3148:
2773:
3143:{\displaystyle {\begin{aligned}k_{s}^{0}&=\Lambda _{0}^{0}k_{\mathrm {obs} }^{0}+\Lambda _{1}^{0}k_{\mathrm {obs} }^{1}+\Lambda _{2}^{0}k_{\mathrm {obs} }^{2}+\Lambda _{3}^{0}k_{\mathrm {obs} }^{3}\\{\frac {\omega _{s}}{c}}&=\gamma {\frac {\omega _{\mathrm {obs} }}{c}}-\beta \gamma k_{\mathrm {obs} }^{1}\\&=\gamma {\frac {\omega _{\mathrm {obs} }}{c}}-\beta \gamma {\frac {\omega _{\mathrm {obs} }}{c}}\cos \theta .\end{aligned}}}
121:
3813:
3601:
1124:
2478:
1619:
2644:
1419:
1859:
2292:
3624:
3412:
714:
945:
842:, the direction of the wavevector is the same as the direction of wave propagation. If the medium is anisotropic, the wave vector in general points in directions other than that of the wave propagation. The wave vector is always perpendicular to surfaces of constant phase.
2124:
2307:
1430:
3927:
2502:
1233:
1709:
3366:
1991:
2649:
In the situation where light is being emitted by a fast moving source and one would like to know the frequency of light detected in an earth (lab) frame, we would apply the
Lorentz transformation as follows. Note that the source is in a frame
2139:
389:
3808:{\displaystyle {\frac {\omega _{\mathrm {obs} }}{\omega _{s}}}={\frac {1}{\gamma (1-\beta )}}={\frac {\sqrt {1-\beta ^{2}}}{1-\beta }}={\frac {\sqrt {(1+\beta )(1-\beta )}}{1-\beta }}={\frac {\sqrt {1+\beta }}{\sqrt {1-\beta }}}}
3596:{\displaystyle {\frac {\omega _{\mathrm {obs} }}{\omega _{s}}}={\frac {1}{\gamma (1+\beta )}}={\frac {\sqrt {1-\beta ^{2}}}{1+\beta }}={\frac {\sqrt {(1+\beta )(1-\beta )}}{1+\beta }}={\frac {\sqrt {1-\beta }}{\sqrt {1+\beta }}}}
2736:
600:
4097:"This effect has been explained by Musgrave (1959) who has shown that the energy of an elastic wave in an anisotropic medium will not, in general, travel along the same path as the normal to the plane wavefront ...",
231:
1119:{\displaystyle K^{\mu }=\left({\frac {\omega }{c}},{\vec {k}}\right)=\left({\frac {\omega }{c}},{\frac {\omega }{v_{p}}}{\hat {n}}\right)=\left({\frac {2\pi }{cT}},{\frac {2\pi {\hat {n}}}{\lambda }}\right)\,}
2778:
293:
1238:
589:
920:
A moving wave surface in special relativity may be regarded as a hypersurface (a 3D subspace) in spacetime, formed by all the events passed by the wave surface. A wavetrain (denoted by some variable
2007:
2473:{\displaystyle K^{\mu }=\left({\frac {\omega }{c}},{\vec {k}}\right)=\left({\frac {\omega _{o}}{c^{2}}}\right)U^{\mu }=\left({\frac {\omega _{o}}{c^{2}}}\right)\gamma \left(c,{\vec {u}}\right)}
3269:
771:
179:
523:
1614:{\displaystyle K^{\mu }K_{\mu }=\left({\frac {\omega }{c}}\right)^{2}-k_{x}^{2}-k_{y}^{2}-k_{z}^{2}=\left({\frac {\omega _{o}}{c}}\right)^{2}=\left({\frac {m_{o}c}{\hbar }}\right)^{2}}
2639:{\displaystyle \Lambda ={\begin{pmatrix}\gamma &-\beta \gamma &\ 0\ &\ 0\ \\-\beta \gamma &\gamma &0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}}
1414:{\displaystyle {\begin{aligned}K^{\mu }&=\left({\frac {\omega }{c}},k_{x},k_{y},k_{z}\right)\,\\K_{\mu }&=\left({\frac {\omega }{c}},-k_{x},-k_{y},-k_{z}\right)\end{aligned}}}
1854:{\displaystyle K^{\mu }=\left({\frac {\omega }{c}},{\vec {k}}\right)=\left({\frac {\omega }{c}},{\frac {\omega }{c}}{\hat {n}}\right)={\frac {\omega }{c}}\left(1,{\hat {n}}\right)\,}
3840:
1156:
3283:
2287:{\displaystyle K^{\mu }=\left({\frac {\omega }{c}},{\vec {k}}\right)=\left({\frac {2\pi }{c}}\right)N^{\mu }=\left({\frac {2\pi }{c}}\right)\left(\nu ,\nu {\vec {n}}\right)}
1871:
3404:
3177:
1185:
4014:
1702:
1663:
2765:
3204:
308:
740:
1224:
1220:
709:{\displaystyle \psi \left(\mathbf {r} ,t\right)=A\cos \left(2\pi \left({\tilde {\boldsymbol {\nu }}}\cdot {\mathbf {r} }-ft\right)+\varphi \right),}
2668:
184:
4187:
4158:
4067:
4040:
3998:
68:), with a typical unit being radian per metre. The wave vector and angular wave vector are related by a fixed constant of proportionality, 2
928:
is a scalar function of position in spacetime. The derivative of this scalar is a vector that characterizes the wave, the four-wavevector.
4211:
4127:
255:
2119:{\displaystyle P^{\mu }=\left({\frac {E}{c}},{\vec {p}}\right)=\hbar K^{\mu }=\hbar \left({\frac {\omega }{c}},{\vec {k}}\right)}
539:
1865:
which would have the following relation between the frequency and the magnitude of the spatial part of the four-wavevector:
532:
is the angular wave vector of the wave, describing how many radians it traverses per unit of distance, and related to the
908:
which is sinusoidal, and the wavevector is defined via that envelope wave, usually using the "physics definition". See
2493:
3209:
747:
155:
485:
792:". The "direction of wave propagation" is the direction of a wave's energy flow, and the direction that a small
3922:{\displaystyle {\frac {\omega _{\mathrm {obs} }}{\omega _{s}}}={\frac {1}{\gamma (1-0)}}={\frac {1}{\gamma }}}
4206:
1132:
53:), and its direction is perpendicular to the wavefront. In isotropic media, this is also the direction of
3967:
3361:{\displaystyle {\frac {\omega _{\mathrm {obs} }}{\omega _{s}}}={\frac {1}{\gamma (1-\beta \cos \theta )}}}
2489:
3380:
As an example, to apply this to a situation where the source is moving directly away from the observer (
1986:{\displaystyle K^{\mu }K_{\mu }=\left({\frac {\omega }{c}}\right)^{2}-k_{x}^{2}-k_{y}^{2}-k_{z}^{2}=0}
3963:
3938:
3823:
To apply this to a situation where the source is moving transversely with respect to the observer (
872:
3383:
3156:
4008:
3943:
97:
34:
1161:
909:
866:
4150:
4144:
924:) can be regarded as a one-parameter family of such hypersurfaces in spacetime. This variable
4183:
4154:
4123:
4115:
4063:
4057:
4036:
4030:
3994:
3988:
1629:
892:
472:
384:{\displaystyle \psi (\mathbf {r} ,t)=A\cos(\mathbf {k} \cdot \mathbf {r} -\omega t+\varphi ),}
1674:
1635:
2744:
1625:
904:
854:
831:
789:
476:
102:
54:
42:
3182:
788:
The direction in which the wave vector points must be distinguished from the "direction of
475:
of the wave, describing how many radians it traverses per unit of time, and related to the
936:
801:
84:
3611:
To apply this to a situation where the source is moving straight towards the observer (
2130:
1199:
850:
809:
805:
797:
783:
725:
299:
115:
4200:
2298:
1998:
1668:
884:
839:
137:
133:
895:
462:
442:
4102:
932:
824:
793:
899:
857:, the wave vector may not point exactly in the direction of wave propagation.
846:
817:
813:
533:
434:
426:
120:
50:
46:
106:
can be defined, combining the (angular) wave vector and (angular) frequency.
2731:{\displaystyle k_{s}^{\mu }=\Lambda _{\nu }^{\mu }k_{\mathrm {obs} }^{\nu }}
452:
125:
4122:(Reprint of Prentice-Hall 1955 2nd ed.). Courier-Dover. p. 624.
226:{\displaystyle {\tilde {\nu }}=\left|{\tilde {\boldsymbol {\nu }}}\right|}
880:
827:
888:
76:
22:
1424:
In general, the
Lorentz scalar magnitude of the wave four-vector is:
425:
describing the disturbance describing the wave (for example, for an
132:, can be measured between any two consecutive points with the same
1667:
An example of a null four-wavevector would be a beam of coherent,
835:
119:
804:. On the other hand, the wave vector points in the direction of
38:
800:. For light waves in vacuum, this is also the direction of the
594:
The equivalent equation using the wave vector and frequency is
288:{\displaystyle \mathbf {k} =2\pi {\tilde {\boldsymbol {\nu }}}}
152:
have distinct meanings. Here, the wave vector is denoted by
136:, such as between adjacent crests, or troughs, or adjacent
584:{\displaystyle |\mathbf {k} |={\tfrac {2\pi }{\lambda }}.}
2662:. Applying the Lorentz transformation to the wave vector
16:
Vector describing a wave; often its propagation direction
4149:(2nd ed.). Oxford Science Publications. pp.
3819:
Source moving tangentially (transverse
Doppler effect)
2517:
1137:
562:
496:
41:, with a typical unit being cycle per metre. It has a
3843:
3627:
3415:
3386:
3286:
3212:
3185:
3159:
2776:
2747:
2671:
2505:
2310:
2142:
2010:
1874:
1712:
1677:
1638:
1433:
1236:
1164:
1158:
is the temporal component, and the wavenumber vector
1135:
948:
750:
728:
603:
542:
488:
311:
258:
187:
158:
3962:
In most contexts, both the radian and the cycle (or
455:
of the wave (the peak magnitude of the oscillation),
3921:
3807:
3595:
3398:
3360:
3263:
3198:
3171:
3142:
2759:
2730:
2638:
2472:
2286:
2118:
1985:
1853:
1696:
1657:
1613:
1413:
1179:
1150:
1118:
765:
734:
708:
583:
517:
433:would be the excess height of the water, or for a
383:
287:
225:
173:
79:to refer to the angular wave vector simply as the
2492:of the four-wavevector is one way to derive the
808:. In other words, the wave vector points in the
3264:{\displaystyle k^{0},k^{1}=k^{0}\cos \theta .}
766:{\displaystyle {\tilde {\boldsymbol {\nu }}}}
174:{\displaystyle {\tilde {\boldsymbol {\nu }}}}
140:with the same direction of transit, as shown.
8:
4180:Modern Problems in Classical Electrodynamics
4143:Wolfgang Rindler (1991). "§24 Wave motion".
4013:: CS1 maint: multiple names: authors list (
845:For example, when a wave travels through an
518:{\displaystyle \omega ={\tfrac {2\pi }{T}},}
4087:. Holt, Rinehart, and Winston. p. 177.
49:of the wave (inversely proportional to the
4056:Vaĭnshteĭn, Boris Konstantinovich (1994).
1632:(photonic) particles, where the rest mass
4120:Fundamental Formulas of Physics, Volume 2
3909:
3879:
3868:
3851:
3850:
3844:
3842:
3779:
3729:
3706:
3693:
3663:
3652:
3635:
3634:
3628:
3626:
3567:
3517:
3494:
3481:
3451:
3440:
3423:
3422:
3416:
3414:
3385:
3322:
3311:
3294:
3293:
3287:
3285:
3243:
3230:
3217:
3211:
3190:
3184:
3158:
3106:
3105:
3099:
3072:
3071:
3065:
3046:
3034:
3033:
3002:
3001:
2995:
2974:
2968:
2958:
2946:
2945:
2935:
2930:
2917:
2905:
2904:
2894:
2889:
2876:
2864:
2863:
2853:
2848:
2835:
2823:
2822:
2812:
2807:
2790:
2785:
2777:
2775:
2746:
2722:
2710:
2709:
2699:
2694:
2681:
2676:
2670:
2512:
2504:
2454:
2453:
2427:
2417:
2411:
2398:
2382:
2372:
2366:
2343:
2342:
2329:
2315:
2309:
2268:
2267:
2234:
2221:
2198:
2175:
2174:
2161:
2147:
2141:
2100:
2099:
2086:
2069:
2043:
2042:
2029:
2015:
2009:
1971:
1966:
1953:
1948:
1935:
1930:
1917:
1903:
1889:
1879:
1873:
1850:
1834:
1833:
1812:
1793:
1792:
1782:
1769:
1745:
1744:
1731:
1717:
1711:
1682:
1676:
1643:
1637:
1605:
1586:
1579:
1565:
1550:
1544:
1530:
1525:
1512:
1507:
1494:
1489:
1476:
1462:
1448:
1438:
1432:
1396:
1380:
1364:
1344:
1326:
1317:
1306:
1293:
1280:
1263:
1245:
1237:
1235:
1166:
1165:
1163:
1136:
1134:
1115:
1093:
1092:
1083:
1060:
1036:
1035:
1027:
1018:
1005:
981:
980:
967:
953:
947:
851:light waves through an asymmetric crystal
752:
751:
749:
727:
672:
671:
657:
656:
612:
602:
561:
553:
548:
543:
541:
495:
487:
355:
347:
318:
310:
274:
273:
259:
257:
233:. The angular wave vector is denoted by
208:
207:
189:
188:
186:
160:
159:
157:
1194:can be written as the angular frequency
898:. These electron waves are not ordinary
3978:
3955:
2078:
2062:
1596:
754:
659:
276:
210:
162:
87:. It is also common to use the symbol
4006:
2297:The four-wavevector is related to the
2129:The four-wavevector is related to the
1997:The four-wavevector is related to the
2656:and earth is in the observing frame,
1151:{\displaystyle {\tfrac {\omega }{c}}}
796:will move, i.e. the direction of the
7:
2496:. The Lorentz matrix is defined as
4146:Introduction to Special Relativity
3987:Harris, Benenson, Stöcker (2002).
3858:
3855:
3852:
3642:
3639:
3636:
3430:
3427:
3424:
3301:
3298:
3295:
3113:
3110:
3107:
3079:
3076:
3073:
3041:
3038:
3035:
3009:
3006:
3003:
2953:
2950:
2947:
2927:
2912:
2909:
2906:
2886:
2871:
2868:
2865:
2845:
2830:
2827:
2824:
2804:
2717:
2714:
2711:
2691:
2506:
902:waves, but they do have a kind of
830:such as air, any gas, any liquid,
75:It is common in several fields of
14:
3607:Source moving towards (blueshift)
2741:and choosing just to look at the
3970:1, reducing this constant to 2π.
3277:
1671:light, which has phase-velocity
1219:When written out explicitly its
1208:, or in terms of inverse period
875:, the "wavevector" (also called
673:
613:
549:
356:
348:
319:
260:
60:A closely related vector is the
83:, in contrast to, for example,
3900:
3888:
3759:
3747:
3744:
3732:
3684:
3672:
3547:
3535:
3532:
3520:
3472:
3460:
3352:
3331:
2459:
2348:
2273:
2180:
2105:
2048:
1839:
1798:
1750:
1171:
1098:
1041:
986:
931:The four-wavevector is a wave
757:
662:
554:
544:
375:
344:
329:
315:
279:
239:and the angular wavenumber by
213:
194:
165:
1:
4085:Introduction to modern optics
3376:Source moving away (redshift)
3399:{\displaystyle \theta =\pi }
3172:{\displaystyle \cos \theta }
1190:Alternately, the wavenumber
1129:where the angular frequency
778:Direction of the wave vector
4182:. Oxford University Press.
3179:is the direction cosine of
2494:relativistic Doppler effect
4230:
4212:Vector physical quantities
1187:is the spatial component.
1180:{\displaystyle {\vec {k}}}
864:
814:surfaces of constant phase
781:
113:
4178:Brau, Charles A. (2004).
4114:Donald H. Menzel (1960).
4027:Crystallography example:
891:is the wavevector of its
853:or sound waves through a
93:for whichever is in use.
72: radians per cycle.
252:. These are related by
1697:{\displaystyle v_{p}=c}
1658:{\displaystyle m_{o}=0}
1624:The four-wavevector is
1212:and inverse wavelength
45:. Its magnitude is the
43:magnitude and direction
4083:Fowles, Grant (1968).
4059:Modern Crystallography
4032:Modern Crystallography
3968:dimensionless quantity
3923:
3809:
3597:
3400:
3362:
3265:
3200:
3173:
3144:
2761:
2760:{\displaystyle \mu =0}
2732:
2640:
2490:Lorentz transformation
2484:Lorentz transformation
2474:
2288:
2120:
1987:
1855:
1698:
1659:
1615:
1415:
1181:
1152:
1120:
861:In solid-state physics
767:
736:
710:
585:
519:
385:
289:
227:
181:and the wavenumber by
175:
141:
4099:Sound waves in solids
3966:) are treated as the
3924:
3810:
3598:
3401:
3363:
3266:
3201:
3199:{\displaystyle k^{1}}
3174:
3145:
2767:component results in
2762:
2733:
2641:
2475:
2289:
2121:
1993:{for light-like/null}
1988:
1861:{for light-like/null}
1856:
1699:
1660:
1616:
1416:
1182:
1153:
1121:
937:Minkowski coordinates
916:In special relativity
912:for further details.
768:
737:
711:
586:
520:
386:
302:follows the equation
290:
228:
176:
123:
37:used in describing a
3939:Plane-wave expansion
3841:
3625:
3413:
3384:
3284:
3210:
3183:
3157:
2774:
2745:
2669:
2503:
2308:
2140:
2008:
1872:
1710:
1675:
1636:
1431:
1234:
1162:
1133:
946:
935:that is defined, in
748:
726:
601:
540:
486:
441:would be the excess
309:
256:
185:
156:
4029:Vaĭnshteĭn (1994).
3990:Handbook of Physics
3051:
2963:
2940:
2922:
2899:
2881:
2858:
2840:
2817:
2795:
2727:
2704:
2686:
1976:
1958:
1940:
1535:
1517:
1499:
873:solid-state physics
150:angular wave vector
62:angular wave vector
4116:"§10.5 Bloch wave"
4101:by Pollard, 1977.
3944:Plane of incidence
3919:
3805:
3593:
3396:
3358:
3261:
3196:
3169:
3140:
3138:
3029:
2941:
2926:
2900:
2885:
2859:
2844:
2818:
2803:
2781:
2757:
2728:
2705:
2690:
2672:
2636:
2630:
2470:
2284:
2116:
1983:
1962:
1944:
1926:
1851:
1694:
1655:
1611:
1521:
1503:
1485:
1411:
1409:
1177:
1148:
1146:
1116:
893:quantum-mechanical
847:anisotropic medium
773:is the wave vector
763:
732:
706:
581:
576:
515:
510:
471:is the (temporal)
381:
285:
223:
171:
142:
98:special relativity
96:In the context of
66:angular wavevector
4189:978-0-19-514665-3
4160:978-0-19-853952-0
4069:978-3-540-56558-1
4042:978-3-540-56558-1
4000:978-0-387-95269-7
3985:Physics example:
3917:
3904:
3874:
3834:), this becomes:
3803:
3802:
3791:
3774:
3762:
3724:
3712:
3688:
3658:
3618:), this becomes:
3591:
3590:
3579:
3562:
3550:
3512:
3500:
3476:
3446:
3406:), this becomes:
3371:
3370:
3356:
3317:
3122:
3088:
3018:
2983:
2555:
2549:
2544:
2538:
2462:
2433:
2388:
2351:
2337:
2276:
2247:
2211:
2183:
2169:
2108:
2094:
2051:
2037:
1911:
1842:
1820:
1801:
1790:
1777:
1753:
1739:
1599:
1559:
1470:
1352:
1271:
1174:
1145:
1108:
1101:
1078:
1044:
1033:
1013:
989:
975:
905:envelope function
760:
735:{\displaystyle f}
665:
575:
509:
473:angular frequency
415:is a function of
282:
216:
197:
168:
4219:
4193:
4165:
4164:
4140:
4134:
4133:
4111:
4105:
4095:
4089:
4088:
4080:
4074:
4073:
4053:
4047:
4046:
4025:
4019:
4018:
4012:
4004:
3983:
3971:
3960:
3928:
3926:
3925:
3920:
3918:
3910:
3905:
3903:
3880:
3875:
3873:
3872:
3863:
3862:
3861:
3845:
3833:
3814:
3812:
3811:
3806:
3804:
3792:
3781:
3780:
3775:
3773:
3731:
3730:
3725:
3723:
3711:
3710:
3695:
3694:
3689:
3687:
3664:
3659:
3657:
3656:
3647:
3646:
3645:
3629:
3617:
3602:
3600:
3599:
3594:
3592:
3580:
3569:
3568:
3563:
3561:
3519:
3518:
3513:
3511:
3499:
3498:
3483:
3482:
3477:
3475:
3452:
3447:
3445:
3444:
3435:
3434:
3433:
3417:
3405:
3403:
3402:
3397:
3367:
3365:
3364:
3359:
3357:
3355:
3323:
3318:
3316:
3315:
3306:
3305:
3304:
3288:
3278:
3270:
3268:
3267:
3262:
3248:
3247:
3235:
3234:
3222:
3221:
3206:with respect to
3205:
3203:
3202:
3197:
3195:
3194:
3178:
3176:
3175:
3170:
3149:
3147:
3146:
3141:
3139:
3123:
3118:
3117:
3116:
3100:
3089:
3084:
3083:
3082:
3066:
3055:
3050:
3045:
3044:
3019:
3014:
3013:
3012:
2996:
2984:
2979:
2978:
2969:
2962:
2957:
2956:
2939:
2934:
2921:
2916:
2915:
2898:
2893:
2880:
2875:
2874:
2857:
2852:
2839:
2834:
2833:
2816:
2811:
2794:
2789:
2766:
2764:
2763:
2758:
2737:
2735:
2734:
2729:
2726:
2721:
2720:
2703:
2698:
2685:
2680:
2661:
2655:
2645:
2643:
2642:
2637:
2635:
2634:
2553:
2547:
2542:
2536:
2479:
2477:
2476:
2471:
2469:
2465:
2464:
2463:
2455:
2438:
2434:
2432:
2431:
2422:
2421:
2412:
2403:
2402:
2393:
2389:
2387:
2386:
2377:
2376:
2367:
2358:
2354:
2353:
2352:
2344:
2338:
2330:
2320:
2319:
2293:
2291:
2290:
2285:
2283:
2279:
2278:
2277:
2269:
2252:
2248:
2243:
2235:
2226:
2225:
2216:
2212:
2207:
2199:
2190:
2186:
2185:
2184:
2176:
2170:
2162:
2152:
2151:
2125:
2123:
2122:
2117:
2115:
2111:
2110:
2109:
2101:
2095:
2087:
2074:
2073:
2058:
2054:
2053:
2052:
2044:
2038:
2030:
2020:
2019:
1992:
1990:
1989:
1984:
1975:
1970:
1957:
1952:
1939:
1934:
1922:
1921:
1916:
1912:
1904:
1894:
1893:
1884:
1883:
1860:
1858:
1857:
1852:
1849:
1845:
1844:
1843:
1835:
1821:
1813:
1808:
1804:
1803:
1802:
1794:
1791:
1783:
1778:
1770:
1760:
1756:
1755:
1754:
1746:
1740:
1732:
1722:
1721:
1703:
1701:
1700:
1695:
1687:
1686:
1664:
1662:
1661:
1656:
1648:
1647:
1620:
1618:
1617:
1612:
1610:
1609:
1604:
1600:
1595:
1591:
1590:
1580:
1570:
1569:
1564:
1560:
1555:
1554:
1545:
1534:
1529:
1516:
1511:
1498:
1493:
1481:
1480:
1475:
1471:
1463:
1453:
1452:
1443:
1442:
1420:
1418:
1417:
1412:
1410:
1406:
1402:
1401:
1400:
1385:
1384:
1369:
1368:
1353:
1345:
1331:
1330:
1316:
1312:
1311:
1310:
1298:
1297:
1285:
1284:
1272:
1264:
1250:
1249:
1215:
1211:
1207:
1197:
1193:
1186:
1184:
1183:
1178:
1176:
1175:
1167:
1157:
1155:
1154:
1149:
1147:
1138:
1125:
1123:
1122:
1117:
1114:
1110:
1109:
1104:
1103:
1102:
1094:
1084:
1079:
1077:
1069:
1061:
1051:
1047:
1046:
1045:
1037:
1034:
1032:
1031:
1019:
1014:
1006:
996:
992:
991:
990:
982:
976:
968:
958:
957:
927:
923:
855:sedimentary rock
832:amorphous solids
828:isotropic medium
810:normal direction
790:wave propagation
772:
770:
769:
764:
762:
761:
753:
742:is the frequency
741:
739:
738:
733:
715:
713:
712:
707:
702:
698:
691:
687:
677:
676:
667:
666:
658:
627:
623:
616:
590:
588:
587:
582:
577:
571:
563:
557:
552:
547:
536:by the equation
531:
524:
522:
521:
516:
511:
505:
497:
482:by the equation
481:
470:
460:
450:
440:
432:
424:
420:
414:
408:
402:
390:
388:
387:
382:
359:
351:
322:
294:
292:
291:
286:
284:
283:
275:
263:
251:
249:
238:
232:
230:
229:
224:
222:
218:
217:
209:
199:
198:
190:
180:
178:
177:
172:
170:
169:
161:
131:
124:Wavelength of a
103:wave four-vector
92:
71:
55:wave propagation
4229:
4228:
4222:
4221:
4220:
4218:
4217:
4216:
4197:
4196:
4190:
4177:
4174:
4172:Further reading
4169:
4168:
4161:
4142:
4141:
4137:
4130:
4113:
4112:
4108:
4096:
4092:
4082:
4081:
4077:
4070:
4062:. p. 259.
4055:
4054:
4050:
4043:
4035:. p. 259.
4028:
4026:
4022:
4005:
4001:
3993:. p. 288.
3986:
3984:
3980:
3975:
3974:
3961:
3957:
3952:
3935:
3884:
3864:
3846:
3839:
3838:
3824:
3821:
3763:
3713:
3702:
3668:
3648:
3630:
3623:
3622:
3612:
3609:
3551:
3501:
3490:
3456:
3436:
3418:
3411:
3410:
3382:
3381:
3378:
3327:
3307:
3289:
3282:
3281:
3239:
3226:
3213:
3208:
3207:
3186:
3181:
3180:
3155:
3154:
3137:
3136:
3101:
3067:
3053:
3052:
2997:
2985:
2970:
2965:
2964:
2796:
2772:
2771:
2743:
2742:
2667:
2666:
2657:
2651:
2629:
2628:
2623:
2618:
2613:
2607:
2606:
2601:
2596:
2591:
2585:
2584:
2579:
2574:
2569:
2557:
2556:
2545:
2534:
2523:
2513:
2501:
2500:
2486:
2446:
2442:
2423:
2413:
2407:
2394:
2378:
2368:
2362:
2328:
2324:
2311:
2306:
2305:
2257:
2253:
2236:
2230:
2217:
2200:
2194:
2160:
2156:
2143:
2138:
2137:
2085:
2081:
2065:
2028:
2024:
2011:
2006:
2005:
1899:
1898:
1885:
1875:
1870:
1869:
1826:
1822:
1768:
1764:
1730:
1726:
1713:
1708:
1707:
1678:
1673:
1672:
1639:
1634:
1633:
1582:
1581:
1575:
1574:
1546:
1540:
1539:
1458:
1457:
1444:
1434:
1429:
1428:
1408:
1407:
1392:
1376:
1360:
1343:
1339:
1332:
1322:
1319:
1318:
1302:
1289:
1276:
1262:
1258:
1251:
1241:
1232:
1231:
1213:
1209:
1206:
1202:
1198:divided by the
1195:
1191:
1160:
1159:
1131:
1130:
1085:
1070:
1062:
1059:
1055:
1023:
1004:
1000:
966:
962:
949:
944:
943:
925:
921:
918:
910:Bloch's theorem
869:
867:Bloch's theorem
863:
802:Poynting vector
786:
780:
746:
745:
724:
723:
655:
651:
644:
640:
611:
607:
599:
598:
564:
538:
537:
527:
498:
484:
483:
479:
468:
458:
448:
438:
430:
422:
416:
412:
406:
398:
307:
306:
254:
253:
245:
240:
234:
203:
183:
182:
154:
153:
129:
118:
112:
88:
85:crystallography
69:
17:
12:
11:
5:
4227:
4226:
4223:
4215:
4214:
4209:
4207:Wave mechanics
4199:
4198:
4195:
4194:
4188:
4173:
4170:
4167:
4166:
4159:
4135:
4129:978-0486605968
4128:
4106:
4090:
4075:
4068:
4048:
4041:
4020:
3999:
3977:
3976:
3973:
3972:
3954:
3953:
3951:
3948:
3947:
3946:
3941:
3934:
3931:
3930:
3929:
3916:
3913:
3908:
3902:
3899:
3896:
3893:
3890:
3887:
3883:
3878:
3871:
3867:
3860:
3857:
3854:
3849:
3820:
3817:
3816:
3815:
3801:
3798:
3795:
3790:
3787:
3784:
3778:
3772:
3769:
3766:
3761:
3758:
3755:
3752:
3749:
3746:
3743:
3740:
3737:
3734:
3728:
3722:
3719:
3716:
3709:
3705:
3701:
3698:
3692:
3686:
3683:
3680:
3677:
3674:
3671:
3667:
3662:
3655:
3651:
3644:
3641:
3638:
3633:
3608:
3605:
3604:
3603:
3589:
3586:
3583:
3578:
3575:
3572:
3566:
3560:
3557:
3554:
3549:
3546:
3543:
3540:
3537:
3534:
3531:
3528:
3525:
3522:
3516:
3510:
3507:
3504:
3497:
3493:
3489:
3486:
3480:
3474:
3471:
3468:
3465:
3462:
3459:
3455:
3450:
3443:
3439:
3432:
3429:
3426:
3421:
3395:
3392:
3389:
3377:
3374:
3373:
3372:
3369:
3368:
3354:
3351:
3348:
3345:
3342:
3339:
3336:
3333:
3330:
3326:
3321:
3314:
3310:
3303:
3300:
3297:
3292:
3260:
3257:
3254:
3251:
3246:
3242:
3238:
3233:
3229:
3225:
3220:
3216:
3193:
3189:
3168:
3165:
3162:
3151:
3150:
3135:
3132:
3129:
3126:
3121:
3115:
3112:
3109:
3104:
3098:
3095:
3092:
3087:
3081:
3078:
3075:
3070:
3064:
3061:
3058:
3056:
3054:
3049:
3043:
3040:
3037:
3032:
3028:
3025:
3022:
3017:
3011:
3008:
3005:
3000:
2994:
2991:
2988:
2986:
2982:
2977:
2973:
2967:
2966:
2961:
2955:
2952:
2949:
2944:
2938:
2933:
2929:
2925:
2920:
2914:
2911:
2908:
2903:
2897:
2892:
2888:
2884:
2879:
2873:
2870:
2867:
2862:
2856:
2851:
2847:
2843:
2838:
2832:
2829:
2826:
2821:
2815:
2810:
2806:
2802:
2799:
2797:
2793:
2788:
2784:
2780:
2779:
2756:
2753:
2750:
2739:
2738:
2725:
2719:
2716:
2713:
2708:
2702:
2697:
2693:
2689:
2684:
2679:
2675:
2647:
2646:
2633:
2627:
2624:
2622:
2619:
2617:
2614:
2612:
2609:
2608:
2605:
2602:
2600:
2597:
2595:
2592:
2590:
2587:
2586:
2583:
2580:
2578:
2575:
2573:
2570:
2568:
2565:
2562:
2559:
2558:
2552:
2546:
2541:
2535:
2533:
2530:
2527:
2524:
2522:
2519:
2518:
2516:
2511:
2508:
2485:
2482:
2481:
2480:
2468:
2461:
2458:
2452:
2449:
2445:
2441:
2437:
2430:
2426:
2420:
2416:
2410:
2406:
2401:
2397:
2392:
2385:
2381:
2375:
2371:
2365:
2361:
2357:
2350:
2347:
2341:
2336:
2333:
2327:
2323:
2318:
2314:
2295:
2294:
2282:
2275:
2272:
2266:
2263:
2260:
2256:
2251:
2246:
2242:
2239:
2233:
2229:
2224:
2220:
2215:
2210:
2206:
2203:
2197:
2193:
2189:
2182:
2179:
2173:
2168:
2165:
2159:
2155:
2150:
2146:
2131:four-frequency
2127:
2126:
2114:
2107:
2104:
2098:
2093:
2090:
2084:
2080:
2077:
2072:
2068:
2064:
2061:
2057:
2050:
2047:
2041:
2036:
2033:
2027:
2023:
2018:
2014:
1995:
1994:
1982:
1979:
1974:
1969:
1965:
1961:
1956:
1951:
1947:
1943:
1938:
1933:
1929:
1925:
1920:
1915:
1910:
1907:
1902:
1897:
1892:
1888:
1882:
1878:
1863:
1862:
1848:
1841:
1838:
1832:
1829:
1825:
1819:
1816:
1811:
1807:
1800:
1797:
1789:
1786:
1781:
1776:
1773:
1767:
1763:
1759:
1752:
1749:
1743:
1738:
1735:
1729:
1725:
1720:
1716:
1693:
1690:
1685:
1681:
1654:
1651:
1646:
1642:
1622:
1621:
1608:
1603:
1598:
1594:
1589:
1585:
1578:
1573:
1568:
1563:
1558:
1553:
1549:
1543:
1538:
1533:
1528:
1524:
1520:
1515:
1510:
1506:
1502:
1497:
1492:
1488:
1484:
1479:
1474:
1469:
1466:
1461:
1456:
1451:
1447:
1441:
1437:
1422:
1421:
1405:
1399:
1395:
1391:
1388:
1383:
1379:
1375:
1372:
1367:
1363:
1359:
1356:
1351:
1348:
1342:
1338:
1335:
1333:
1329:
1325:
1321:
1320:
1315:
1309:
1305:
1301:
1296:
1292:
1288:
1283:
1279:
1275:
1270:
1267:
1261:
1257:
1254:
1252:
1248:
1244:
1240:
1239:
1204:
1200:phase-velocity
1173:
1170:
1144:
1141:
1127:
1126:
1113:
1107:
1100:
1097:
1091:
1088:
1082:
1076:
1073:
1068:
1065:
1058:
1054:
1050:
1043:
1040:
1030:
1026:
1022:
1017:
1012:
1009:
1003:
999:
995:
988:
985:
979:
974:
971:
965:
961:
956:
952:
917:
914:
865:Main article:
862:
859:
840:cubic crystals
816:, also called
806:phase velocity
798:group velocity
784:Group velocity
782:Main article:
779:
776:
775:
774:
759:
756:
743:
731:
717:
716:
705:
701:
697:
694:
690:
686:
683:
680:
675:
670:
664:
661:
654:
650:
647:
643:
639:
636:
633:
630:
626:
622:
619:
615:
610:
606:
592:
591:
580:
574:
570:
567:
560:
556:
551:
546:
525:
514:
508:
504:
501:
494:
491:
466:
456:
446:
410:
404:
392:
391:
380:
377:
374:
371:
368:
365:
362:
358:
354:
350:
346:
343:
340:
337:
334:
331:
328:
325:
321:
317:
314:
300:traveling wave
281:
278:
272:
269:
266:
262:
221:
215:
212:
206:
202:
196:
193:
167:
164:
138:zero crossings
116:Traveling wave
111:
108:
15:
13:
10:
9:
6:
4:
3:
2:
4225:
4224:
4213:
4210:
4208:
4205:
4204:
4202:
4191:
4185:
4181:
4176:
4175:
4171:
4162:
4156:
4152:
4148:
4147:
4139:
4136:
4131:
4125:
4121:
4117:
4110:
4107:
4104:
4100:
4094:
4091:
4086:
4079:
4076:
4071:
4065:
4061:
4060:
4052:
4049:
4044:
4038:
4034:
4033:
4024:
4021:
4016:
4010:
4002:
3996:
3992:
3991:
3982:
3979:
3969:
3965:
3959:
3956:
3949:
3945:
3942:
3940:
3937:
3936:
3932:
3914:
3911:
3906:
3897:
3894:
3891:
3885:
3881:
3876:
3869:
3865:
3847:
3837:
3836:
3835:
3831:
3827:
3818:
3799:
3796:
3793:
3788:
3785:
3782:
3776:
3770:
3767:
3764:
3756:
3753:
3750:
3741:
3738:
3735:
3726:
3720:
3717:
3714:
3707:
3703:
3699:
3696:
3690:
3681:
3678:
3675:
3669:
3665:
3660:
3653:
3649:
3631:
3621:
3620:
3619:
3615:
3606:
3587:
3584:
3581:
3576:
3573:
3570:
3564:
3558:
3555:
3552:
3544:
3541:
3538:
3529:
3526:
3523:
3514:
3508:
3505:
3502:
3495:
3491:
3487:
3484:
3478:
3469:
3466:
3463:
3457:
3453:
3448:
3441:
3437:
3419:
3409:
3408:
3407:
3393:
3390:
3387:
3375:
3349:
3346:
3343:
3340:
3337:
3334:
3328:
3324:
3319:
3312:
3308:
3290:
3280:
3279:
3276:
3275:
3274:
3271:
3258:
3255:
3252:
3249:
3244:
3240:
3236:
3231:
3227:
3223:
3218:
3214:
3191:
3187:
3166:
3163:
3160:
3133:
3130:
3127:
3124:
3119:
3102:
3096:
3093:
3090:
3085:
3068:
3062:
3059:
3057:
3047:
3030:
3026:
3023:
3020:
3015:
2998:
2992:
2989:
2987:
2980:
2975:
2971:
2959:
2942:
2936:
2931:
2923:
2918:
2901:
2895:
2890:
2882:
2877:
2860:
2854:
2849:
2841:
2836:
2819:
2813:
2808:
2800:
2798:
2791:
2786:
2782:
2770:
2769:
2768:
2754:
2751:
2748:
2723:
2706:
2700:
2695:
2687:
2682:
2677:
2673:
2665:
2664:
2663:
2660:
2654:
2631:
2625:
2620:
2615:
2610:
2603:
2598:
2593:
2588:
2581:
2576:
2571:
2566:
2563:
2560:
2550:
2539:
2531:
2528:
2525:
2520:
2514:
2509:
2499:
2498:
2497:
2495:
2491:
2483:
2466:
2456:
2450:
2447:
2443:
2439:
2435:
2428:
2424:
2418:
2414:
2408:
2404:
2399:
2395:
2390:
2383:
2379:
2373:
2369:
2363:
2359:
2355:
2345:
2339:
2334:
2331:
2325:
2321:
2316:
2312:
2304:
2303:
2302:
2300:
2299:four-velocity
2280:
2270:
2264:
2261:
2258:
2254:
2249:
2244:
2240:
2237:
2231:
2227:
2222:
2218:
2213:
2208:
2204:
2201:
2195:
2191:
2187:
2177:
2171:
2166:
2163:
2157:
2153:
2148:
2144:
2136:
2135:
2134:
2132:
2112:
2102:
2096:
2091:
2088:
2082:
2075:
2070:
2066:
2059:
2055:
2045:
2039:
2034:
2031:
2025:
2021:
2016:
2012:
2004:
2003:
2002:
2000:
1999:four-momentum
1980:
1977:
1972:
1967:
1963:
1959:
1954:
1949:
1945:
1941:
1936:
1931:
1927:
1923:
1918:
1913:
1908:
1905:
1900:
1895:
1890:
1886:
1880:
1876:
1868:
1867:
1866:
1846:
1836:
1830:
1827:
1823:
1817:
1814:
1809:
1805:
1795:
1787:
1784:
1779:
1774:
1771:
1765:
1761:
1757:
1747:
1741:
1736:
1733:
1727:
1723:
1718:
1714:
1706:
1705:
1704:
1691:
1688:
1683:
1679:
1670:
1669:monochromatic
1665:
1652:
1649:
1644:
1640:
1631:
1627:
1606:
1601:
1592:
1587:
1583:
1576:
1571:
1566:
1561:
1556:
1551:
1547:
1541:
1536:
1531:
1526:
1522:
1518:
1513:
1508:
1504:
1500:
1495:
1490:
1486:
1482:
1477:
1472:
1467:
1464:
1459:
1454:
1449:
1445:
1439:
1435:
1427:
1426:
1425:
1403:
1397:
1393:
1389:
1386:
1381:
1377:
1373:
1370:
1365:
1361:
1357:
1354:
1349:
1346:
1340:
1336:
1334:
1327:
1323:
1313:
1307:
1303:
1299:
1294:
1290:
1286:
1281:
1277:
1273:
1268:
1265:
1259:
1255:
1253:
1246:
1242:
1230:
1229:
1228:
1226:
1222:
1221:contravariant
1217:
1201:
1188:
1168:
1142:
1139:
1111:
1105:
1095:
1089:
1086:
1080:
1074:
1071:
1066:
1063:
1056:
1052:
1048:
1038:
1028:
1024:
1020:
1015:
1010:
1007:
1001:
997:
993:
983:
977:
972:
969:
963:
959:
954:
950:
942:
941:
940:
938:
934:
929:
915:
913:
911:
907:
906:
901:
897:
894:
890:
886:
882:
878:
874:
868:
860:
858:
856:
852:
848:
843:
841:
837:
833:
829:
826:
821:
819:
815:
811:
807:
803:
799:
795:
791:
785:
777:
744:
729:
722:
721:
720:
703:
699:
695:
692:
688:
684:
681:
678:
668:
652:
648:
645:
641:
637:
634:
631:
628:
624:
620:
617:
608:
604:
597:
596:
595:
578:
572:
568:
565:
558:
535:
530:
526:
512:
506:
502:
499:
492:
489:
478:
474:
467:
464:
457:
454:
447:
444:
436:
428:
419:
411:
405:
401:
397:
396:
395:
378:
372:
369:
366:
363:
360:
352:
341:
338:
335:
332:
326:
323:
312:
305:
304:
303:
301:
298:A sinusoidal
296:
270:
267:
264:
248:
243:
237:
219:
204:
200:
191:
151:
147:
139:
135:
127:
122:
117:
109:
107:
105:
104:
99:
94:
91:
86:
82:
78:
73:
67:
63:
58:
56:
52:
48:
44:
40:
36:
32:
28:
24:
19:
4179:
4145:
4138:
4119:
4109:
4098:
4093:
4084:
4078:
4058:
4051:
4031:
4023:
3989:
3981:
3958:
3829:
3825:
3822:
3613:
3610:
3379:
3272:
3152:
2740:
2658:
2652:
2648:
2487:
2301:as follows:
2296:
2133:as follows:
2128:
2001:as follows:
1996:
1864:
1666:
1623:
1423:
1227:forms are:
1218:
1189:
1128:
930:
919:
903:
896:wavefunction
876:
870:
844:
822:
787:
718:
593:
528:
463:phase offset
443:air pressure
417:
403:is position,
399:
393:
297:
246:
241:
235:
149:
145:
143:
101:
95:
89:
80:
74:
65:
61:
59:
30:
26:
20:
18:
2488:Taking the
933:four-vector
794:wave packet
146:wave vector
81:wave vector
27:wave vector
4201:Categories
3950:References
900:sinusoidal
849:, such as
818:wavefronts
534:wavelength
435:sound wave
427:ocean wave
144:The terms
114:See also:
110:Definition
51:wavelength
47:wavenumber
31:wavevector
4009:cite book
3915:γ
3895:−
3886:γ
3866:ω
3848:ω
3800:β
3797:−
3789:β
3771:β
3768:−
3757:β
3754:−
3742:β
3721:β
3718:−
3704:β
3700:−
3682:β
3679:−
3670:γ
3650:ω
3632:ω
3588:β
3577:β
3574:−
3559:β
3545:β
3542:−
3530:β
3509:β
3492:β
3488:−
3470:β
3458:γ
3438:ω
3420:ω
3394:π
3388:θ
3350:θ
3347:
3341:β
3338:−
3329:γ
3309:ω
3291:ω
3256:θ
3253:
3167:θ
3164:
3131:θ
3128:
3103:ω
3097:γ
3094:β
3091:−
3069:ω
3063:γ
3027:γ
3024:β
3021:−
2999:ω
2993:γ
2972:ω
2928:Λ
2887:Λ
2846:Λ
2805:Λ
2749:μ
2724:ν
2701:μ
2696:ν
2692:Λ
2683:μ
2572:γ
2567:γ
2564:β
2561:−
2532:γ
2529:β
2526:−
2521:γ
2507:Λ
2460:→
2440:γ
2415:ω
2400:μ
2370:ω
2349:→
2332:ω
2317:μ
2274:→
2265:ν
2259:ν
2241:π
2223:μ
2205:π
2181:→
2164:ω
2149:μ
2106:→
2089:ω
2079:ℏ
2071:μ
2063:ℏ
2049:→
2017:μ
1960:−
1942:−
1924:−
1906:ω
1891:μ
1881:μ
1840:^
1815:ω
1799:^
1785:ω
1772:ω
1751:→
1734:ω
1719:μ
1597:ℏ
1548:ω
1519:−
1501:−
1483:−
1465:ω
1450:μ
1440:μ
1390:−
1374:−
1358:−
1347:ω
1328:μ
1266:ω
1247:μ
1225:covariant
1172:→
1140:ω
1106:λ
1099:^
1090:π
1067:π
1042:^
1021:ω
1008:ω
987:→
970:ω
955:μ
834:(such as
758:~
755:ν
696:φ
679:−
669:⋅
663:~
660:ν
649:π
638:
605:ψ
573:λ
569:π
503:π
490:ω
453:amplitude
373:φ
364:ω
361:−
353:⋅
342:
313:ψ
280:~
277:ν
271:π
214:~
211:ν
195:~
192:ν
166:~
163:ν
126:sine wave
3933:See also
1630:massless
881:electron
879:) of an
877:k-vector
825:lossless
409:is time,
244:= |
889:crystal
838:), and
812:to the
719:where:
451:is the
394:where:
77:physics
33:) is a
23:physics
4186:
4157:
4126:
4066:
4039:
3997:
3964:period
3830:π
3826:θ
3614:θ
3153:where
2554:
2548:
2543:
2537:
1214:λ
1196:ω
939:, as:
477:period
469:ω
459:φ
439:ψ
431:ψ
413:ψ
250:|
35:vector
4151:60–65
887:in a
836:glass
823:In a
461:is a
134:phase
4184:ISBN
4155:ISBN
4124:ISBN
4103:link
4064:ISBN
4037:ISBN
4015:link
3995:ISBN
1628:for
1626:null
1223:and
885:hole
421:and
148:and
100:, a
64:(or
39:wave
29:(or
25:, a
3616:= 0
3344:cos
3273:So
3250:cos
3161:cos
3125:cos
883:or
871:In
635:cos
339:cos
21:In
4203::
4153:.
4118:.
4011:}}
4007:{{
3832:/2
3828:=
1216:.
820:.
445:).
437:,
429:,
295:.
128:,
57:.
4192:.
4163:.
4132:.
4072:.
4045:.
4017:)
4003:.
3912:1
3907:=
3901:)
3898:0
3892:1
3889:(
3882:1
3877:=
3870:s
3859:s
3856:b
3853:o
3794:1
3786:+
3783:1
3777:=
3765:1
3760:)
3751:1
3748:(
3745:)
3739:+
3736:1
3733:(
3727:=
3715:1
3708:2
3697:1
3691:=
3685:)
3676:1
3673:(
3666:1
3661:=
3654:s
3643:s
3640:b
3637:o
3585:+
3582:1
3571:1
3565:=
3556:+
3553:1
3548:)
3539:1
3536:(
3533:)
3527:+
3524:1
3521:(
3515:=
3506:+
3503:1
3496:2
3485:1
3479:=
3473:)
3467:+
3464:1
3461:(
3454:1
3449:=
3442:s
3431:s
3428:b
3425:o
3391:=
3353:)
3335:1
3332:(
3325:1
3320:=
3313:s
3302:s
3299:b
3296:o
3259:.
3245:0
3241:k
3237:=
3232:1
3228:k
3224:,
3219:0
3215:k
3192:1
3188:k
3134:.
3120:c
3114:s
3111:b
3108:o
3086:c
3080:s
3077:b
3074:o
3060:=
3048:1
3042:s
3039:b
3036:o
3031:k
3016:c
3010:s
3007:b
3004:o
2990:=
2981:c
2976:s
2960:3
2954:s
2951:b
2948:o
2943:k
2937:0
2932:3
2924:+
2919:2
2913:s
2910:b
2907:o
2902:k
2896:0
2891:2
2883:+
2878:1
2872:s
2869:b
2866:o
2861:k
2855:0
2850:1
2842:+
2837:0
2831:s
2828:b
2825:o
2820:k
2814:0
2809:0
2801:=
2792:0
2787:s
2783:k
2755:0
2752:=
2718:s
2715:b
2712:o
2707:k
2688:=
2678:s
2674:k
2659:S
2653:S
2632:)
2626:1
2621:0
2616:0
2611:0
2604:0
2599:1
2594:0
2589:0
2582:0
2577:0
2551:0
2540:0
2515:(
2510:=
2467:)
2457:u
2451:,
2448:c
2444:(
2436:)
2429:2
2425:c
2419:o
2409:(
2405:=
2396:U
2391:)
2384:2
2380:c
2374:o
2364:(
2360:=
2356:)
2346:k
2340:,
2335:c
2326:(
2322:=
2313:K
2281:)
2271:n
2262:,
2255:(
2250:)
2245:c
2238:2
2232:(
2228:=
2219:N
2214:)
2209:c
2202:2
2196:(
2192:=
2188:)
2178:k
2172:,
2167:c
2158:(
2154:=
2145:K
2113:)
2103:k
2097:,
2092:c
2083:(
2076:=
2067:K
2060:=
2056:)
2046:p
2040:,
2035:c
2032:E
2026:(
2022:=
2013:P
1981:0
1978:=
1973:2
1968:z
1964:k
1955:2
1950:y
1946:k
1937:2
1932:x
1928:k
1919:2
1914:)
1909:c
1901:(
1896:=
1887:K
1877:K
1847:)
1837:n
1831:,
1828:1
1824:(
1818:c
1810:=
1806:)
1796:n
1788:c
1780:,
1775:c
1766:(
1762:=
1758:)
1748:k
1742:,
1737:c
1728:(
1724:=
1715:K
1692:c
1689:=
1684:p
1680:v
1653:0
1650:=
1645:o
1641:m
1607:2
1602:)
1593:c
1588:o
1584:m
1577:(
1572:=
1567:2
1562:)
1557:c
1552:o
1542:(
1537:=
1532:2
1527:z
1523:k
1514:2
1509:y
1505:k
1496:2
1491:x
1487:k
1478:2
1473:)
1468:c
1460:(
1455:=
1446:K
1436:K
1404:)
1398:z
1394:k
1387:,
1382:y
1378:k
1371:,
1366:x
1362:k
1355:,
1350:c
1341:(
1337:=
1324:K
1314:)
1308:z
1304:k
1300:,
1295:y
1291:k
1287:,
1282:x
1278:k
1274:,
1269:c
1260:(
1256:=
1243:K
1210:T
1205:p
1203:v
1192:k
1169:k
1143:c
1112:)
1096:n
1087:2
1081:,
1075:T
1072:c
1064:2
1057:(
1053:=
1049:)
1039:n
1029:p
1025:v
1016:,
1011:c
1002:(
998:=
994:)
984:k
978:,
973:c
964:(
960:=
951:K
926:X
922:X
730:f
704:,
700:)
693:+
689:)
685:t
682:f
674:r
653:(
646:2
642:(
632:A
629:=
625:)
621:t
618:,
614:r
609:(
579:.
566:2
559:=
555:|
550:k
545:|
529:k
513:,
507:T
500:2
493:=
480:T
465:,
449:A
423:t
418:r
407:t
400:r
379:,
376:)
370:+
367:t
357:r
349:k
345:(
336:A
333:=
330:)
327:t
324:,
320:r
316:(
268:2
265:=
261:k
247:k
242:k
236:k
220:|
205:|
201:=
130:λ
90:k
70:π
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