2626:
1131:
1409:, 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
1139:
1674:
1466:
2045:
1123:
3887:
1529:
1422:
1357:
38:
447:
439:
1387:
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
2056:
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
1433:
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
474:), 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.
1493:
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
1328:
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
1992:
1694:. 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,
1473:
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
653:
1168:
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.
2211:
973:
892:
811:
477:
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
1853:
2434:
1323:
508:
1677:
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.
1681:
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
1646:
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
1540:. 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.
1372:
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.
1038:
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.
2538:
1831:
1233:
1112:
2478:
2260:
292:
2075:
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 (
1757:
2079:). 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.
1340:. 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
3187:
3832:
416:
2988:
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,
1482:) 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
470:
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
2104:
339:
315:
250:
2012:, 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
1429:
Waves in crystalline solids are not continuous, because they are composed of vibrations of discrete particles arranged in a regular lattice. This produces
903:
822:
698:
2574:
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
1987:{\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)\ ,}
1549:
3760:
1650:
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
442:
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.
3825:
1376:
1344:
at different angles. The mathematical relationship that describes how the speed of light within a medium varies with wavelength is known as a
3770:
3743:
3716:
3660:
3573:
3544:
3482:
3367:
3340:
3313:
3284:
3085:
3058:
3031:
3001:
2973:
2912:
2879:
1150:, 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.
85:
is the distance over which the wave's shape repeats. In other words, it is the distance between consecutive corresponding points of the same
1570:
that describes the overall amplitude of the wave; within the envelope, the distance between adjacent peaks or troughs is sometimes called a
2576:
3706:
2562:
As with other diffraction patterns, the pattern scales in proportion to wavelength, so shorter wavelengths can lead to higher resolution.
2370:
3650:
648:{\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)}
2869:
1267:
3251:
3218:
2596:
1566:, "bursts" of wave action where each wave packet travels as a unit, find application in many fields of physics. A wave packet has an
3818:
3689:
3633:
3606:
3511:
3446:
3394:
3166:
3135:
3115:
2946:
2852:
2827:
2800:
2771:
3796:
2963:
2360:, 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
1146:
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
3357:
1651:
1668:
435:
so they can resolve targets smaller than 17 mm. Wavelengths in audible sound are much longer than those in visible light.
129:
3405:
of a dispersing wave is twice the distance between two successive zeros. ... the local wavelength and the local wave number
463:
is an undulatory motion that stays in one place. A sinusoidal standing wave includes stationary points of no motion, called
3470:
3453:(p. 61) ... the individual waves move more slowly than the packet and therefore pass back through the packet as it advances
1368:
in space. For example, in an ocean wave approaching shore, shown in the figure, the incoming wave undulates with a varying
2501:
1053:
phase when describing a wave is based on the fact that the cosine is the real part of the complex exponential in the wave
227:
media, any wave pattern can be described in terms of the independent propagation of sinusoidal components. The wavelength
3075:
3048:
146:
Wavelength depends on the medium (for example, vacuum, air, or water) that a wave travels through. Examples of waves are
2625:
1454:
3933:
1836:
Thus, if the wavelength of the light is known, the slit separation can be determined from the interference pattern or
1772:
31:
1184:
1059:
1434:
provide a description of all possible waves in a crystalline medium corresponds to the wave vectors confined to the
450:
A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue)
143:
of the wave: waves with higher frequencies have shorter wavelengths, and lower frequencies have longer wavelengths.
3928:
2442:
2298:
2273:
is the distance of the pattern (on the screen) from the slit, and λ is the wavelength of light used. The function
2216:
1261:, where the latter is measured in vacuum rather than in the medium. The corresponding wavelength in the medium is
259:
2083:
357:
175:
1718:
4228:
3841:
2495:) of the image diffracted by a circular aperture, a measure most commonly used for telescopes and cameras, is:
1687:
1499:
401:
204:
2595:
are holes smaller than the wavelength of light propagating through them. Such structures have applications in
1336:, and is also responsible for the familiar phenomenon in which light is separated into component colours by a
2064:
Two types of diffraction are distinguished, depending upon the separation between the source and the screen:
4278:
2677:
1487:
1130:
3359:
The
Quantum Theory of Motion: An Account of the de Broglie–Bohm Causal Interpretation of Quantum Mechanics
2723:
2065:
1691:
1410:
125:
100:
2587:
A subwavelength particle is a particle smaller than the wavelength of light with which it interacts (see
1525:
with certain shapes can propagate unchanged, because of properties of the nonlinear surface-wave medium.
1383:
Waves that are sinusoidal in time but propagate through a medium whose properties vary with position (an
2614:
2039:
1627:
1392:
1158:
1138:
4238:
4233:
2481:
159:
3077:
Time-frequency and time-scale methods: adaptive decompositions, uncertainty principles, and sampling
4273:
4268:
4243:
4223:
3974:
2600:
2588:
2069:
2044:
1673:
1465:
1345:
1333:
1122:
471:
350:
2788:
1585:, wave packets can be analyzed into infinite sums (or integrals) of sinusoidal waves of different
1441:
This indeterminacy in wavelength in solids is important in the analysis of wave phenomena such as
4263:
4111:
4084:
3994:
3979:
2728:
2662:
2361:
2357:
2304:
2294:
212:
2720:– dark lines in the solar spectrum, traditionally used as standard optical wavelength references
1157:, or a change in direction of waves that encounter the interface between media at an angle. For
1041:
Generalizations to sinusoids of other phases, and to complex exponentials, are also common; see
482:
can be determined from observation of standing waves in a metal box containing an ideal vacuum.
3527:
Greenfield Sluder & David E. Wolf (2007). "IV. Young's
Experiment: Two-Slit Interference".
4119:
4115:
4096:
4092:
4088:
3766:
3739:
3733:
3712:
3685:
3656:
3629:
3602:
3590:
3569:
3561:
3540:
3507:
3499:
3478:
3442:
3390:
3384:
3363:
3336:
3309:
3280:
3247:
3235:
3214:
3206:
3162:
3152:
3131:
3111:
3081:
3054:
3027:
3021:
2997:
2991:
2969:
2942:
2932:
2908:
2904:
2875:
2848:
2823:
2817:
2796:
2767:
2707:
2685:
1623:
1397:
1365:
686:
346:
187:
105:
82:
3677:
3623:
3536:
3430:
3301:
3125:
4253:
4024:
3999:
3867:
3330:
3268:
2717:
2712:
2676:). It is usually encountered in quantum mechanics, where it is used in combination with the
2049:
1635:
1592:
1582:
1337:
1174:
200:
3886:
2206:{\displaystyle S(u)=\mathrm {sinc} ^{2}(u)=\left({\frac {\sin \pi u}{\pi u}}\right)^{2}\ ;}
490:
Traveling sinusoidal waves are often represented mathematically in terms of their velocity
4338:
4302:
4215:
4057:
3989:
3800:
2346:
1619:
1548:
1528:
1421:
136:
3788:
3793:
2026:
the light, so the energy contained in the light is not altered, just where it shows up.
1134:
Refraction: upon entering a medium where its speed is lower, the wave changes direction.
4333:
4248:
4032:
3806:
The visible electromagnetic spectrum displayed in web colors with according wavelengths
2552:
2312:
1683:
1435:
1243:
1162:
479:
464:
412:
397:
365:
324:
318:
300:
235:
208:
183:
179:
167:
1375:
1360:
Various local wavelengths on a crest-to-crest basis in an ocean wave approaching shore
4327:
4258:
4195:
4176:
4080:
4014:
4009:
4004:
3529:
3183:
3104:
2897:
2733:
2646:
2581:
2095:
1643:
968:{\displaystyle \lambda ={\frac {2\pi }{k}}={\frac {2\pi v}{\omega }}={\frac {v}{f}}.}
887:{\displaystyle k={\frac {2\pi }{\lambda }}={\frac {2\pi f}{v}}={\frac {\omega }{v}},}
460:
96:
92:
87:
54:
50:
2082:
In the
Fraunhofer diffraction pattern sufficiently far from a single slit, within a
4134:
3437:(Reprint of Academic Press 1981 ed.). Courier Dover Publications. pp. 59
2738:
1495:
1442:
1425:
A wave on a line of atoms can be interpreted according to a variety of wavelengths.
110:
1142:
Separation of colors by a prism (click for animation if it is not already playing)
806:{\displaystyle y(x,\ t)=A\cos \left(kx-\omega t\right)=A\cos \left(k(x-vt)\right)}
3762:
Reflecting
Telescope Optics I: Basic Design Theory and Its Historical Development
4307:
3984:
3966:
3862:
3789:
Conversion: Wavelength to
Frequency and vice versa – Sound waves and radio waves
2316:
2058:
2035:
1563:
1557:
1536:
If a traveling wave has a fixed shape that repeats in space or in time, it is a
1479:
1011:
186:
vary. Water waves are variations in the height of a body of water. In a crystal
66:
2629:
Relationship between wavelength, angular wavelength, and other wave properties.
1356:
376:. Thus the wavelength of a 100 MHz electromagnetic (radio) wave is about:
3956:
3951:
3882:
2555:
diameter of the imaging system, in the same units, and the angular resolution
1647:
1586:
1522:
1475:
1406:
1154:
1042:
1015:
679:
446:
428:
408:
361:
155:
147:
121:
37:
2613:
may also refer to a phenomenon involving subwavelength objects; for example,
2319:. For a circular aperture, the diffraction-limited image spot is known as an
816:
in which wavelength and wavenumber are related to velocity and frequency as:
17:
4297:
4144:
3920:
3898:
3877:
3852:
2604:
2584:
whose diameter is less than the wavelength of light propagating through it.
2492:
2320:
2308:
675:
393:
349:, the phase speed itself depends upon the frequency of the wave, making the
342:
140:
53:, such as between crests (on top), or troughs (on bottom), or corresponding
42:
3810:
1578:
of the wave packet moves at a speed different from the constituent waves.
1494:
sharper crests and flatter troughs than those of a sinusoid, typical of a
438:
419:). The wavelengths of sound frequencies audible to the human ear (20
139:
moving at a fixed wave speed, wavelength is inversely proportional to the
4312:
4129:
4107:
4102:
4072:
3872:
3805:
2743:
2592:
1631:
1596:
1450:
1430:
1034:, is still in the same relationship with wavelength as shown above, with
195:
3593:. In John W Harris; Walter Benenson; Horst Stöcker; Holger Lutz (eds.).
2356:
size of objects viewed through a microscope is limited according to the
1702:) then the paths are nearly parallel, and the path difference is simply
4181:
4167:
1446:
1341:
1019:
193:
The range of wavelengths or frequencies for wave phenomena is called a
62:
2931:
David C. Cassidy; Gerald James Holton; Floyd James
Rutherford (2002).
2327:
in the single-slit diffraction formula is replaced by radial distance
4205:
4200:
4190:
4162:
4157:
4152:
4123:
4052:
2649:). It is equal to the ordinary wavelength reduced by a factor of 2π (
1147:
1046:
224:
113:
2429:{\displaystyle r_{Airy}=1.22{\frac {\lambda }{2\,\mathrm {NA} }}\ ,}
1172:
For electromagnetic waves the speed in a medium is governed by its
4047:
3943:
3857:
1672:
1547:
1527:
1464:
1420:
1374:
1355:
1137:
1129:
1121:
445:
437:
424:
420:
171:
163:
151:
36:
1318:{\displaystyle \lambda ={\frac {\lambda _{0}}{n(\lambda _{0})}}.}
4042:
4037:
3269:"Chapter 1: Brief history and overview of nonlinear water waves"
2491:
size of the central bright portion (radius to first null of the
2057:
image on the screen. This distribution of wave energy is called
1483:
1050:
78:
3814:
3794:
Teaching resource for 14–16 years on sound including wavelength
2480:
for θ being the half-angle of the cone of rays accepted by the
1712:. Accordingly, the condition for constructive interference is:
4062:
1379:
A sinusoidal wave travelling in a nonuniform medium, with loss
432:
389:
2547:
is the wavelength of the waves that are focused for imaging,
388:= 3 m. The wavelength of visible light ranges from deep
95:. Wavelength is a characteristic of both traveling waves and
3885:
3240:
Nonlinear Waves and
Solitons on Contours and Closed Surfaces
3207:"Figure 4.4: Transition from quasi-harmonic to cnoidal wave"
1332:
The variation in speed of light with wavelength is known as
1126:
Wavelength is decreased in a medium with slower propagation.
2660:), with SI units of meter per radian. It is the inverse of
1498:, a traveling wave so named because it is described by the
1405:). The method integrates phase through space using a local
1364:
Wavelength can be a useful concept even if the wave is not
3386:
Introduction to partial differential equations with MATLAB
3273:
Nonlinear Ocean Waves and the
Inverse Scattering Transform
678:
of the wave. They are also commonly expressed in terms of
467:, and the wavelength is twice the distance between nodes.
2871:
Electromagnetic Theory for
Microwaves and Optoelectronics
2795:(2nd ed.). Cambridge University Press. p. 473.
1486:
solutions, and more complex solutions can be built up by
1161:, this change in the angle of propagation is governed by
3130:(4th ed.). Cambridge University Press. p. 22.
431:, respectively. Somewhat higher frequencies are used by
3625:
Fundamentals of light microscopy and electronic imaging
3194:(9th ed.). The Henry G Allen Company. p. 422.
2048:
Diffraction pattern of a double slit has a single-slit
1595:
postulated that all particles with a specific value of
1395:
of such systems is often done approximately, using the
1257:) is the refractive index of the medium at wavelength λ
49:, can be measured between any two points with the same
120:). The term "wavelength" is also sometimes applied to
91:
on the wave, such as two adjacent crests, troughs, or
3435:
Quantum Mechanics for Applied Physics and Engineering
2822:(4th ed.). Cengage Learning. pp. 404, 440.
2504:
2445:
2373:
2219:
2107:
1856:
1775:
1721:
1574:. An example is shown in the figure. In general, the
1532:
Wavelength of a periodic but non-sinusoidal waveform.
1270:
1187:
1062:
906:
825:
701:
511:
423:–20 kHz) are thus between approximately 17
327:
303:
262:
238:
231:
of a sinusoidal waveform traveling at constant speed
3211:
Nonlinear Dynamics: Between Linear and Impact Limits
2533:{\displaystyle \delta =1.22{\frac {\lambda }{D}}\ ,}
1766:
is an integer, and for destructive interference is:
4287:
4214:
4143:
4071:
4023:
3965:
3942:
3919:
3568:(Extended 8th ed.). Wiley-India. p. 965.
2919:
wavelength lambda light sound frequency wave speed.
2019:has been set to unity, a very rough approximation.
1153:This change in speed upon entering a medium causes
3528:
3103:
2896:
2532:
2472:
2428:
2254:
2205:
2068:or far-field diffraction at large separations and
1986:
1825:
1751:
1317:
1227:
1106:
967:
886:
805:
647:
333:
309:
286:
244:
2303:Diffraction is the fundamental limitation on the
2285:values at a separation proportion to wavelength.
1014:that specifies the direction and wavenumber of a
3711:. Courier Dover Publications. pp. 117–120.
2766:(2nd ed.). Addison Wesley. pp. 15–16.
2072:or near-field diffraction at close separations.
1826:{\displaystyle d\sin \theta =(m+1/2)\lambda \ .}
3074:Jeffrey A. Hogan & Joseph D. Lakey (2005).
1228:{\displaystyle v={\frac {c}{n(\lambda _{0})}},}
1107:{\displaystyle Ae^{i\left(kx-\omega t\right)}.}
400:, roughly 400 nm (for other examples, see
3306:Introduction to Macromolecular Crystallography
3026:. Jones & Bartlett Learning. p. 242.
1638:display have a De Broglie wavelength of about
99:, as well as other spatial wave patterns. The
3826:
3506:. Vol. 53. Academic Press. p. 271.
2473:{\displaystyle \mathrm {NA} =n\sin \theta \;}
351:relationship between wavelength and frequency
8:
3652:Optical scattering: measurement and analysis
3504:Advances in Electronics and Electron Physics
2633:A quantity related to the wavelength is the
2255:{\displaystyle u={\frac {xL}{\lambda R}}\ ,}
685:(2π times the reciprocal of wavelength) and
317:is called the phase speed (magnitude of the
287:{\displaystyle \lambda ={\frac {v}{f}}\,\,,}
3362:. Cambridge University Press. p. 160.
2996:. Cambridge University Press. p. 327.
2903:. Jones & Bartlett Publishers. p.
2793:An introduction to numerical methods in C++
2439:where the numerical aperture is defined as
2005:is the grating constant. The first factor,
1686:. A simple example is an experiment due to
109:. Wavelength is commonly designated by the
3902:
3833:
3819:
3811:
3735:Handbook of biological confocal microscopy
3463:
3461:
3110:. Cambridge University Press. p. 97.
2816:Raymond A. Serway; John W. Jewett (2006).
2469:
1752:{\displaystyle d\sin \theta =m\lambda \ ,}
1626:. Nowadays, this wavelength is called the
978:In the second form given above, the phase
27:Distance over which a wave's shape repeats
3157:(2nd ed.). Birkhäuser. pp. 165
3015:
3013:
2895:Theo Koupelis & Karl F. Kuhn (2007).
2514:
2503:
2446:
2444:
2409:
2408:
2399:
2378:
2372:
2226:
2218:
2191:
2161:
2138:
2124:
2106:
1950:
1937:
1928:
1897:
1884:
1874:
1861:
1855:
1803:
1774:
1720:
1698:is large compared to the slit separation
1449:. It is mathematically equivalent to the
1300:
1283:
1277:
1269:
1210:
1194:
1186:
1070:
1061:
952:
931:
913:
905:
871:
850:
832:
824:
700:
662:is the value of the wave at any position
607:
561:
510:
417:room temperature and atmospheric pressure
326:
302:
280:
279:
269:
261:
237:
3655:(2nd ed.). SPIE Press. p. 64.
3535:(3rd ed.). Academic Press. p.
2624:
2043:
1388:with a high loss and the wave dies out.
3738:(2nd ed.). Springer. p. 112.
3562:"§35-4 Young's interference experiment"
3242:(2nd ed.). Springer. pp. 469
3154:Fundamentals of solid state engineering
2754:
3502:. In L. Marton; Claire Marton (eds.).
3053:. New Age International. p. 454.
2845:The surface physics of liquid crystals
1622:. This hypothesis was at the basis of
1469:Near-periodic waves over shallow water
1045:. The typical convention of using the
128:of modulated waves or waves formed by
203:but now can be applied to the entire
158:and periodic electrical signals in a
7:
3471:"Heisenberg's uncertainty principle"
3467:See, for example, Figs. 2.8–2.10 in
3335:. John Wiley & Sons. p. 1.
2868:Keqian Zhang & Dejie Li (2007).
2847:. Taylor & Francis. p. 17.
2577:subwavelength-diameter optical fibre
2281:is a non-zero integer, where are at
1847:For multiple slits, the pattern is
1022:, parameterized by position vector
3560:Halliday; Resnick; Walker (2008).
2597:extraordinary optical transmission
2450:
2447:
2413:
2410:
2134:
2131:
2128:
2125:
25:
3591:"§9.8.2 Diffraction by a grating"
3308:(2 ed.). Wiley. p. 77.
2990:Raymond T. Pierrehumbert (2010).
2022:The effect of interference is to
3475:Quantum Physics: An Introduction
3127:Introduction to lattice dynamics
3106:Introduction to mineral sciences
2307:of optical instruments, such as
1652:Heisenberg uncertainty principle
494:(in the x direction), frequency
103:of the wavelength is called the
3047:Bishwanath Chakraborty (2007).
2993:Principles of Planetary Climate
2968:. Nelson Thornes. p. 460.
1669:Interference (wave propagation)
1026:. In that case, the wavenumber
199:. The name originated with the
3050:Principles of Plasma Mechanics
2289:Diffraction-limited resolution
2150:
2144:
2117:
2111:
1811:
1791:
1690:where light is passed through
1306:
1293:
1216:
1203:
1006:, by replacing the wavenumber
795:
780:
720:
705:
637:
622:
530:
515:
1:
3708:Introduction to Modern Optics
3477:. CRC Press. pp. 53–56.
3275:. Academic Press. pp. 3
2787:Brian Hilton Flowers (2000).
2331:and the sine is replaced by 2
1506:th order, usually denoted as
692:(2π times the frequency) as:
124:waves, and to the sinusoidal
3302:"Waves and their properties"
3300:Alexander McPherson (2009).
3205:Valery N. Pilipchuk (2010).
2001:is the number of slits, and
1658:Interference and diffraction
3934:Ultra-high-energy gamma ray
3080:. Birkhäuser. p. 348.
2937:. Birkhäuser. pp. 339
486:Mathematical representation
166:wave is a variation in air
32:Wavelength (disambiguation)
4355:
3929:Very-high-energy gamma ray
3628:. Wiley/IEEE. p. 64.
3622:Douglas B. Murphy (2002).
3589:Kordt Griepenkerl (2002).
2789:"§21.2 Periodic functions"
2299:Diffraction-limited system
2292:
2033:
1666:
1555:
29:
3848:
3765:. Springer. p. 302.
3684:. CRC Press. p. 57.
3597:. Springer. pp. 307
3389:. Springer. p. 272.
3356:Peter R. Holland (1995).
3236:"§18.3 Special functions"
3213:. Springer. p. 127.
2874:. Springer. p. 533.
2641:), usually symbolized by
2084:small-angle approximation
1552:A propagating wave packet
364:, the phase speed is the
358:electromagnetic radiation
190:, atomic positions vary.
176:electromagnetic radiation
3842:Electromagnetic spectrum
3732:James B. Pawley (1995).
3705:Grant R. Fowles (1989).
3678:"Diffraction limitation"
3151:Manijeh Razeghi (2006).
2899:In Quest of the Universe
1663:Double-slit interference
1500:Jacobi elliptic function
990:is often generalized to
402:electromagnetic spectrum
205:electromagnetic spectrum
3649:John C. Stover (1995).
3566:Fundamentals of Physics
3500:"Electron Interference"
3498:Ming Chiang Li (1980).
3431:"Wave packet solutions"
3429:A. T. Fromhold (1991).
3383:Jeffery Cooper (1998).
3267:Alfred Osborne (2010).
3192:Encyclopædia Britannica
3124:Martin T. Dove (1993).
2678:reduced Planck constant
2603:, among other areas of
2090:is related to position
2086:, the intensity spread
2030:Single-slit diffraction
1457:at discrete intervals.
3890:
3759:Ray N. Wilson (2004).
3682:The science of imaging
2762:Hecht, Eugene (1987).
2724:Index of wave articles
2630:
2534:
2474:
2430:
2256:
2207:
2066:Fraunhofer diffraction
2053:
1988:
1827:
1753:
1678:
1553:
1533:
1470:
1461:More general waveforms
1426:
1411:conservation of energy
1403:Liouville–Green method
1393:differential equations
1380:
1361:
1319:
1229:
1143:
1135:
1127:
1108:
969:
888:
807:
649:
451:
443:
335:
311:
288:
246:
201:visible light spectrum
132:of several sinusoids.
58:
3889:
3676:Graham Saxby (2002).
3401:The local wavelength
3020:Paul R Pinet (2009).
2934:Understanding physics
2819:Principles of physics
2628:
2615:subwavelength imaging
2535:
2475:
2431:
2257:
2208:
2047:
2040:Diffraction formalism
1989:
1828:
1754:
1676:
1628:de Broglie wavelength
1551:
1531:
1468:
1424:
1378:
1359:
1320:
1230:
1159:electromagnetic waves
1141:
1133:
1125:
1109:
1049:phase instead of the
970:
889:
808:
650:
449:
441:
336:
312:
289:
247:
40:
3469:Joy Manners (2000).
3234:Andrei Ludu (2012).
2965:The World of Physics
2962:John Avison (1999).
2843:A. A. Sonin (1995).
2601:zero-mode waveguides
2502:
2482:microscope objective
2443:
2371:
2217:
2105:
1854:
1773:
1719:
1453:of a signal that is
1268:
1185:
1060:
904:
823:
699:
509:
415:is 343 m/s (at
325:
301:
260:
236:
178:the strength of the
41:The wavelength of a
30:For other uses, see
3975:Extreme ultraviolet
3595:Handbook of physics
3329:Eric Stade (2011).
3100:See Figure 4.20 in
2589:Rayleigh scattering
2269:is the slit width,
2213: with
2070:Fresnel diffraction
1630:. For example, the
1401:(also known as the
1346:dispersion relation
1030:, the magnitude of
472:boundary conditions
392:, roughly 700
3980:Vacuum ultraviolet
3891:
3799:2012-03-13 at the
3531:Digital microscopy
3122:and Figure 2.3 in
3102:A. Putnis (1992).
2729:Length measurement
2663:angular wavenumber
2639:reduced wavelength
2635:angular wavelength
2631:
2621:Angular wavelength
2530:
2470:
2426:
2362:numerical aperture
2358:Rayleigh criterion
2295:Angular resolution
2252:
2203:
2054:
1984:
1823:
1749:
1679:
1602:have a wavelength
1554:
1534:
1521:. Large-amplitude
1471:
1447:lattice vibrations
1427:
1381:
1362:
1329:specified medium.
1315:
1225:
1144:
1136:
1128:
1104:
965:
884:
803:
645:
452:
444:
360:—such as light—in
331:
321:) of the wave and
307:
284:
242:
213:vibration spectrum
59:
4321:
4320:
4025:Visible (optical)
3772:978-3-540-40106-3
3745:978-0-306-44826-3
3718:978-0-486-65957-2
3662:978-0-8194-1934-7
3575:978-81-265-1442-7
3546:978-0-12-374025-0
3484:978-0-7503-0720-8
3369:978-0-521-48543-2
3342:978-1-118-16551-5
3315:978-0-470-18590-2
3286:978-0-12-528629-9
3087:978-0-8176-4276-1
3060:978-81-224-1446-2
3033:978-0-7637-5993-3
3003:978-0-521-86556-2
2975:978-0-17-438733-6
2914:978-0-7637-4387-1
2881:978-3-540-74295-1
2708:Emission spectrum
2686:angular frequency
2684:, h-bar) and the
2645:("lambda-bar" or
2591:). Subwavelength
2526:
2522:
2422:
2418:
2345:is a first order
2248:
2244:
2199:
2185:
2077:Huygens' wavelets
1980:
1972:
1922:
1819:
1745:
1642:. To prevent the
1624:quantum mechanics
1310:
1220:
960:
947:
926:
879:
866:
845:
716:
687:angular frequency
620:
569:
526:
347:dispersive medium
334:{\displaystyle f}
310:{\displaystyle v}
277:
245:{\displaystyle v}
188:lattice vibration
106:spatial frequency
83:periodic function
16:(Redirected from
4346:
3912:
3910:
3903:
3896:
3835:
3828:
3821:
3812:
3777:
3776:
3756:
3750:
3749:
3729:
3723:
3722:
3702:
3696:
3695:
3673:
3667:
3666:
3646:
3640:
3639:
3619:
3613:
3612:
3586:
3580:
3579:
3557:
3551:
3550:
3534:
3524:
3518:
3517:
3495:
3489:
3488:
3465:
3456:
3455:
3426:
3420:
3419:
3380:
3374:
3373:
3353:
3347:
3346:
3332:Fourier Analysis
3326:
3320:
3319:
3297:
3291:
3290:
3264:
3258:
3257:
3231:
3225:
3224:
3202:
3196:
3195:
3179:
3173:
3172:
3148:
3142:
3141:
3121:
3109:
3098:
3092:
3091:
3071:
3065:
3064:
3044:
3038:
3037:
3017:
3008:
3007:
2986:
2980:
2979:
2959:
2953:
2952:
2928:
2922:
2921:
2902:
2892:
2886:
2885:
2865:
2859:
2858:
2840:
2834:
2833:
2813:
2807:
2806:
2784:
2778:
2777:
2759:
2718:Fraunhofer lines
2713:Envelope (waves)
2697:
2675:
2659:
2539:
2537:
2536:
2531:
2524:
2523:
2515:
2479:
2477:
2476:
2471:
2453:
2435:
2433:
2432:
2427:
2420:
2419:
2417:
2416:
2400:
2392:
2391:
2277:has zeros where
2261:
2259:
2258:
2253:
2246:
2245:
2243:
2235:
2227:
2212:
2210:
2209:
2204:
2197:
2196:
2195:
2190:
2186:
2184:
2176:
2162:
2143:
2142:
2137:
1993:
1991:
1990:
1985:
1978:
1977:
1973:
1968:
1951:
1942:
1941:
1932:
1927:
1923:
1918:
1898:
1889:
1888:
1879:
1878:
1866:
1865:
1832:
1830:
1829:
1824:
1817:
1807:
1758:
1756:
1755:
1750:
1743:
1711:
1641:
1593:Louis de Broglie
1589:or wavelengths.
1583:Fourier analysis
1572:local wavelength
1520:
1391:The analysis of
1352:Nonuniform media
1324:
1322:
1321:
1316:
1311:
1309:
1305:
1304:
1288:
1287:
1278:
1234:
1232:
1231:
1226:
1221:
1219:
1215:
1214:
1195:
1175:refractive index
1113:
1111:
1110:
1105:
1100:
1099:
1098:
1094:
1005:
989:
974:
972:
971:
966:
961:
953:
948:
943:
932:
927:
922:
914:
893:
891:
890:
885:
880:
872:
867:
862:
851:
846:
841:
833:
812:
810:
809:
804:
802:
798:
759:
755:
714:
654:
652:
651:
646:
644:
640:
621:
616:
608:
589:
585:
584:
580:
570:
562:
524:
387:
383:
381:
375:
373:
340:
338:
337:
332:
316:
314:
313:
308:
293:
291:
290:
285:
278:
270:
251:
249:
248:
243:
219:Sinusoidal waves
207:as well as to a
21:
4354:
4353:
4349:
4348:
4347:
4345:
4344:
4343:
4324:
4323:
4322:
4317:
4283:
4210:
4185:
4171:
4139:
4067:
4019:
3961:
3938:
3915:
3908:
3901:
3894:
3892:
3844:
3839:
3801:Wayback Machine
3785:
3780:
3773:
3758:
3757:
3753:
3746:
3731:
3730:
3726:
3719:
3704:
3703:
3699:
3692:
3675:
3674:
3670:
3663:
3648:
3647:
3643:
3636:
3621:
3620:
3616:
3609:
3588:
3587:
3583:
3576:
3559:
3558:
3554:
3547:
3526:
3525:
3521:
3514:
3497:
3496:
3492:
3485:
3468:
3466:
3459:
3449:
3428:
3427:
3423:
3409:are related by
3397:
3382:
3381:
3377:
3370:
3355:
3354:
3350:
3343:
3328:
3327:
3323:
3316:
3299:
3298:
3294:
3287:
3266:
3265:
3261:
3254:
3233:
3232:
3228:
3221:
3204:
3203:
3199:
3182:
3180:
3176:
3169:
3150:
3149:
3145:
3138:
3123:
3118:
3101:
3099:
3095:
3088:
3073:
3072:
3068:
3061:
3046:
3045:
3041:
3034:
3019:
3018:
3011:
3004:
2989:
2987:
2983:
2976:
2961:
2960:
2956:
2949:
2930:
2929:
2925:
2915:
2894:
2893:
2889:
2882:
2867:
2866:
2862:
2855:
2842:
2841:
2837:
2830:
2815:
2814:
2810:
2803:
2786:
2785:
2781:
2774:
2761:
2760:
2756:
2752:
2704:
2689:
2667:
2650:
2637:(also known as
2623:
2568:
2559:is in radians.
2500:
2499:
2441:
2440:
2404:
2374:
2369:
2368:
2352:The resolvable
2347:Bessel function
2344:
2337:
2323:; the distance
2313:radiotelescopes
2305:resolving power
2301:
2293:Main articles:
2291:
2236:
2228:
2215:
2214:
2177:
2163:
2157:
2156:
2123:
2103:
2102:
2042:
2034:Main articles:
2032:
2018:
2011:
1952:
1946:
1933:
1899:
1893:
1880:
1870:
1857:
1852:
1851:
1771:
1770:
1717:
1716:
1703:
1671:
1665:
1660:
1639:
1620:Planck constant
1560:
1546:
1507:
1463:
1419:
1354:
1296:
1289:
1279:
1266:
1265:
1260:
1256:
1206:
1199:
1183:
1182:
1120:
1078:
1074:
1066:
1058:
1057:
991:
979:
933:
915:
902:
901:
852:
834:
821:
820:
776:
772:
739:
735:
697:
696:
609:
606:
602:
560:
556:
549:
545:
507:
506:
498:and wavelength
488:
457:
385:
379:
377:
371:
369:
356:In the case of
323:
322:
299:
298:
258:
257:
234:
233:
221:
137:sinusoidal wave
35:
28:
23:
22:
15:
12:
11:
5:
4352:
4350:
4342:
4341:
4336:
4326:
4325:
4319:
4318:
4316:
4315:
4310:
4305:
4300:
4294:
4292:
4285:
4284:
4282:
4281:
4276:
4271:
4266:
4261:
4256:
4251:
4246:
4241:
4236:
4231:
4226:
4220:
4218:
4212:
4211:
4209:
4208:
4203:
4198:
4193:
4188:
4183:
4179:
4174:
4169:
4165:
4160:
4155:
4149:
4147:
4141:
4140:
4138:
4137:
4132:
4127:
4105:
4100:
4077:
4075:
4069:
4068:
4066:
4065:
4060:
4055:
4050:
4045:
4040:
4035:
4029:
4027:
4021:
4020:
4018:
4017:
4012:
4007:
4002:
3997:
3992:
3987:
3982:
3977:
3971:
3969:
3963:
3962:
3960:
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3954:
3948:
3946:
3940:
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3925:
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3845:
3840:
3838:
3837:
3830:
3823:
3815:
3809:
3808:
3803:
3791:
3784:
3783:External links
3781:
3779:
3778:
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3724:
3717:
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3690:
3668:
3661:
3641:
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3457:
3447:
3421:
3395:
3375:
3368:
3348:
3341:
3321:
3314:
3292:
3285:
3259:
3253:978-3642228940
3252:
3226:
3220:978-3642127984
3219:
3197:
3174:
3167:
3143:
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3116:
3093:
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2553:entrance pupil
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2149:
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2141:
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2133:
2130:
2127:
2122:
2119:
2116:
2113:
2110:
2094:via a squared
2031:
2028:
2016:
2009:
1995:
1994:
1983:
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1971:
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1684:interferometer
1667:Main article:
1664:
1661:
1659:
1656:
1556:Main article:
1545:
1542:
1484:traveling wave
1462:
1459:
1436:Brillouin zone
1418:
1415:
1353:
1350:
1326:
1325:
1314:
1308:
1303:
1299:
1295:
1292:
1286:
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1258:
1254:
1246:in vacuum and
1244:speed of light
1236:
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768:
765:
762:
758:
754:
751:
748:
745:
742:
738:
734:
731:
728:
725:
722:
719:
713:
710:
707:
704:
656:
655:
643:
639:
636:
633:
630:
627:
624:
619:
615:
612:
605:
601:
598:
595:
592:
588:
583:
579:
576:
573:
568:
565:
559:
555:
552:
548:
544:
541:
538:
535:
532:
529:
523:
520:
517:
514:
487:
484:
480:speed of light
456:
455:Standing waves
453:
413:speed of sound
366:speed of light
341:is the wave's
330:
319:phase velocity
306:
295:
294:
283:
276:
273:
268:
265:
241:
220:
217:
209:sound spectrum
184:magnetic field
97:standing waves
93:zero crossings
75:spatial period
55:zero crossings
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4351:
4340:
4337:
4335:
4332:
4331:
4329:
4314:
4311:
4309:
4306:
4304:
4301:
4299:
4296:
4295:
4293:
4290:
4286:
4280:
4277:
4275:
4272:
4270:
4267:
4265:
4262:
4260:
4257:
4255:
4252:
4250:
4247:
4245:
4242:
4240:
4237:
4235:
4232:
4230:
4227:
4225:
4222:
4221:
4219:
4217:
4213:
4207:
4204:
4202:
4199:
4197:
4194:
4192:
4189:
4187:
4180:
4178:
4175:
4173:
4166:
4164:
4161:
4159:
4156:
4154:
4151:
4150:
4148:
4146:
4142:
4136:
4133:
4131:
4128:
4125:
4121:
4117:
4113:
4109:
4106:
4104:
4101:
4098:
4094:
4090:
4086:
4082:
4079:
4078:
4076:
4074:
4070:
4064:
4061:
4059:
4056:
4054:
4051:
4049:
4046:
4044:
4041:
4039:
4036:
4034:
4031:
4030:
4028:
4026:
4022:
4016:
4013:
4011:
4008:
4006:
4003:
4001:
3998:
3996:
3993:
3991:
3988:
3986:
3983:
3981:
3978:
3976:
3973:
3972:
3970:
3968:
3964:
3958:
3955:
3953:
3950:
3949:
3947:
3945:
3941:
3935:
3932:
3930:
3927:
3926:
3924:
3922:
3918:
3911:
3907:
3900:
3888:
3884:
3881:
3879:
3876:
3874:
3871:
3869:
3866:
3864:
3861:
3859:
3856:
3854:
3851:
3850:
3847:
3843:
3836:
3831:
3829:
3824:
3822:
3817:
3816:
3813:
3807:
3804:
3802:
3798:
3795:
3792:
3790:
3787:
3786:
3782:
3774:
3768:
3764:
3763:
3755:
3752:
3747:
3741:
3737:
3736:
3728:
3725:
3720:
3714:
3710:
3709:
3701:
3698:
3693:
3691:0-7503-0734-X
3687:
3683:
3679:
3672:
3669:
3664:
3658:
3654:
3653:
3645:
3642:
3637:
3635:0-471-23429-X
3631:
3627:
3626:
3618:
3615:
3610:
3608:0-387-95269-1
3604:
3600:
3596:
3592:
3585:
3582:
3577:
3571:
3567:
3563:
3556:
3553:
3548:
3542:
3538:
3533:
3532:
3523:
3520:
3515:
3513:0-12-014653-3
3509:
3505:
3501:
3494:
3491:
3486:
3480:
3476:
3472:
3464:
3462:
3458:
3454:
3450:
3448:0-486-66741-3
3444:
3440:
3436:
3432:
3425:
3422:
3418:
3416:
3412:
3408:
3404:
3398:
3396:0-8176-3967-5
3392:
3388:
3387:
3379:
3376:
3371:
3365:
3361:
3360:
3352:
3349:
3344:
3338:
3334:
3333:
3325:
3322:
3317:
3311:
3307:
3303:
3296:
3293:
3288:
3282:
3278:
3274:
3270:
3263:
3260:
3255:
3249:
3245:
3241:
3237:
3230:
3227:
3222:
3216:
3212:
3208:
3201:
3198:
3193:
3189:
3188:"Wave theory"
3185:
3184:Lord Rayleigh
3178:
3175:
3170:
3168:0-387-28152-5
3164:
3160:
3156:
3155:
3147:
3144:
3139:
3137:0-521-39293-4
3133:
3129:
3128:
3119:
3117:0-521-42947-1
3113:
3108:
3107:
3097:
3094:
3089:
3083:
3079:
3078:
3070:
3067:
3062:
3056:
3052:
3051:
3043:
3040:
3035:
3029:
3025:
3024:
3016:
3014:
3010:
3005:
2999:
2995:
2994:
2985:
2982:
2977:
2971:
2967:
2966:
2958:
2955:
2950:
2948:0-387-98756-8
2944:
2940:
2936:
2935:
2927:
2924:
2920:
2916:
2910:
2906:
2901:
2900:
2891:
2888:
2883:
2877:
2873:
2872:
2864:
2861:
2856:
2854:2-88124-995-7
2850:
2846:
2839:
2836:
2831:
2829:0-534-49143-X
2825:
2821:
2820:
2812:
2809:
2804:
2802:0-19-850693-7
2798:
2794:
2790:
2783:
2780:
2775:
2773:0-201-11609-X
2769:
2765:
2758:
2755:
2749:
2745:
2742:
2740:
2737:
2735:
2734:Spectral line
2732:
2730:
2727:
2725:
2722:
2719:
2716:
2714:
2711:
2709:
2706:
2705:
2701:
2699:
2696:
2692:
2688:(symbol
2687:
2683:
2680:(symbol
2679:
2674:
2670:
2665:
2664:
2657:
2653:
2648:
2647:barred lambda
2644:
2640:
2636:
2627:
2620:
2618:
2616:
2612:
2611:Subwavelength
2608:
2606:
2602:
2598:
2594:
2590:
2585:
2583:
2582:optical fibre
2579:
2578:
2573:
2572:subwavelength
2566:Subwavelength
2565:
2563:
2560:
2558:
2554:
2550:
2546:
2527:
2519:
2516:
2511:
2508:
2505:
2498:
2497:
2496:
2494:
2490:
2485:
2483:
2466:
2463:
2460:
2457:
2454:
2423:
2405:
2401:
2396:
2393:
2388:
2385:
2382:
2379:
2375:
2367:
2366:
2365:
2363:
2359:
2355:
2350:
2348:
2341:
2334:
2330:
2326:
2322:
2318:
2314:
2310:
2306:
2300:
2296:
2288:
2286:
2284:
2280:
2276:
2272:
2268:
2249:
2240:
2237:
2232:
2229:
2223:
2220:
2200:
2192:
2187:
2181:
2178:
2173:
2170:
2167:
2164:
2158:
2153:
2147:
2139:
2120:
2114:
2108:
2101:
2100:
2099:
2097:
2096:sinc function
2093:
2089:
2085:
2080:
2078:
2073:
2071:
2067:
2062:
2060:
2051:
2046:
2041:
2037:
2029:
2027:
2025:
2020:
2015:
2008:
2004:
2000:
1981:
1974:
1969:
1965:
1962:
1959:
1956:
1953:
1947:
1943:
1938:
1934:
1929:
1924:
1919:
1915:
1912:
1909:
1906:
1903:
1900:
1894:
1890:
1885:
1881:
1875:
1871:
1867:
1862:
1858:
1850:
1849:
1848:
1845:
1843:
1839:
1820:
1814:
1808:
1804:
1800:
1797:
1794:
1788:
1785:
1782:
1779:
1776:
1769:
1768:
1767:
1765:
1746:
1740:
1737:
1734:
1731:
1728:
1725:
1722:
1715:
1714:
1713:
1710:
1706:
1701:
1697:
1693:
1689:
1685:
1675:
1670:
1662:
1657:
1655:
1653:
1649:
1645:
1644:wave function
1637:
1633:
1629:
1625:
1621:
1617:
1613:
1609:
1605:
1601:
1598:
1594:
1590:
1588:
1584:
1579:
1577:
1573:
1569:
1565:
1559:
1550:
1543:
1541:
1539:
1538:periodic wave
1530:
1526:
1524:
1518:
1514:
1510:
1505:
1501:
1497:
1491:
1489:
1488:superposition
1485:
1481:
1477:
1467:
1460:
1458:
1456:
1452:
1448:
1444:
1439:
1437:
1432:
1423:
1416:
1414:
1413:in the wave.
1412:
1408:
1404:
1400:
1399:
1394:
1389:
1386:
1385:inhomogeneous
1377:
1373:
1371:
1367:
1358:
1351:
1349:
1347:
1343:
1339:
1335:
1330:
1312:
1301:
1297:
1290:
1284:
1280:
1274:
1271:
1264:
1263:
1262:
1253:
1249:
1245:
1241:
1222:
1211:
1207:
1200:
1196:
1191:
1188:
1181:
1180:
1179:
1178:according to
1177:
1176:
1170:
1166:
1164:
1160:
1156:
1151:
1149:
1140:
1132:
1124:
1118:General media
1117:
1101:
1095:
1091:
1088:
1085:
1082:
1079:
1075:
1071:
1067:
1063:
1056:
1055:
1054:
1052:
1048:
1044:
1039:
1037:
1033:
1029:
1025:
1021:
1017:
1013:
1009:
1003:
999:
995:
987:
983:
962:
957:
954:
949:
944:
940:
937:
934:
928:
923:
919:
916:
910:
907:
900:
899:
898:
881:
876:
873:
868:
863:
859:
856:
853:
847:
842:
838:
835:
829:
826:
819:
818:
817:
799:
792:
789:
786:
783:
777:
773:
769:
766:
763:
760:
756:
752:
749:
746:
743:
740:
736:
732:
729:
726:
723:
717:
711:
708:
702:
695:
694:
693:
691:
688:
684:
681:
677:
673:
669:
665:
661:
641:
634:
631:
628:
625:
617:
613:
610:
603:
599:
596:
593:
590:
586:
581:
577:
574:
571:
566:
563:
557:
553:
550:
546:
542:
539:
536:
533:
527:
521:
518:
512:
505:
504:
503:
501:
497:
493:
485:
483:
481:
475:
473:
468:
466:
462:
461:standing wave
454:
448:
440:
436:
434:
430:
426:
422:
418:
414:
410:
405:
403:
399:
395:
391:
367:
363:
359:
354:
352:
348:
344:
328:
320:
304:
281:
274:
271:
266:
263:
256:
255:
254:
252:
239:
230:
226:
218:
216:
214:
210:
206:
202:
198:
197:
191:
189:
185:
181:
177:
173:
169:
165:
161:
157:
153:
149:
144:
142:
138:
133:
131:
127:
123:
119:
115:
112:
108:
107:
102:
98:
94:
90:
89:
84:
80:
76:
72:
68:
64:
56:
52:
48:
44:
39:
33:
19:
18:Subwavelength
4288:
3905:
3893:
3761:
3754:
3734:
3727:
3707:
3700:
3681:
3671:
3651:
3644:
3624:
3617:
3598:
3594:
3584:
3565:
3555:
3530:
3522:
3503:
3493:
3474:
3452:
3438:
3434:
3424:
3414:
3410:
3406:
3402:
3400:
3385:
3378:
3358:
3351:
3331:
3324:
3305:
3295:
3276:
3272:
3262:
3243:
3239:
3229:
3210:
3200:
3191:
3177:
3158:
3153:
3146:
3126:
3105:
3096:
3076:
3069:
3049:
3042:
3022:
2992:
2984:
2964:
2957:
2938:
2933:
2926:
2918:
2898:
2890:
2870:
2863:
2844:
2838:
2818:
2811:
2792:
2782:
2763:
2757:
2739:Spectroscopy
2694:
2690:
2681:
2672:
2668:
2661:
2655:
2651:
2642:
2638:
2634:
2632:
2610:
2609:
2586:
2575:
2571:
2569:
2561:
2556:
2548:
2544:
2542:
2488:
2486:
2438:
2353:
2351:
2339:
2332:
2328:
2324:
2302:
2282:
2278:
2274:
2270:
2266:
2264:
2091:
2087:
2081:
2076:
2074:
2063:
2055:
2024:redistribute
2023:
2021:
2013:
2006:
2002:
1998:
1996:
1846:
1841:
1837:
1835:
1763:
1761:
1708:
1704:
1699:
1695:
1680:
1615:
1611:
1607:
1603:
1599:
1591:
1580:
1575:
1571:
1567:
1564:wave packets
1561:
1544:Wave packets
1537:
1535:
1516:
1512:
1508:
1503:
1496:cnoidal wave
1492:
1472:
1443:energy bands
1440:
1428:
1402:
1396:
1390:
1384:
1382:
1369:
1363:
1331:
1327:
1251:
1247:
1239:
1237:
1173:
1171:
1167:
1152:
1145:
1040:
1035:
1031:
1027:
1023:
1007:
1001:
997:
993:
985:
981:
977:
896:
815:
689:
682:
671:
667:
663:
659:
657:
499:
495:
491:
489:
476:
469:
458:
427:and 17
411:in air, the
406:
355:
296:
253:is given by
232:
228:
222:
194:
192:
145:
134:
130:interference
117:
111:Greek letter
104:
86:
74:
70:
60:
46:
4308:Medium wave
3985:Lyman-alpha
3967:Ultraviolet
3906:wavelengths
3899:frequencies
3863:Ultraviolet
2317:microscopes
2311:(including
2059:diffraction
2036:Diffraction
1648:wavenumbers
1587:wavenumbers
1558:Wave packet
1523:ocean waves
1480:wave vector
1163:Snell's law
1012:wave vector
409:sound waves
384:divided by
382:10 m/s
374:10 m/s
353:nonlinear.
170:, while in
156:water waves
148:sound waves
135:Assuming a
67:mathematics
4328:Categories
4289:Wavelength
4145:Microwaves
3957:Hard X-ray
3952:Soft X-ray
3921:Gamma rays
3853:Gamma rays
2750:References
2309:telescopes
1842:vice versa
1562:Localized
1476:wavenumber
1407:wavenumber
1398:WKB method
1334:dispersion
1155:refraction
1043:plane wave
1016:plane wave
680:wavenumber
386:10 Hz
362:free space
174:and other
71:wavelength
4303:Shortwave
4298:Microwave
3878:Microwave
2605:photonics
2593:apertures
2580:means an
2570:The term
2517:λ
2506:δ
2493:Airy disk
2467:θ
2464:
2402:λ
2321:Airy disk
2238:λ
2179:π
2171:π
2168:
1970:λ
1966:α
1963:
1954:π
1944:
1920:λ
1916:α
1913:
1904:π
1891:
1815:λ
1786:θ
1783:
1741:λ
1732:θ
1729:
1692:two slits
1640:10 m
1632:electrons
1298:λ
1281:λ
1272:λ
1208:λ
1089:ω
1086:−
945:ω
938:π
920:π
908:λ
874:ω
857:π
843:λ
839:π
787:−
770:
750:ω
747:−
733:
676:amplitude
666:and time
629:−
618:λ
614:π
600:
572:−
567:λ
554:π
543:
343:frequency
264:λ
160:conductor
141:frequency
126:envelopes
122:modulated
57:as shown.
43:sine wave
4313:Longwave
4073:Infrared
3873:Infrared
3797:Archived
3186:(1890).
2744:Spectrum
2702:See also
2338:, where
2050:envelope
1614:, where
1597:momentum
1576:envelope
1568:envelope
1451:aliasing
1431:aliasing
1417:Crystals
1366:periodic
368:, about
196:spectrum
182:and the
180:electric
168:pressure
3904:longer
3897:higher
3868:Visible
3413:= 2π /
3023:op. cit
2489:angular
2354:spatial
1838:fringes
1618:is the
1455:sampled
1342:refract
1242:is the
1020:3-space
1010:with a
674:is the
345:. In a
101:inverse
63:physics
4339:Length
4206:L band
4201:S band
4196:C band
4191:X band
4177:K band
4163:Q band
4158:V band
4153:W band
4058:Orange
4053:Yellow
4033:Violet
3944:X-rays
3858:X-rays
3769:
3742:
3715:
3688:
3659:
3632:
3605:
3572:
3543:
3510:
3481:
3445:
3393:
3366:
3339:
3312:
3283:
3250:
3217:
3165:
3134:
3114:
3084:
3057:
3030:
3000:
2972:
2945:
2911:
2878:
2851:
2826:
2799:
2770:
2764:Optics
2599:, and
2543:where
2525:
2421:
2315:) and
2265:where
2247:
2198:
1997:where
1979:
1840:, and
1818:
1762:where
1744:
1581:Using
1238:where
1148:vacuum
1047:cosine
715:
670:, and
658:where
525:
398:violet
297:where
225:linear
114:lambda
4334:Waves
4291:types
4216:Radio
4112:Bands
4085:Bands
4048:Green
3883:Radio
2671:= 2π/
1688:Young
1634:in a
1370:local
1338:prism
465:nodes
396:, to
172:light
164:sound
152:light
88:phase
77:of a
51:phase
4186:band
4172:band
4130:LWIR
4108:MWIR
4103:SWIR
4043:Cyan
4038:Blue
3767:ISBN
3740:ISBN
3713:ISBN
3686:ISBN
3657:ISBN
3630:ISBN
3603:ISBN
3570:ISBN
3541:ISBN
3508:ISBN
3479:ISBN
3443:ISBN
3391:ISBN
3364:ISBN
3337:ISBN
3310:ISBN
3281:ISBN
3248:ISBN
3215:ISBN
3181:See
3163:ISBN
3132:ISBN
3112:ISBN
3082:ISBN
3055:ISBN
3028:ISBN
2998:ISBN
2970:ISBN
2943:ISBN
2909:ISBN
2876:ISBN
2849:ISBN
2824:ISBN
2797:ISBN
2768:ISBN
2693:= 2π
2551:the
2512:1.22
2487:The
2397:1.22
2297:and
2038:and
1707:sin
1445:and
1051:sine
502:as:
433:bats
407:For
162:. A
79:wave
65:and
4279:ELF
4274:SLF
4269:ULF
4264:VLF
4244:VHF
4239:UHF
4234:SHF
4229:EHF
4224:THF
4135:FIR
4081:NIR
4063:Red
4015:UVA
4010:UVB
4005:UVC
4000:NUV
3995:MUV
3990:FUV
2905:102
2698:).
2658:/2π
2461:sin
2165:sin
1960:sin
1935:sin
1910:sin
1882:sin
1780:sin
1726:sin
1636:CRT
1502:of
1478:or
1018:in
897:or
767:cos
730:cos
597:cos
540:cos
404:).
390:red
223:In
211:or
81:or
73:or
61:In
4330::
4259:LF
4254:MF
4249:HF
4122:,
4118:,
4114::
4095:,
4091:,
4087::
3680:.
3601:.
3599:ff
3564:.
3539:.
3537:15
3473:.
3460:^
3451:.
3441:.
3439:ff
3433:.
3399:.
3304:.
3279:.
3277:ff
3271:.
3246:.
3244:ff
3238:.
3209:.
3190:.
3161:.
3159:ff
3012:^
2941:.
2939:ff
2917:.
2907:.
2791:.
2654:=
2617:.
2607:.
2484:.
2364::
2349:.
2098::
2061:.
1844:.
1654:.
1606:=
1515:;
1509:cn
1490:.
1438:.
1348:.
1165:.
1002:ωt
1000:−
996:⋅
986:ωt
984:−
982:kx
459:A
429:mm
421:Hz
394:nm
215:.
154:,
150:,
69:,
45:,
4184:u
4182:K
4170:a
4168:K
4126:)
4124:N
4120:M
4116:L
4110:(
4099:)
4097:H
4093:K
4089:J
4083:(
3909:→
3895:←
3834:e
3827:t
3820:v
3775:.
3748:.
3721:.
3694:.
3665:.
3638:.
3611:.
3578:.
3549:.
3516:.
3487:.
3417:.
3415:λ
3411:k
3407:k
3403:λ
3372:.
3345:.
3318:.
3289:.
3256:.
3223:.
3171:.
3140:.
3120:.
3090:.
3063:.
3036:.
3006:.
2978:.
2951:.
2884:.
2857:.
2832:.
2805:.
2776:.
2695:f
2691:ω
2682:ħ
2673:λ
2669:k
2666:(
2656:λ
2652:ƛ
2643:ƛ
2557:δ
2549:D
2545:λ
2528:,
2520:D
2509:=
2458:n
2455:=
2451:A
2448:N
2424:,
2414:A
2411:N
2406:2
2394:=
2389:y
2386:r
2383:i
2380:A
2376:r
2343:1
2340:J
2336:1
2333:J
2329:r
2325:x
2283:x
2279:u
2275:S
2271:R
2267:L
2250:,
2241:R
2233:L
2230:x
2224:=
2221:u
2201:;
2193:2
2188:)
2182:u
2174:u
2159:(
2154:=
2151:)
2148:u
2145:(
2140:2
2135:c
2132:n
2129:i
2126:s
2121:=
2118:)
2115:u
2112:(
2109:S
2092:x
2088:S
2052:.
2017:1
2014:I
2010:1
2007:I
2003:g
1999:q
1982:,
1975:)
1957:g
1948:(
1939:2
1930:/
1925:)
1907:g
1901:q
1895:(
1886:2
1876:1
1872:I
1868:=
1863:q
1859:I
1821:.
1812:)
1809:2
1805:/
1801:1
1798:+
1795:m
1792:(
1789:=
1777:d
1764:m
1747:,
1738:m
1735:=
1723:d
1709:θ
1705:d
1700:d
1696:s
1616:h
1612:p
1610:/
1608:h
1604:λ
1600:p
1519:)
1517:m
1513:x
1511:(
1504:m
1313:.
1307:)
1302:0
1294:(
1291:n
1285:0
1275:=
1259:0
1255:0
1252:λ
1250:(
1248:n
1240:c
1223:,
1217:)
1212:0
1204:(
1201:n
1197:c
1192:=
1189:v
1102:.
1096:)
1092:t
1083:x
1080:k
1076:(
1072:i
1068:e
1064:A
1036:v
1032:k
1028:k
1024:r
1008:k
1004:)
998:r
994:k
992:(
988:)
980:(
963:.
958:f
955:v
950:=
941:v
935:2
929:=
924:k
917:2
911:=
882:,
877:v
869:=
864:v
860:f
854:2
848:=
836:2
830:=
827:k
800:)
796:)
793:t
790:v
784:x
781:(
778:k
774:(
764:A
761:=
757:)
753:t
744:x
741:k
737:(
727:A
724:=
721:)
718:t
712:,
709:x
706:(
703:y
690:ω
683:k
672:A
668:t
664:x
660:y
642:)
638:)
635:t
632:v
626:x
623:(
611:2
604:(
594:A
591:=
587:)
582:)
578:t
575:f
564:x
558:(
551:2
547:(
537:A
534:=
531:)
528:t
522:,
519:x
516:(
513:y
500:λ
496:f
492:v
425:m
380:×
378:3
372:×
370:3
329:f
305:v
282:,
275:f
272:v
267:=
240:v
229:λ
118:λ
116:(
47:λ
34:.
20:)
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