 29: 16:Type of plane wave 1188:Mathematics stubs 1145: 1144: 1097:978-3-642-20122-6 1080:Sound and Signals 1021: 1001: 996: 954:{\displaystyle F} 934:{\displaystyle G} 871: 856: 833: 809: 771: 745:{\displaystyle z} 686: 648: 617:{\displaystyle u} 557:{\displaystyle G} 529:{\displaystyle F} 506:{\displaystyle c} 480: 416: 393:{\displaystyle d} 367: 352: 293:{\displaystyle G} 185: 170: 138: 97: 69:{\displaystyle c} 1195: 1166: 1159: 1152: 1124: 1117: 1107: 1106: 1105: 1104: 1071: 1032: 1030: 1029: 1024: 1022: 1020: 1012: 1004: 1002: 997: 989: 987: 960: 958: 957: 952: 940: 938: 937: 932: 920: 918: 917: 912: 910: 895: 893: 892: 887: 873: 872: 864: 858: 857: 849: 843: 835: 834: 826: 811: 810: 802: 784:; specifically, 783: 781: 780: 775: 773: 772: 764: 751: 749: 748: 743: 731: 729: 728: 723: 688: 687: 679: 660: 658: 657: 652: 650: 649: 641: 623: 621: 620: 615: 599: 597: 596: 591: 563: 561: 560: 555: 535: 533: 532: 527: 512: 510: 509: 504: 492: 490: 489: 484: 482: 481: 473: 457: 455: 454: 449: 428: 426: 425: 420: 418: 417: 409: 399: 397: 396: 391: 379: 377: 376: 371: 369: 368: 360: 354: 353: 345: 329: 327: 326: 321: 299: 297: 296: 291: 280:. The function 279: 277: 276: 271: 244: 242: 241: 236: 212: 210: 209: 204: 201: 197: 187: 186: 178: 172: 171: 163: 140: 139: 131: 109: 107: 106: 101: 99: 98: 90: 76:, along a fixed 75: 73: 72: 67: 1203: 1202: 1198: 1197: 1196: 1194: 1193: 1192: 1173: 1172: 1171: 1170: 1113: 1111: 1110: 1102: 1100: 1098: 1073: 1072: 1068: 1063: 1046: 1013: 1005: 969: 968: 943: 942: 923: 922: 903: 898: 897: 836: 786: 785: 757: 756: 734: 733: 663: 662: 634: 633: 630: 606: 605: 573: 572: 546: 545: 518: 517: 495: 494: 466: 465: 431: 430: 402: 401: 382: 381: 332: 331: 306: 305: 282: 281: 247: 246: 218: 217: 160: 156: 118: 117: 83: 82: 58: 57: 17: 12: 11: 5: 1201: 1199: 1191: 1190: 1185: 1175: 1174: 1169: 1168: 1161: 1154: 1146: 1143: 1142: 1125: 1109: 1108: 1096: 1065: 1064: 1062: 1059: 1058: 1057: 1052: 1050:Spherical wave 1045: 1042: 1034: 1033: 1019: 1016: 1011: 1008: 1000: 995: 992: 985: 982: 979: 976: 950: 930: 909: 906: 885: 882: 879: 876: 870: 867: 861: 855: 852: 846: 842: 839: 832: 829: 823: 820: 817: 814: 808: 805: 799: 796: 793: 770: 767: 741: 721: 718: 715: 712: 709: 706: 703: 700: 697: 694: 691: 685: 682: 676: 673: 670: 647: 644: 629: 626: 613: 589: 586: 583: 580: 553: 525: 502: 493:with velocity 479: 476: 447: 444: 441: 438: 415: 412: 389: 366: 363: 357: 351: 348: 342: 339: 319: 316: 313: 300:describes the 289: 269: 266: 263: 260: 257: 254: 234: 231: 228: 225: 214: 213: 200: 196: 193: 190: 184: 181: 175: 169: 166: 159: 155: 152: 149: 146: 143: 137: 134: 128: 125: 96: 93: 65: 15: 13: 10: 9: 6: 4: 3: 2: 1200: 1189: 1186: 1184: 1181: 1180: 1178: 1167: 1162: 1160: 1155: 1153: 1148: 1147: 1141: 1139: 1135: 1131: 1126: 1123: 1119: 1114: 1099: 1093: 1089: 1085: 1081: 1077: 1070: 1067: 1060: 1056: 1055:Standing wave 1053: 1051: 1048: 1047: 1043: 1041: 1039: 1038:wave equation 1017: 1009: 998: 990: 983: 980: 977: 967: 966: 965: 964: 948: 928: 907: 904: 880: 877: 874: 865: 859: 850: 840: 837: 827: 821: 815: 812: 803: 794: 765: 753: 739: 716: 713: 710: 707: 701: 698: 692: 689: 680: 674: 668: 642: 627: 625: 611: 603: 584: 578: 570: 565: 551: 543: 539: 523: 514: 500: 474: 463: 462: 445: 442: 439: 436: 410: 387: 361: 355: 346: 340: 337: 317: 314: 311: 303: 287: 267: 264: 261: 258: 255: 252: 229: 223: 198: 194: 191: 188: 179: 173: 164: 157: 153: 150: 144: 141: 132: 123: 116: 115: 114: 111: 91: 81: 80: 63: 56: 55: 50: 46: 42: 38: 34: 26: 21: 1138:expanding it 1127: 1112: 1101:, retrieved 1079: 1069: 1035: 754: 631: 604:function of 566: 542:vector field 515: 459: 429:at distance 301: 215: 112: 77: 52: 40: 30: 1130:mathematics 47:, namely a 33:mathematics 1177:Categories 1103:2024-08-05 1061:References 628:Properties 602:sinusoidal 54:wave speed 45:plane wave 25:wavefronts 1015:∂ 1007:∂ 994:→ 984:− 975:∇ 875:− 869:→ 860:⋅ 854:→ 831:→ 807:→ 792:∇ 769:→ 711:− 684:→ 646:→ 536:may be a 516:The wave 478:→ 461:wavefront 414:→ 365:→ 356:⋅ 350:→ 262:− 189:− 183:→ 174:⋅ 168:→ 136:→ 95:→ 1044:See also 908:′ 896:, where 841:′ 302:profile 37:physics 1094:  538:scalar 216:where 1183:Waves 1128:This 600:is a 49:field 1134:stub 1092:ISBN 39:, a 35:and 23:The 1084:doi 540:or 31:In 1179:: 1090:, 1078:, 752:. 624:. 567:A 564:. 110:. 1165:e 1158:t 1151:v 1140:. 1086:: 1018:t 1010:F 999:c 991:n 981:= 978:F 949:F 929:G 905:G 884:) 881:t 878:c 866:n 851:x 845:( 838:G 828:n 822:= 819:) 816:t 813:, 804:x 798:( 795:F 766:n 740:z 720:) 717:t 714:c 708:z 705:( 702:G 699:= 696:) 693:t 690:, 681:n 675:z 672:( 669:F 643:n 612:u 588:) 585:u 582:( 579:G 552:G 524:F 501:c 475:n 446:t 443:c 440:+ 437:d 411:n 388:d 362:n 347:x 341:= 338:d 318:0 315:= 312:t 288:G 268:t 265:c 259:d 256:= 253:u 233:) 230:u 227:( 224:G 199:) 195:t 192:c 180:n 165:x 158:( 154:G 151:= 148:) 145:t 142:, 133:x 127:( 124:F 92:n 64:c

Index


wavefronts
mathematics
physics
plane wave
field
wave speed
direction of propagation
wavefront
scalar
vector field
sinusoidal plane wave
sinusoidal
partial differential equation
wave equation
Spherical wave
Standing wave
"Waves and Speed of Sound in the Air"
doi
10.1007/978-3-642-20122-6_6#citeas
ISBN
978-3-642-20122-6
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