2615:
1120:
1128:
1663:
1455:
1112:
2034:
3876:
1398:, which can be interpreted as indicating a "local wavelength" of the solution as a function of time and space. This method treats the system locally as if it were uniform with the local properties; in particular, the local wave velocity associated with a frequency is the only thing needed to estimate the corresponding local wavenumber or wavelength. In addition, the method computes a slowly changing amplitude to satisfy other constraints of the equations or of the physical system, such as for
1518:
1411:
1346:
27:
436:
428:
1376:
medium) may propagate at a velocity that varies with position, and as a result may not be sinusoidal in space. The figure at right shows an example. As the wave slows down, the wavelength gets shorter and the amplitude increases; after a place of maximum response, the short wavelength is associated
2045:
The notion of path difference and constructive or destructive interference used above for the double-slit experiment applies as well to the display of a single slit of light intercepted on a screen. The main result of this interference is to spread out the light from the narrow slit into a broader
1422:
because the same vibration can be considered to have a variety of different wavelengths, as shown in the figure. Descriptions using more than one of these wavelengths are redundant; it is conventional to choose the longest wavelength that fits the phenomenon. The range of wavelengths sufficient to
463:), thus determining the allowed wavelengths. For example, for an electromagnetic wave, if the box has ideal conductive walls, the condition for nodes at the walls results because the conductive walls cannot support a tangential electric field, forcing the wave to have zero amplitude at the wall.
1482:
In the special case of dispersion-free and uniform media, waves other than sinusoids propagate with unchanging shape and constant velocity. In certain circumstances, waves of unchanging shape also can occur in nonlinear media; for example, the figure shows ocean waves in shallow water that have
1317:
When wavelengths of electromagnetic radiation are quoted, the wavelength in vacuum usually is intended unless the wavelength is specifically identified as the wavelength in some other medium. In acoustics, where a medium is essential for the waves to exist, the wavelength value is given for a
1981:
1683:. As shown in the figure, light is passed through two slits and shines on a screen. The path of the light to a position on the screen is different for the two slits, and depends upon the angle θ the path makes with the screen. If we suppose the screen is far enough from the slits (that is,
1462:
The concept of wavelength is most often applied to sinusoidal, or nearly sinusoidal, waves, because in a linear system the sinusoid is the unique shape that propagates with no shape change – just a phase change and potentially an amplitude change. The wavelength (or alternatively
642:
1157:
The wave velocity in one medium not only may differ from that in another, but the velocity typically varies with wavelength. As a result, the change in direction upon entering a different medium changes with the wavelength of the wave.
2200:
962:
881:
800:
466:
The stationary wave can be viewed as the sum of two traveling sinusoidal waves of oppositely directed velocities. Consequently, wavelength, period, and wave velocity are related just as for a traveling wave. For example, the
1842:
2423:
1312:
497:
1666:
Pattern of light intensity on a screen for light passing through two slits. The labels on the right refer to the difference of the path lengths from the two slits, which are idealized here as point sources.
1670:
When sinusoidal waveforms add, they may reinforce each other (constructive interference) or cancel each other (destructive interference) depending upon their relative phase. This phenomenon is used in the
1635:
for such a particle being spread over all space, de
Broglie proposed using wave packets to represent particles that are localized in space. The spatial spread of the wave packet, and the spread of the
1529:. Such waves are sometimes regarded as having a wavelength even though they are not sinusoidal. As shown in the figure, wavelength is measured between consecutive corresponding points on the waveform.
1361:
wavelength that depends in part on the depth of the sea floor compared to the wave height. The analysis of the wave can be based upon comparison of the local wavelength with the local water depth.
1027:
being interpreted as scalar speed in the direction of the wave vector. The first form, using reciprocal wavelength in the phase, does not generalize as easily to a wave in an arbitrary direction.
2527:
1538:
1820:
1222:
1101:
2467:
2249:
1365:
281:
2064:
In the analysis of the single slit, the non-zero width of the slit is taken into account, and each point in the aperture is taken as the source of one contribution to the beam of light (
1746:
2068:). On the screen, the light arriving from each position within the slit has a different path length, albeit possibly a very small difference. Consequently, interference occurs.
1329:. Separation occurs when the refractive index inside the prism varies with wavelength, so different wavelengths propagate at different speeds inside the prism, causing them to
3176:
2977:
To aid imagination, this bending of the wave often is compared to the analogy of a column of marching soldiers crossing from solid ground into mud. See, for example,
1471:) is a characterization of the wave in space, that is functionally related to its frequency, as constrained by the physics of the system. Sinusoids are the simplest
459:
The upper figure shows three standing waves in a box. The walls of the box are considered to require the wave to have nodes at the walls of the box (an example of
2093:
328:
304:
239:
2001:, is the single-slit result, which modulates the more rapidly varying second factor that depends upon the number of slits and their spacing. In the figure
1418:
Waves in crystalline solids are not continuous, because they are composed of vibrations of discrete particles arranged in a regular lattice. This produces
892:
811:
687:
3821:
2563:
is used to describe an object having one or more dimensions smaller than the length of the wave with which the object interacts. For example, the term
1976:{\displaystyle I_{q}=I_{1}\sin ^{2}\left({\frac {q\pi g\sin \alpha }{\lambda }}\right)/\sin ^{2}\left({\frac {\pi g\sin \alpha }{\lambda }}\right)\ ,}
405:
3749:
1639:
of sinusoids that make up the packet, correspond to the uncertainties in the particle's position and momentum, the product of which is bounded by
431:
Sinusoidal standing waves in a box that constrains the end points to be nodes will have an integer number of half wavelengths fitting in the box.
1333:
at different angles. The mathematical relationship that describes how the speed of light within a medium varies with wavelength is known as a
1139:, which means that the same frequency will correspond to a shorter wavelength in the medium than in vacuum, as shown in the figure at right.
74:
is the distance over which the wave's shape repeats. In other words, it is the distance between consecutive corresponding points of the same
1559:
that describes the overall amplitude of the wave; within the envelope, the distance between adjacent peaks or troughs is sometimes called a
3695:
2551:
As with other diffraction patterns, the pattern scales in proportion to wavelength, so shorter wavelengths can lead to higher resolution.
2359:
3639:
637:{\displaystyle y(x,\ t)=A\cos \left(2\pi \left({\frac {x}{\lambda }}-ft\right)\right)=A\cos \left({\frac {2\pi }{\lambda }}(x-vt)\right)}
2858:
1256:
1555:, "bursts" of wave action where each wave packet travels as a unit, find application in many fields of physics. A wave packet has an
3785:
2952:
2349:, the radius to the first null of the Airy disk, to a size proportional to the wavelength of the light used, and depending on the
1135:
The speed of a wave depends upon the medium in which it propagates. In particular, the speed of light in a medium is less than in
3346:
3814:
424:
so they can resolve targets smaller than 17 mm. Wavelengths in audible sound are much longer than those in visible light.
3759:
3732:
3705:
3649:
3562:
3533:
3471:
3356:
3329:
3302:
3273:
3074:
3047:
3020:
2990:
2962:
2901:
2868:
3394:
of a dispersing wave is twice the distance between two successive zeros. ... the local wavelength and the local wave number
452:
is an undulatory motion that stays in one place. A sinusoidal standing wave includes stationary points of no motion, called
3459:
3442:(p. 61) ... the individual waves move more slowly than the packet and therefore pass back through the packet as it advances
2565:
1357:
in space. For example, in an ocean wave approaching shore, shown in the figure, the incoming wave undulates with a varying
2490:
1042:
phase when describing a wave is based on the fact that the cosine is the real part of the complex exponential in the wave
216:
media, any wave pattern can be described in terms of the independent propagation of sinusoidal components. The wavelength
3064:
3037:
135:
Wavelength depends on the medium (for example, vacuum, air, or water) that a wave travels through. Examples of waves are
3240:
3207:
2585:
1825:
Thus, if the wavelength of the light is known, the slit separation can be determined from the interference pattern or
1761:
3807:
3678:
3622:
3595:
3500:
3435:
3383:
3155:
3124:
3104:
2935:
2841:
2816:
2789:
2760:
1173:
1048:
1423:
provide a description of all possible waves in a crystalline medium corresponds to the wave vectors confined to the
439:
A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue)
132:
of the wave: waves with higher frequencies have shorter wavelengths, and lower frequencies have longer wavelengths.
2431:
1640:
2262:
is the distance of the pattern (on the screen) from the slit, and λ is the wavelength of light used. The function
2205:
1250:, where the latter is measured in vacuum rather than in the medium. The corresponding wavelength in the medium is
248:
1657:
118:
1707:
2484:) of the image diffracted by a circular aperture, a measure most commonly used for telescopes and cameras, is:
2584:
are holes smaller than the wavelength of light propagating through them. Such structures have applications in
2346:
1325:, and is also responsible for the familiar phenomenon in which light is separated into component colours by a
2053:
Two types of diffraction are distinguished, depending upon the separation between the source and the screen:
1443:
3348:
The
Quantum Theory of Motion: An Account of the de Broglie–Bohm Causal Interpretation of Quantum Mechanics
3922:
2576:
A subwavelength particle is a particle smaller than the wavelength of light with which it interacts (see
1514:
with certain shapes can propagate unchanged, because of properties of the nonlinear surface-wave medium.
1372:
Waves that are sinusoidal in time but propagate through a medium whose properties vary with position (an
20:
3917:
2287:
2072:
346:
164:
3066:
Time-frequency and time-scale methods: adaptive decompositions, uncertainty principles, and sampling
4217:
3830:
1676:
1488:
390:
193:
2777:
2614:
1574:, wave packets can be analyzed into infinite sums (or integrals) of sinusoidal waves of different
1430:
This indeterminacy in wavelength in solids is important in the analysis of wave phenomena such as
4267:
2666:
1476:
2709:– dark lines in the solar spectrum, traditionally used as standard optical wavelength references
1146:, or a change in direction of waves that encounter the interface between media at an angle. For
1030:
Generalizations to sinusoids of other phases, and to complex exponentials, are also common; see
471:
can be determined from observation of standing waves in a metal box containing an ideal vacuum.
468:
3516:
Greenfield Sluder & David E. Wolf (2007). "IV. Young's
Experiment: Two-Slit Interference".
2712:
2054:
1680:
1399:
114:
89:
3722:
3579:
3550:
3488:
3373:
3224:
3195:
3141:
3010:
2980:
2921:
2893:
2806:
3666:
3612:
3525:
3419:
3290:
3114:
2603:
2028:
1616:
1381:
1147:
3319:
3257:
2665:). It is usually encountered in quantum mechanics, where it is used in combination with the
4227:
4222:
2470:
2195:{\displaystyle S(u)=\mathrm {sinc} ^{2}(u)=\left({\frac {\sin \pi u}{\pi u}}\right)^{2}\ ;}
479:
Traveling sinusoidal waves are often represented mathematically in terms of their velocity
148:
8:
4262:
4257:
4232:
4212:
3963:
3777:
2919:
2589:
2577:
2058:
1334:
1322:
1119:
460:
339:
3782:
2015:
the light, so the energy contained in the light is not altered, just where it shows up.
1123:
Refraction: upon entering a medium where its speed is lower, the wave changes direction.
4252:
4100:
4073:
3983:
3968:
3795:
The visible electromagnetic spectrum displayed in web colors with according wavelengths
2717:
2651:
2350:
2293:
2283:
313:
289:
224:
201:
1349:
Various local wavelengths on a crest-to-crest basis in an ocean wave approaching shore
4108:
4104:
4085:
4081:
4077:
3755:
3728:
3701:
3674:
3645:
3618:
3591:
3558:
3529:
3518:
3496:
3467:
3431:
3379:
3352:
3325:
3298:
3269:
3236:
3203:
3151:
3120:
3100:
3093:
3070:
3043:
3016:
2986:
2958:
2931:
2897:
2886:
2864:
2837:
2812:
2785:
2756:
2696:
2674:
1612:
1386:
1354:
957:{\displaystyle \lambda ={\frac {2\pi }{k}}={\frac {2\pi v}{\omega }}={\frac {v}{f}}.}
876:{\displaystyle k={\frac {2\pi }{\lambda }}={\frac {2\pi f}{v}}={\frac {\omega }{v}},}
675:
335:
176:
94:
71:
2071:
In the
Fraunhofer diffraction pattern sufficiently far from a single slit, within a
1127:
4242:
4013:
3988:
3856:
3426:(Reprint of Academic Press 1981 ed.). Courier Dover Publications. pp. 59
2706:
2701:
2038:
1624:
1581:
1571:
1414:
A wave on a line of atoms can be interpreted according to a variety of wavelengths.
1326:
1163:
1136:
189:
2033:
1662:
1454:
1131:
Separation of colors by a prism (click for animation if it is not already playing)
1111:
795:{\displaystyle y(x,\ t)=A\cos \left(kx-\omega t\right)=A\cos \left(k(x-vt)\right)}
4327:
4291:
4204:
4046:
3978:
3789:
3751:
Reflecting
Telescope Optics I: Basic Design Theory and Its Historical Development
2335:
1608:
125:
3778:
Conversion: Wavelength to
Frequency and vice versa – Sound waves and radio waves
1525:
If a traveling wave has a fixed shape that repeats in space or in time, it is a
175:
vary. Water waves are variations in the height of a body of water. In a crystal
4322:
4237:
4021:
2618:
Relationship between wavelength, angular wavelength, and other wave properties.
2541:
2301:
1672:
1424:
1232:
1151:
453:
401:
386:
365:. Thus the wavelength of a 100 MHz electromagnetic (radio) wave is about:
354:
307:
197:
172:
168:
156:
2544:
diameter of the imaging system, in the same units, and the angular resolution
4316:
4247:
4184:
4165:
4069:
4003:
3998:
3993:
3172:
2722:
2635:
2602:
may also refer to a phenomenon involving subwavelength objects; for example,
2570:
2308:. For a circular aperture, the diffraction-limited image spot is known as an
2084:
1632:
805:
in which wavelength and wavenumber are related to velocity and frequency as:
449:
85:
81:
76:
43:
39:
3875:
2573:
whose diameter is less than the wavelength of light propagating through it.
338:, the phase speed itself depends upon the frequency of the wave, making the
42:, such as between crests (on top), or troughs (on bottom), or corresponding
4123:
4118:
4096:
4091:
3799:
2727:
1537:
1517:
1484:
1431:
1410:
99:
1567:
of the wave packet moves at a speed different from the constituent waves.
1483:
sharper crests and flatter troughs than those of a sinusoid, typical of a
408:). The wavelengths of sound frequencies audible to the human ear (20
128:
moving at a fixed wave speed, wavelength is inversely proportional to the
4296:
3973:
3955:
3851:
3794:
2305:
2047:
2024:
1552:
1546:
1468:
1023:, is still in the same relationship with wavelength as shown above, with
1000:
55:
3582:. In John W Harris; Walter Benenson; Horst Stöcker; Holger Lutz (eds.).
2345:
size of objects viewed through a microscope is limited according to the
1691:) then the paths are nearly parallel, and the path difference is simply
1364:
3945:
3940:
3871:
1636:
1575:
1511:
1464:
1395:
1143:
1031:
1004:
668:
417:
397:
350:
182:
The range of wavelengths or frequencies for wave phenomena is called a
144:
136:
110:
2920:
David C. Cassidy; Gerald James Holton; Floyd James
Rutherford (2002).
2316:
in the single-slit diffraction formula is replaced by radial distance
4286:
4133:
3909:
3887:
3866:
3841:
2638:). It is equal to the ordinary wavelength reduced by a factor of 2π (
2593:
2481:
2309:
2297:
664:
382:
331:
129:
31:
2418:{\displaystyle r_{Airy}=1.22{\frac {\lambda }{2\,\mathrm {NA} }}\ ,}
4301:
4061:
3861:
2732:
2581:
1620:
1585:
1439:
1419:
1345:
1161:
For electromagnetic waves the speed in a medium is governed by its
184:
435:
26:
4170:
4156:
1435:
1330:
1307:{\displaystyle \lambda ={\frac {\lambda _{0}}{n(\lambda _{0})}}.}
1008:
51:
3258:"Chapter 1: Brief history and overview of nonlinear water waves"
2480:
size of the central bright portion (radius to first null of the
2046:
image on the screen. This distribution of wave energy is called
4194:
4189:
4179:
4151:
4146:
4141:
4112:
4041:
3783:
Teaching resource for 14–16 years on sound including wavelength
3062:
2804:
2469:
for θ being the half-angle of the cone of rays accepted by the
1701:. Accordingly, the condition for constructive interference is:
1035:
427:
213:
102:
3515:
1368:
A sinusoidal wave travelling in a nonuniform medium, with loss
4036:
3932:
3846:
2536:
is the wavelength of the waves that are focused for imaging,
413:
409:
377:= 3 m. The wavelength of visible light ranges from deep
160:
152:
140:
84:. Wavelength is a characteristic of both traveling waves and
3229:
Nonlinear Waves and
Solitons on Contours and Closed Surfaces
3196:"Figure 4.4: Transition from quasi-harmonic to cnoidal wave"
1321:
The variation in speed of light with wavelength is known as
1115:
Wavelength is decreased in a medium with slower propagation.
4031:
4026:
2649:), with SI units of meter per radian. It is the inverse of
1487:, a traveling wave so named because it is described by the
1472:
1394:). The method integrates phase through space using a local
1353:
Wavelength can be a useful concept even if the wave is not
1039:
67:
3375:
Introduction to partial differential equations with MATLAB
3262:
Nonlinear Ocean Waves and the
Inverse Scattering Transform
667:
of the wave. They are also commonly expressed in terms of
456:, and the wavelength is twice the distance between nodes.
4051:
2883:
2860:
Electromagnetic Theory for
Microwaves and Optoelectronics
2784:(2nd ed.). Cambridge University Press. p. 473.
1475:
solutions, and more complex solutions can be built up by
1150:, this change in the angle of propagation is governed by
421:
378:
3119:(4th ed.). Cambridge University Press. p. 22.
2978:
420:, respectively. Somewhat higher frequencies are used by
3614:
Fundamentals of light microscopy and electronic imaging
3183:(9th ed.). The Henry G Allen Company. p. 422.
2037:
Diffraction pattern of a double slit has a single-slit
1584:
postulated that all particles with a specific value of
1384:
of such systems is often done approximately, using the
1246:) is the refractive index of the medium at wavelength λ
38:, can be measured between any two points with the same
3035:
109:). The term "wavelength" is also sometimes applied to
80:
on the wave, such as two adjacent crests, troughs, or
3548:
3424:
Quantum Mechanics for Applied Physics and Engineering
2811:(4th ed.). Cengage Learning. pp. 404, 440.
2493:
2434:
2362:
2208:
2096:
1845:
1764:
1710:
1563:. An example is shown in the figure. In general, the
1521:
Wavelength of a periodic but non-sinusoidal waveform.
1259:
1176:
1051:
895:
814:
690:
500:
412:–20 kHz) are thus between approximately 17
316:
292:
251:
227:
220:
of a sinusoidal waveform traveling at constant speed
3200:
Nonlinear Dynamics: Between Linear and Impact Limits
2856:
2775:
2522:{\displaystyle \delta =1.22{\frac {\lambda }{D}}\ ,}
1755:
is an integer, and for destructive interference is:
3557:(Extended 8th ed.). Wiley-India. p. 965.
3288:
3193:
2908:
wavelength lambda light sound frequency wave speed.
2008:has been set to unity, a very rough approximation.
1142:This change in speed upon entering a medium causes
3517:
3092:
2885:
2521:
2461:
2417:
2243:
2194:
2057:or far-field diffraction at large separations and
1975:
1814:
1740:
1306:
1216:
1095:
956:
875:
794:
636:
322:
298:
275:
233:
3610:
3577:
2292:Diffraction is the fundamental limitation on the
2277:
2274:values at a separation proportion to wavelength.
1003:that specifies the direction and wavenumber of a
4314:
3700:. Courier Dover Publications. pp. 117–120.
3344:
2755:(2nd ed.). Addison Wesley. pp. 15–16.
2061:or near-field diffraction at close separations.
1815:{\displaystyle d\sin \theta =(m+1/2)\lambda \ .}
3720:
3693:
3139:
3063:Jeffrey A. Hogan & Joseph D. Lakey (2005).
1646:
1217:{\displaystyle v={\frac {c}{n(\lambda _{0})}},}
1096:{\displaystyle Ae^{i\left(kx-\omega t\right)}.}
389:, roughly 400 nm (for other examples, see
3637:
3486:
3417:
3371:
3295:Introduction to Macromolecular Crystallography
3255:
3112:
3015:. Jones & Bartlett Learning. p. 242.
1627:display have a De Broglie wavelength of about
474:
88:, as well as other spatial wave patterns. The
3815:
3747:
3495:. Vol. 53. Academic Press. p. 271.
3171:
2462:{\displaystyle \mathrm {NA} =n\sin \theta \;}
340:relationship between wavelength and frequency
3829:
3664:
3641:Optical scattering: measurement and analysis
3493:Advances in Electronics and Electron Physics
3008:
2622:A quantity related to the wavelength is the
2244:{\displaystyle u={\frac {xL}{\lambda R}}\ ,}
674:(2π times the reciprocal of wavelength) and
306:is called the phase speed (magnitude of the
276:{\displaystyle \lambda ={\frac {v}{f}}\,\,,}
3457:
3351:. Cambridge University Press. p. 160.
3222:
2985:. Cambridge University Press. p. 327.
2950:
2892:. Jones & Bartlett Publishers. p.
2831:
2782:An introduction to numerical methods in C++
2428:where the numerical aperture is defined as
1994:is the grating constant. The first factor,
1675:. A simple example is an experiment due to
1651:
98:. Wavelength is commonly designated by the
3891:
3822:
3808:
3724:Handbook of biological confocal microscopy
3452:
3450:
3317:
3099:. Cambridge University Press. p. 97.
2805:Raymond A. Serway; John W. Jewett (2006).
2458:
2018:
1741:{\displaystyle d\sin \theta =m\lambda \ ,}
1615:. Nowadays, this wavelength is called the
967:In the second form given above, the phase
16:Distance over which a wave's shape repeats
3282:
3187:
3146:(2nd ed.). Birkhäuser. pp. 165
3090:
3004:
3002:
2884:Theo Koupelis & Karl F. Kuhn (2007).
2397:
1687:is large compared to the slit separation
1449:
1438:. It is mathematically equivalent to the
651:is the value of the wave at any position
406:room temperature and atmospheric pressure
269:
268:
3644:(2nd ed.). SPIE Press. p. 64.
3542:
3524:(3rd ed.). Academic Press. p.
3411:
3164:
2613:
2032:
1661:
1536:
1516:
1453:
1409:
1377:with a high loss and the wave dies out.
1363:
1344:
1126:
1118:
1110:
434:
426:
25:
3727:(2nd ed.). Springer. p. 112.
3551:"§35-4 Young's interference experiment"
3447:
3231:(2nd ed.). Springer. pp. 469
3143:Fundamentals of solid state engineering
3133:
4315:
3604:
3571:
3509:
3491:. In L. Marton; Claire Marton (eds.).
3480:
3365:
3083:
3042:. New Age International. p. 454.
2999:
2913:
2834:The surface physics of liquid crystals
2769:
1611:. This hypothesis was at the basis of
1458:Near-periodic waves over shallow water
1034:. The typical convention of using the
117:of modulated waves or waves formed by
3803:
3658:
2798:
2750:
2609:
192:but now can be applied to the entire
147:and periodic electrical signals in a
3460:"Heisenberg's uncertainty principle"
3456:See, for example, Figs. 2.8–2.10 in
3324:. John Wiley & Sons. p. 1.
3216:
2857:Keqian Zhang & Dejie Li (2007).
2836:. Taylor & Francis. p. 17.
2744:
2566:subwavelength-diameter optical fibre
2270:is a non-zero integer, where are at
3311:
1836:For multiple slits, the pattern is
1340:
1011:, parameterized by position vector
207:
13:
3874:
3549:Halliday; Resnick; Walker (2008).
2971:
2586:extraordinary optical transmission
2439:
2436:
2402:
2399:
2123:
2120:
2117:
2114:
14:
4339:
3771:
3580:"§9.8.2 Diffraction by a grating"
3297:(2 ed.). Wiley. p. 77.
2979:Raymond T. Pierrehumbert (2010).
2011:The effect of interference is to
443:
3464:Quantum Physics: An Introduction
3116:Introduction to lattice dynamics
3095:Introduction to mineral sciences
2554:
2296:of optical instruments, such as
1641:Heisenberg uncertainty principle
1106:
483:(in the x direction), frequency
92:of the wavelength is called the
3741:
3714:
3687:
3631:
3338:
3249:
3056:
3036:Bishwanath Chakraborty (2007).
3029:
2982:Principles of Planetary Climate
2957:. Nelson Thornes. p. 460.
1658:Interference (wave propagation)
1532:
1015:. In that case, the wavenumber
188:. The name originated with the
3039:Principles of Plasma Mechanics
2944:
2877:
2850:
2825:
2278:Diffraction-limited resolution
2139:
2133:
2106:
2100:
1800:
1780:
1679:where light is passed through
1295:
1282:
1205:
1192:
995:, by replacing the wavenumber
784:
769:
709:
694:
626:
611:
519:
504:
1:
3697:Introduction to Modern Optics
3466:. CRC Press. pp. 53–56.
3264:. Academic Press. pp. 3
2776:Brian Hilton Flowers (2000).
2738:
2320:and the sine is replaced by 2
1495:th order, usually denoted as
681:(2π times the frequency) as:
113:waves, and to the sinusoidal
3291:"Waves and their properties"
3289:Alexander McPherson (2009).
3194:Valery N. Pilipchuk (2010).
1990:is the number of slits, and
1647:Interference and diffraction
7:
3923:Ultra-high-energy gamma ray
3069:. Birkhäuser. p. 348.
2926:. Birkhäuser. pp. 339
2690:
1405:
475:Mathematical representation
155:wave is a variation in air
21:Wavelength (disambiguation)
10:
4344:
3918:Very-high-energy gamma ray
3617:. Wiley/IEEE. p. 64.
3611:Douglas B. Murphy (2002).
3578:Kordt Griepenkerl (2002).
2778:"§21.2 Periodic functions"
2288:Diffraction-limited system
2281:
2022:
1655:
1544:
18:
4276:
4203:
4132:
4060:
4012:
3954:
3931:
3908:
3837:
3754:. Springer. p. 302.
3673:. CRC Press. p. 57.
3586:. Springer. pp. 307
3378:. Springer. p. 272.
3345:Peter R. Holland (1995).
3225:"§18.3 Special functions"
3202:. Springer. p. 127.
2863:. Springer. p. 533.
2630:), usually symbolized by
2073:small-angle approximation
1541:A propagating wave packet
353:, the phase speed is the
347:electromagnetic radiation
179:, atomic positions vary.
165:electromagnetic radiation
3831:Electromagnetic spectrum
3721:James B. Pawley (1995).
3694:Grant R. Fowles (1989).
3667:"Diffraction limitation"
3140:Manijeh Razeghi (2006).
2888:In Quest of the Universe
1652:Double-slit interference
1489:Jacobi elliptic function
979:is often generalized to
391:electromagnetic spectrum
194:electromagnetic spectrum
3638:John C. Stover (1995).
3555:Fundamentals of Physics
3489:"Electron Interference"
3487:Ming Chiang Li (1980).
3420:"Wave packet solutions"
3418:A. T. Fromhold (1991).
3372:Jeffery Cooper (1998).
3256:Alfred Osborne (2010).
3181:Encyclopædia Britannica
3113:Martin T. Dove (1993).
2667:reduced Planck constant
2592:, among other areas of
2079:is related to position
2075:, the intensity spread
2019:Single-slit diffraction
1446:at discrete intervals.
3879:
3748:Ray N. Wilson (2004).
3671:The science of imaging
2751:Hecht, Eugene (1987).
2713:Index of wave articles
2619:
2523:
2463:
2419:
2245:
2196:
2055:Fraunhofer diffraction
2042:
1977:
1816:
1742:
1667:
1542:
1522:
1459:
1450:More general waveforms
1415:
1400:conservation of energy
1392:Liouville–Green method
1382:differential equations
1369:
1350:
1308:
1218:
1132:
1124:
1116:
1097:
958:
877:
796:
638:
440:
432:
324:
300:
277:
235:
190:visible light spectrum
121:of several sinusoids.
47:
3878:
3665:Graham Saxby (2002).
3390:The local wavelength
3009:Paul R Pinet (2009).
2923:Understanding physics
2808:Principles of physics
2617:
2604:subwavelength imaging
2524:
2464:
2420:
2246:
2197:
2036:
2029:Diffraction formalism
1978:
1817:
1743:
1665:
1617:de Broglie wavelength
1540:
1520:
1457:
1413:
1367:
1348:
1309:
1219:
1148:electromagnetic waves
1130:
1122:
1114:
1098:
1038:phase instead of the
959:
878:
797:
639:
438:
430:
325:
301:
278:
236:
29:
3458:Joy Manners (2000).
3223:Andrei Ludu (2012).
2954:The World of Physics
2951:John Avison (1999).
2832:A. A. Sonin (1995).
2590:zero-mode waveguides
2491:
2471:microscope objective
2432:
2360:
2206:
2094:
1843:
1762:
1708:
1442:of a signal that is
1257:
1174:
1049:
893:
812:
688:
498:
404:is 343 m/s (at
314:
290:
249:
225:
167:the strength of the
30:The wavelength of a
19:For other uses, see
3964:Extreme ultraviolet
3584:Handbook of physics
3318:Eric Stade (2011).
3089:See Figure 4.20 in
2578:Rayleigh scattering
2258:is the slit width,
2202: with
2059:Fresnel diffraction
1619:. For example, the
1390:(also known as the
1335:dispersion relation
1019:, the magnitude of
461:boundary conditions
381:, roughly 700
3969:Vacuum ultraviolet
3880:
3788:2012-03-13 at the
3520:Digital microscopy
3111:and Figure 2.3 in
3091:A. Putnis (1992).
2718:Length measurement
2652:angular wavenumber
2628:reduced wavelength
2624:angular wavelength
2620:
2610:Angular wavelength
2519:
2459:
2415:
2351:numerical aperture
2347:Rayleigh criterion
2284:Angular resolution
2241:
2192:
2043:
1973:
1812:
1738:
1668:
1591:have a wavelength
1543:
1523:
1510:. Large-amplitude
1460:
1436:lattice vibrations
1416:
1370:
1351:
1318:specified medium.
1304:
1214:
1133:
1125:
1117:
1093:
954:
873:
792:
634:
441:
433:
349:—such as light—in
320:
310:) of the wave and
296:
273:
231:
202:vibration spectrum
48:
4310:
4309:
4014:Visible (optical)
3761:978-3-540-40106-3
3734:978-0-306-44826-3
3707:978-0-486-65957-2
3651:978-0-8194-1934-7
3564:978-81-265-1442-7
3535:978-0-12-374025-0
3473:978-0-7503-0720-8
3358:978-0-521-48543-2
3331:978-1-118-16551-5
3304:978-0-470-18590-2
3275:978-0-12-528629-9
3076:978-0-8176-4276-1
3049:978-81-224-1446-2
3022:978-0-7637-5993-3
2992:978-0-521-86556-2
2964:978-0-17-438733-6
2903:978-0-7637-4387-1
2870:978-3-540-74295-1
2697:Emission spectrum
2675:angular frequency
2673:, h-bar) and the
2634:("lambda-bar" or
2580:). Subwavelength
2515:
2511:
2411:
2407:
2334:is a first order
2237:
2233:
2188:
2174:
2066:Huygens' wavelets
1969:
1961:
1911:
1808:
1734:
1631:. To prevent the
1613:quantum mechanics
1299:
1209:
949:
936:
915:
868:
855:
834:
705:
676:angular frequency
609:
558:
515:
336:dispersive medium
323:{\displaystyle f}
299:{\displaystyle v}
266:
234:{\displaystyle v}
177:lattice vibration
95:spatial frequency
72:periodic function
4335:
3901:
3899:
3892:
3885:
3824:
3817:
3810:
3801:
3800:
3766:
3765:
3745:
3739:
3738:
3718:
3712:
3711:
3691:
3685:
3684:
3662:
3656:
3655:
3635:
3629:
3628:
3608:
3602:
3601:
3575:
3569:
3568:
3546:
3540:
3539:
3523:
3513:
3507:
3506:
3484:
3478:
3477:
3454:
3445:
3444:
3415:
3409:
3408:
3369:
3363:
3362:
3342:
3336:
3335:
3321:Fourier Analysis
3315:
3309:
3308:
3286:
3280:
3279:
3253:
3247:
3246:
3220:
3214:
3213:
3191:
3185:
3184:
3168:
3162:
3161:
3137:
3131:
3130:
3110:
3098:
3087:
3081:
3080:
3060:
3054:
3053:
3033:
3027:
3026:
3006:
2997:
2996:
2975:
2969:
2968:
2948:
2942:
2941:
2917:
2911:
2910:
2891:
2881:
2875:
2874:
2854:
2848:
2847:
2829:
2823:
2822:
2802:
2796:
2795:
2773:
2767:
2766:
2748:
2707:Fraunhofer lines
2702:Envelope (waves)
2686:
2664:
2648:
2528:
2526:
2525:
2520:
2513:
2512:
2504:
2468:
2466:
2465:
2460:
2442:
2424:
2422:
2421:
2416:
2409:
2408:
2406:
2405:
2389:
2381:
2380:
2266:has zeros where
2250:
2248:
2247:
2242:
2235:
2234:
2232:
2224:
2216:
2201:
2199:
2198:
2193:
2186:
2185:
2184:
2179:
2175:
2173:
2165:
2151:
2132:
2131:
2126:
1982:
1980:
1979:
1974:
1967:
1966:
1962:
1957:
1940:
1931:
1930:
1921:
1916:
1912:
1907:
1887:
1878:
1877:
1868:
1867:
1855:
1854:
1821:
1819:
1818:
1813:
1806:
1796:
1747:
1745:
1744:
1739:
1732:
1700:
1630:
1582:Louis de Broglie
1578:or wavelengths.
1572:Fourier analysis
1561:local wavelength
1509:
1380:The analysis of
1341:Nonuniform media
1313:
1311:
1310:
1305:
1300:
1298:
1294:
1293:
1277:
1276:
1267:
1223:
1221:
1220:
1215:
1210:
1208:
1204:
1203:
1184:
1164:refractive index
1102:
1100:
1099:
1094:
1089:
1088:
1087:
1083:
994:
978:
963:
961:
960:
955:
950:
942:
937:
932:
921:
916:
911:
903:
882:
880:
879:
874:
869:
861:
856:
851:
840:
835:
830:
822:
801:
799:
798:
793:
791:
787:
748:
744:
703:
643:
641:
640:
635:
633:
629:
610:
605:
597:
578:
574:
573:
569:
559:
551:
513:
376:
372:
370:
364:
362:
329:
327:
326:
321:
305:
303:
302:
297:
282:
280:
279:
274:
267:
259:
240:
238:
237:
232:
208:Sinusoidal waves
196:as well as to a
4343:
4342:
4338:
4337:
4336:
4334:
4333:
4332:
4313:
4312:
4311:
4306:
4272:
4199:
4174:
4160:
4128:
4056:
4008:
3950:
3927:
3904:
3897:
3890:
3883:
3881:
3833:
3828:
3790:Wayback Machine
3774:
3769:
3762:
3746:
3742:
3735:
3719:
3715:
3708:
3692:
3688:
3681:
3663:
3659:
3652:
3636:
3632:
3625:
3609:
3605:
3598:
3576:
3572:
3565:
3547:
3543:
3536:
3514:
3510:
3503:
3485:
3481:
3474:
3455:
3448:
3438:
3416:
3412:
3398:are related by
3386:
3370:
3366:
3359:
3343:
3339:
3332:
3316:
3312:
3305:
3287:
3283:
3276:
3254:
3250:
3243:
3221:
3217:
3210:
3192:
3188:
3169:
3165:
3158:
3138:
3134:
3127:
3107:
3088:
3084:
3077:
3061:
3057:
3050:
3034:
3030:
3023:
3007:
3000:
2993:
2976:
2972:
2965:
2949:
2945:
2938:
2918:
2914:
2904:
2882:
2878:
2871:
2855:
2851:
2844:
2830:
2826:
2819:
2803:
2799:
2792:
2774:
2770:
2763:
2749:
2745:
2741:
2693:
2678:
2656:
2639:
2626:(also known as
2612:
2557:
2548:is in radians.
2503:
2492:
2489:
2488:
2435:
2433:
2430:
2429:
2398:
2393:
2388:
2367:
2363:
2361:
2358:
2357:
2341:The resolvable
2336:Bessel function
2333:
2326:
2312:; the distance
2302:radiotelescopes
2294:resolving power
2290:
2282:Main articles:
2280:
2225:
2217:
2215:
2207:
2204:
2203:
2180:
2166:
2152:
2150:
2146:
2145:
2127:
2113:
2112:
2095:
2092:
2091:
2031:
2023:Main articles:
2021:
2007:
2000:
1941:
1939:
1935:
1926:
1922:
1917:
1888:
1886:
1882:
1873:
1869:
1863:
1859:
1850:
1846:
1844:
1841:
1840:
1792:
1763:
1760:
1759:
1709:
1706:
1705:
1692:
1660:
1654:
1649:
1628:
1609:Planck constant
1549:
1535:
1496:
1452:
1408:
1343:
1289:
1285:
1278:
1272:
1268:
1266:
1258:
1255:
1254:
1249:
1245:
1199:
1195:
1188:
1183:
1175:
1172:
1171:
1109:
1067:
1063:
1059:
1055:
1050:
1047:
1046:
980:
968:
941:
922:
920:
904:
902:
894:
891:
890:
860:
841:
839:
823:
821:
813:
810:
809:
765:
761:
728:
724:
689:
686:
685:
598:
596:
595:
591:
550:
549:
545:
538:
534:
499:
496:
495:
487:and wavelength
477:
446:
374:
368:
366:
360:
358:
345:In the case of
315:
312:
311:
291:
288:
287:
258:
250:
247:
246:
226:
223:
222:
210:
126:sinusoidal wave
24:
17:
12:
11:
5:
4341:
4331:
4330:
4325:
4308:
4307:
4305:
4304:
4299:
4294:
4289:
4283:
4281:
4274:
4273:
4271:
4270:
4265:
4260:
4255:
4250:
4245:
4240:
4235:
4230:
4225:
4220:
4215:
4209:
4207:
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4200:
4198:
4197:
4192:
4187:
4182:
4177:
4172:
4168:
4163:
4158:
4154:
4149:
4144:
4138:
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4130:
4129:
4127:
4126:
4121:
4116:
4094:
4089:
4066:
4064:
4058:
4057:
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4007:
4006:
4001:
3996:
3991:
3986:
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3849:
3844:
3838:
3835:
3834:
3827:
3826:
3819:
3812:
3804:
3798:
3797:
3792:
3780:
3773:
3772:External links
3770:
3768:
3767:
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3733:
3713:
3706:
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3384:
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3310:
3303:
3281:
3274:
3248:
3242:978-3642228940
3241:
3215:
3209:978-3642127984
3208:
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2138:
2135:
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2111:
2108:
2105:
2102:
2099:
2083:via a squared
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2017:
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1673:interferometer
1656:Main article:
1653:
1650:
1648:
1645:
1545:Main article:
1534:
1531:
1473:traveling wave
1451:
1448:
1425:Brillouin zone
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1235:in vacuum and
1233:speed of light
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1079:
1076:
1073:
1070:
1066:
1062:
1058:
1054:
965:
964:
953:
948:
945:
940:
935:
931:
928:
925:
919:
914:
910:
907:
901:
898:
884:
883:
872:
867:
864:
859:
854:
850:
847:
844:
838:
833:
829:
826:
820:
817:
803:
802:
790:
786:
783:
780:
777:
774:
771:
768:
764:
760:
757:
754:
751:
747:
743:
740:
737:
734:
731:
727:
723:
720:
717:
714:
711:
708:
702:
699:
696:
693:
645:
644:
632:
628:
625:
622:
619:
616:
613:
608:
604:
601:
594:
590:
587:
584:
581:
577:
572:
568:
565:
562:
557:
554:
548:
544:
541:
537:
533:
530:
527:
524:
521:
518:
512:
509:
506:
503:
476:
473:
469:speed of light
445:
444:Standing waves
442:
402:speed of sound
355:speed of light
330:is the wave's
319:
308:phase velocity
295:
284:
283:
272:
265:
262:
257:
254:
230:
209:
206:
198:sound spectrum
173:magnetic field
86:standing waves
82:zero crossings
64:spatial period
44:zero crossings
15:
9:
6:
4:
3:
2:
4340:
4329:
4326:
4324:
4321:
4320:
4318:
4303:
4300:
4298:
4295:
4293:
4290:
4288:
4285:
4284:
4282:
4279:
4275:
4269:
4266:
4264:
4261:
4259:
4256:
4254:
4251:
4249:
4246:
4244:
4241:
4239:
4236:
4234:
4231:
4229:
4226:
4224:
4221:
4219:
4216:
4214:
4211:
4210:
4208:
4206:
4202:
4196:
4193:
4191:
4188:
4186:
4183:
4181:
4178:
4176:
4169:
4167:
4164:
4162:
4155:
4153:
4150:
4148:
4145:
4143:
4140:
4139:
4137:
4135:
4131:
4125:
4122:
4120:
4117:
4114:
4110:
4106:
4102:
4098:
4095:
4093:
4090:
4087:
4083:
4079:
4075:
4071:
4068:
4067:
4065:
4063:
4059:
4053:
4050:
4048:
4045:
4043:
4040:
4038:
4035:
4033:
4030:
4028:
4025:
4023:
4020:
4019:
4017:
4015:
4011:
4005:
4002:
4000:
3997:
3995:
3992:
3990:
3987:
3985:
3982:
3980:
3977:
3975:
3972:
3970:
3967:
3965:
3962:
3961:
3959:
3957:
3953:
3947:
3944:
3942:
3939:
3938:
3936:
3934:
3930:
3924:
3921:
3919:
3916:
3915:
3913:
3911:
3907:
3900:
3896:
3889:
3877:
3873:
3870:
3868:
3865:
3863:
3860:
3858:
3855:
3853:
3850:
3848:
3845:
3843:
3840:
3839:
3836:
3832:
3825:
3820:
3818:
3813:
3811:
3806:
3805:
3802:
3796:
3793:
3791:
3787:
3784:
3781:
3779:
3776:
3775:
3763:
3757:
3753:
3752:
3744:
3736:
3730:
3726:
3725:
3717:
3709:
3703:
3699:
3698:
3690:
3682:
3680:0-7503-0734-X
3676:
3672:
3668:
3661:
3653:
3647:
3643:
3642:
3634:
3626:
3624:0-471-23429-X
3620:
3616:
3615:
3607:
3599:
3597:0-387-95269-1
3593:
3589:
3585:
3581:
3574:
3566:
3560:
3556:
3552:
3545:
3537:
3531:
3527:
3522:
3521:
3512:
3504:
3502:0-12-014653-3
3498:
3494:
3490:
3483:
3475:
3469:
3465:
3461:
3453:
3451:
3443:
3439:
3437:0-486-66741-3
3433:
3429:
3425:
3421:
3414:
3407:
3405:
3401:
3397:
3393:
3387:
3385:0-8176-3967-5
3381:
3377:
3376:
3368:
3360:
3354:
3350:
3349:
3341:
3333:
3327:
3323:
3322:
3314:
3306:
3300:
3296:
3292:
3285:
3277:
3271:
3267:
3263:
3259:
3252:
3244:
3238:
3234:
3230:
3226:
3219:
3211:
3205:
3201:
3197:
3190:
3182:
3178:
3177:"Wave theory"
3174:
3173:Lord Rayleigh
3167:
3159:
3157:0-387-28152-5
3153:
3149:
3145:
3144:
3136:
3128:
3126:0-521-39293-4
3122:
3118:
3117:
3108:
3106:0-521-42947-1
3102:
3097:
3096:
3086:
3078:
3072:
3068:
3067:
3059:
3051:
3045:
3041:
3040:
3032:
3024:
3018:
3014:
3013:
3005:
3003:
2994:
2988:
2984:
2983:
2974:
2966:
2960:
2956:
2955:
2947:
2939:
2937:0-387-98756-8
2933:
2929:
2925:
2924:
2916:
2909:
2905:
2899:
2895:
2890:
2889:
2880:
2872:
2866:
2862:
2861:
2853:
2845:
2843:2-88124-995-7
2839:
2835:
2828:
2820:
2818:0-534-49143-X
2814:
2810:
2809:
2801:
2793:
2791:0-19-850693-7
2787:
2783:
2779:
2772:
2764:
2762:0-201-11609-X
2758:
2754:
2747:
2743:
2734:
2731:
2729:
2726:
2724:
2723:Spectral line
2721:
2719:
2716:
2714:
2711:
2708:
2705:
2703:
2700:
2698:
2695:
2694:
2688:
2685:
2681:
2677:(symbol
2676:
2672:
2669:(symbol
2668:
2663:
2659:
2654:
2653:
2646:
2642:
2637:
2636:barred lambda
2633:
2629:
2625:
2616:
2607:
2605:
2601:
2600:Subwavelength
2597:
2595:
2591:
2587:
2583:
2579:
2574:
2572:
2571:optical fibre
2568:
2567:
2562:
2561:subwavelength
2555:Subwavelength
2552:
2549:
2547:
2543:
2539:
2535:
2516:
2508:
2505:
2500:
2497:
2494:
2487:
2486:
2485:
2483:
2479:
2474:
2472:
2455:
2452:
2449:
2446:
2443:
2412:
2394:
2390:
2385:
2382:
2377:
2374:
2371:
2368:
2364:
2356:
2355:
2354:
2352:
2348:
2344:
2339:
2337:
2330:
2323:
2319:
2315:
2311:
2307:
2303:
2299:
2295:
2289:
2285:
2275:
2273:
2269:
2265:
2261:
2257:
2238:
2229:
2226:
2221:
2218:
2212:
2209:
2189:
2181:
2176:
2170:
2167:
2162:
2159:
2156:
2153:
2147:
2142:
2136:
2128:
2109:
2103:
2097:
2090:
2089:
2088:
2086:
2085:sinc function
2082:
2078:
2074:
2069:
2067:
2062:
2060:
2056:
2051:
2049:
2040:
2035:
2030:
2026:
2016:
2014:
2009:
2004:
1997:
1993:
1989:
1970:
1963:
1958:
1954:
1951:
1948:
1945:
1942:
1936:
1932:
1927:
1923:
1918:
1913:
1908:
1904:
1901:
1898:
1895:
1892:
1889:
1883:
1879:
1874:
1870:
1864:
1860:
1856:
1851:
1847:
1839:
1838:
1837:
1834:
1832:
1828:
1809:
1803:
1797:
1793:
1789:
1786:
1783:
1777:
1774:
1771:
1768:
1765:
1758:
1757:
1756:
1754:
1735:
1729:
1726:
1723:
1720:
1717:
1714:
1711:
1704:
1703:
1702:
1699:
1695:
1690:
1686:
1682:
1678:
1674:
1664:
1659:
1644:
1642:
1638:
1634:
1633:wave function
1626:
1622:
1618:
1614:
1610:
1606:
1602:
1598:
1594:
1590:
1587:
1583:
1579:
1577:
1573:
1568:
1566:
1562:
1558:
1554:
1548:
1539:
1530:
1528:
1527:periodic wave
1519:
1515:
1513:
1507:
1503:
1499:
1494:
1490:
1486:
1480:
1478:
1477:superposition
1474:
1470:
1466:
1456:
1447:
1445:
1441:
1437:
1433:
1428:
1426:
1421:
1412:
1403:
1402:in the wave.
1401:
1397:
1393:
1389:
1388:
1383:
1378:
1375:
1374:inhomogeneous
1366:
1362:
1360:
1356:
1347:
1338:
1336:
1332:
1328:
1324:
1319:
1301:
1290:
1286:
1279:
1273:
1269:
1263:
1260:
1253:
1252:
1251:
1242:
1238:
1234:
1230:
1211:
1200:
1196:
1189:
1185:
1180:
1177:
1170:
1169:
1168:
1167:according to
1166:
1165:
1159:
1155:
1153:
1149:
1145:
1140:
1138:
1129:
1121:
1113:
1107:General media
1090:
1084:
1080:
1077:
1074:
1071:
1068:
1064:
1060:
1056:
1052:
1045:
1044:
1043:
1041:
1037:
1033:
1028:
1026:
1022:
1018:
1014:
1010:
1006:
1002:
998:
992:
988:
984:
976:
972:
951:
946:
943:
938:
933:
929:
926:
923:
917:
912:
908:
905:
899:
896:
889:
888:
887:
870:
865:
862:
857:
852:
848:
845:
842:
836:
831:
827:
824:
818:
815:
808:
807:
806:
788:
781:
778:
775:
772:
766:
762:
758:
755:
752:
749:
745:
741:
738:
735:
732:
729:
725:
721:
718:
715:
712:
706:
700:
697:
691:
684:
683:
682:
680:
677:
673:
670:
666:
662:
658:
654:
650:
630:
623:
620:
617:
614:
606:
602:
599:
592:
588:
585:
582:
579:
575:
570:
566:
563:
560:
555:
552:
546:
542:
539:
535:
531:
528:
525:
522:
516:
510:
507:
501:
494:
493:
492:
490:
486:
482:
472:
470:
464:
462:
457:
455:
451:
450:standing wave
437:
429:
425:
423:
419:
415:
411:
407:
403:
399:
394:
392:
388:
384:
380:
356:
352:
348:
343:
341:
337:
333:
317:
309:
293:
270:
263:
260:
255:
252:
245:
244:
243:
241:
228:
219:
215:
205:
203:
199:
195:
191:
187:
186:
180:
178:
174:
170:
166:
162:
158:
154:
150:
146:
142:
138:
133:
131:
127:
122:
120:
116:
112:
108:
104:
101:
97:
96:
91:
87:
83:
79:
78:
73:
69:
65:
61:
57:
53:
45:
41:
37:
33:
28:
22:
4277:
3894:
3882:
3750:
3743:
3723:
3716:
3696:
3689:
3670:
3660:
3640:
3633:
3613:
3606:
3587:
3583:
3573:
3554:
3544:
3519:
3511:
3492:
3482:
3463:
3441:
3427:
3423:
3413:
3403:
3399:
3395:
3391:
3389:
3374:
3367:
3347:
3340:
3320:
3313:
3294:
3284:
3265:
3261:
3251:
3232:
3228:
3218:
3199:
3189:
3180:
3166:
3147:
3142:
3135:
3115:
3094:
3085:
3065:
3058:
3038:
3031:
3011:
2981:
2973:
2953:
2946:
2927:
2922:
2915:
2907:
2887:
2879:
2859:
2852:
2833:
2827:
2807:
2800:
2781:
2771:
2752:
2746:
2728:Spectroscopy
2683:
2679:
2670:
2661:
2657:
2650:
2644:
2640:
2631:
2627:
2623:
2621:
2599:
2598:
2575:
2564:
2560:
2558:
2550:
2545:
2537:
2533:
2531:
2477:
2475:
2427:
2342:
2340:
2328:
2321:
2317:
2313:
2291:
2271:
2267:
2263:
2259:
2255:
2253:
2080:
2076:
2070:
2065:
2063:
2052:
2044:
2013:redistribute
2012:
2010:
2002:
1995:
1991:
1987:
1985:
1835:
1830:
1826:
1824:
1752:
1750:
1697:
1693:
1688:
1684:
1669:
1604:
1600:
1596:
1592:
1588:
1580:
1569:
1564:
1560:
1556:
1553:wave packets
1550:
1533:Wave packets
1526:
1524:
1505:
1501:
1497:
1492:
1485:cnoidal wave
1481:
1461:
1432:energy bands
1429:
1417:
1391:
1385:
1379:
1373:
1371:
1358:
1352:
1320:
1316:
1240:
1236:
1228:
1226:
1162:
1160:
1156:
1141:
1134:
1029:
1024:
1020:
1016:
1012:
996:
990:
986:
982:
974:
970:
966:
885:
804:
678:
671:
660:
656:
652:
648:
646:
488:
484:
480:
478:
465:
458:
447:
416:and 17
400:in air, the
395:
344:
285:
242:is given by
221:
217:
211:
183:
181:
134:
123:
119:interference
106:
100:Greek letter
93:
75:
63:
59:
49:
35:
4297:Medium wave
3974:Lyman-alpha
3956:Ultraviolet
3895:wavelengths
3888:frequencies
3852:Ultraviolet
2306:microscopes
2300:(including
2048:diffraction
2025:Diffraction
1637:wavenumbers
1576:wavenumbers
1547:Wave packet
1512:ocean waves
1469:wave vector
1152:Snell's law
1001:wave vector
398:sound waves
373:divided by
371:10 m/s
363:10 m/s
342:nonlinear.
159:, while in
145:water waves
137:sound waves
124:Assuming a
56:mathematics
4317:Categories
4278:Wavelength
4134:Microwaves
3946:Hard X-ray
3941:Soft X-ray
3910:Gamma rays
3842:Gamma rays
2739:References
2298:telescopes
1831:vice versa
1551:Localized
1465:wavenumber
1396:wavenumber
1387:WKB method
1323:dispersion
1144:refraction
1032:plane wave
1005:plane wave
669:wavenumber
375:10 Hz
351:free space
163:and other
60:wavelength
4292:Shortwave
4287:Microwave
3867:Microwave
2594:photonics
2582:apertures
2569:means an
2559:The term
2506:λ
2495:δ
2482:Airy disk
2456:θ
2453:
2391:λ
2310:Airy disk
2227:λ
2168:π
2160:π
2157:
1959:λ
1955:α
1952:
1943:π
1933:
1909:λ
1905:α
1902:
1893:π
1880:
1804:λ
1775:θ
1772:
1730:λ
1721:θ
1718:
1681:two slits
1629:10 m
1621:electrons
1287:λ
1270:λ
1261:λ
1197:λ
1078:ω
1075:−
934:ω
927:π
909:π
897:λ
863:ω
846:π
832:λ
828:π
776:−
759:
739:ω
736:−
722:
665:amplitude
655:and time
618:−
607:λ
603:π
589:
561:−
556:λ
543:π
532:
332:frequency
253:λ
149:conductor
130:frequency
115:envelopes
111:modulated
46:as shown.
32:sine wave
4302:Longwave
4062:Infrared
3862:Infrared
3786:Archived
3175:(1890).
2733:Spectrum
2691:See also
2327:, where
2039:envelope
1603:, where
1586:momentum
1565:envelope
1557:envelope
1440:aliasing
1420:aliasing
1406:Crystals
1355:periodic
357:, about
185:spectrum
171:and the
169:electric
157:pressure
3893:longer
3886:higher
3857:Visible
3402:= 2π /
3012:op. cit
2478:angular
2343:spatial
1827:fringes
1607:is the
1444:sampled
1331:refract
1231:is the
1009:3-space
999:with a
663:is the
334:. In a
90:inverse
52:physics
4328:Length
4195:L band
4190:S band
4185:C band
4180:X band
4166:K band
4152:Q band
4147:V band
4142:W band
4047:Orange
4042:Yellow
4022:Violet
3933:X-rays
3847:X-rays
3758:
3731:
3704:
3677:
3648:
3621:
3594:
3561:
3532:
3499:
3470:
3434:
3382:
3355:
3328:
3301:
3272:
3239:
3206:
3154:
3123:
3103:
3073:
3046:
3019:
2989:
2961:
2934:
2900:
2867:
2840:
2815:
2788:
2759:
2753:Optics
2588:, and
2532:where
2514:
2410:
2304:) and
2254:where
2236:
2187:
1986:where
1968:
1829:, and
1807:
1751:where
1733:
1570:Using
1227:where
1137:vacuum
1036:cosine
704:
659:, and
647:where
514:
387:violet
286:where
214:linear
103:lambda
4323:Waves
4280:types
4205:Radio
4101:Bands
4074:Bands
4037:Green
3872:Radio
2660:= 2π/
1677:Young
1623:in a
1359:local
1327:prism
454:nodes
385:, to
161:light
153:sound
141:light
77:phase
66:of a
40:phase
4175:band
4161:band
4119:LWIR
4097:MWIR
4092:SWIR
4032:Cyan
4027:Blue
3756:ISBN
3729:ISBN
3702:ISBN
3675:ISBN
3646:ISBN
3619:ISBN
3592:ISBN
3559:ISBN
3530:ISBN
3497:ISBN
3468:ISBN
3432:ISBN
3380:ISBN
3353:ISBN
3326:ISBN
3299:ISBN
3270:ISBN
3237:ISBN
3204:ISBN
3170:See
3152:ISBN
3121:ISBN
3101:ISBN
3071:ISBN
3044:ISBN
3017:ISBN
2987:ISBN
2959:ISBN
2932:ISBN
2898:ISBN
2865:ISBN
2838:ISBN
2813:ISBN
2786:ISBN
2757:ISBN
2682:= 2π
2540:the
2501:1.22
2476:The
2386:1.22
2286:and
2027:and
1696:sin
1434:and
1040:sine
491:as:
422:bats
396:For
151:. A
68:wave
54:and
4268:ELF
4263:SLF
4258:ULF
4253:VLF
4233:VHF
4228:UHF
4223:SHF
4218:EHF
4213:THF
4124:FIR
4070:NIR
4052:Red
4004:UVA
3999:UVB
3994:UVC
3989:NUV
3984:MUV
3979:FUV
2894:102
2687:).
2647:/2π
2450:sin
2154:sin
1949:sin
1924:sin
1899:sin
1871:sin
1769:sin
1715:sin
1625:CRT
1491:of
1467:or
1007:in
886:or
756:cos
719:cos
586:cos
529:cos
393:).
379:red
212:In
200:or
70:or
62:or
50:In
4319::
4248:LF
4243:MF
4238:HF
4111:,
4107:,
4103::
4084:,
4080:,
4076::
3669:.
3590:.
3588:ff
3553:.
3528:.
3526:15
3462:.
3449:^
3440:.
3430:.
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3422:.
3388:.
3293:.
3268:.
3266:ff
3260:.
3235:.
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3227:.
3198:.
3179:.
3150:.
3148:ff
3001:^
2930:.
2928:ff
2906:.
2896:.
2780:.
2643:=
2606:.
2596:.
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2353::
2338:.
2087::
2050:.
1833:.
1643:.
1595:=
1504:;
1498:cn
1479:.
1427:.
1337:.
1154:.
991:ωt
989:−
985:⋅
975:ωt
973:−
971:kx
448:A
418:mm
410:Hz
383:nm
204:.
143:,
139:,
58:,
34:,
4173:u
4171:K
4159:a
4157:K
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4113:N
4109:M
4105:L
4099:(
4088:)
4086:H
4082:K
4078:J
4072:(
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3884:←
3823:e
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3809:v
3764:.
3737:.
3710:.
3683:.
3654:.
3627:.
3600:.
3567:.
3538:.
3505:.
3476:.
3406:.
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3400:k
3396:k
3392:λ
3361:.
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3245:.
3212:.
3160:.
3129:.
3109:.
3079:.
3052:.
3025:.
2995:.
2967:.
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2873:.
2846:.
2821:.
2794:.
2765:.
2684:f
2680:ω
2671:ħ
2662:λ
2658:k
2655:(
2645:λ
2641:ƛ
2632:ƛ
2546:δ
2538:D
2534:λ
2517:,
2509:D
2498:=
2447:n
2444:=
2440:A
2437:N
2413:,
2403:A
2400:N
2395:2
2383:=
2378:y
2375:r
2372:i
2369:A
2365:r
2332:1
2329:J
2325:1
2322:J
2318:r
2314:x
2272:x
2268:u
2264:S
2260:R
2256:L
2239:,
2230:R
2222:L
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2213:=
2210:u
2190:;
2182:2
2177:)
2171:u
2163:u
2148:(
2143:=
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2137:u
2134:(
2129:2
2124:c
2121:n
2118:i
2115:s
2110:=
2107:)
2104:u
2101:(
2098:S
2081:x
2077:S
2041:.
2006:1
2003:I
1999:1
1996:I
1992:g
1988:q
1971:,
1964:)
1946:g
1937:(
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1914:)
1896:g
1890:q
1884:(
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1810:.
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1798:2
1794:/
1790:1
1787:+
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1781:(
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1766:d
1753:m
1736:,
1727:m
1724:=
1712:d
1698:θ
1694:d
1689:d
1685:s
1605:h
1601:p
1599:/
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1593:λ
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1500:(
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1302:.
1296:)
1291:0
1283:(
1280:n
1274:0
1264:=
1248:0
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1241:λ
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1229:c
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987:r
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866:v
858:=
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763:(
753:A
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726:(
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695:(
692:y
679:ω
672:k
661:A
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485:f
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367:3
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318:f
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