Wavelength

In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.[1] It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns.[2][3] Wavelength is commonly designated by the Greek letter lambda (λ). The concept can also be applied to periodic waves of non-sinusoidal shape.[1][4] The term wavelength is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids.[5] The SI unit of wavelength is the meter.

Assuming a sinusoidal wave moving at a fixed wave speed, wavelength is inversely proportional to frequency: waves with higher frequencies have shorter wavelengths, and lower frequencies have longer wavelengths.[6]

Examples of wave-like phenomena are sound waves, light, and water waves. A sound wave is a periodic variation in air pressure, while in light and other electromagnetic radiation the strength of the electric and the magnetic field vary. Water waves are periodic variations in the height of a body of water. In a crystal lattice vibration, atomic positions vary periodically in both lattice position and time.

Wavelength is a measure of the distance between repetitions of a shape feature such as peaks, valleys, or zero-crossings, not a measure of how far any given particle moves. For example, in waves over deep water a particle in the water moves in a circle of the same diameter as the wave height, unrelated to wavelength.[7]

Contents

Sinusoidal waves

In linear media, any wave pattern can be described in terms of the independent propagation of sinusoidal components.

The wavelength λ of a sinusoidal waveform traveling at constant speed v is given by:[8]

\lambda = \frac{v}{f},

where v is called the phase speed (magnitude of the phase velocity) of the wave and f is the wave's frequency.

In the case of electromagnetic radiation—such as light—in free space, the phase speed is the speed of light, about 3×108 m/s. Thus the wavelength of a 100 MHz electromagnetic (radio) wave is about: 3×108 m/s divided by 108 Hz = 3 metres. Visible light ranges from deep red, roughly 700 nm, to violet, roughly 400 nm (430–750 THz) (for other examples, see electromagnetic spectrum).

For sound waves in air, the speed of sound is 343 m/s (1238 km/h) (at room temperature and atmospheric pressure). The wavelengths of sound frequencies audible to the human ear (20 Hz–20 kHz) are between approximately 17 m and 17 mm, respectively, assuming a typical speed of sound of about 343 m/s. Note that the wavelengths in audible sound are much longer than those in visible light.

Frequency and wavelength can change independently, but only when the speed of the wave changes. For example, when light enters another medium, its speed and wavelength change while its frequency does not; this change of wavelength causes refraction, or a change in propagation direction of waves that encounter the interface between media at an angle.

Standing waves

A standing wave is an undulatory motion that stays in one place. A sinusoidal standing wave includes stationary points of no motion, called nodes, and the wavelength is twice the distance between nodes. The wavelength, period, and wave velocity are related as before, if the stationary wave is viewed as the sum of two traveling sinusoidal waves of oppositely directed velocities.[9]

Mathematical representation

Traveling sinusoidal waves are often represented mathematically in terms of their velocity v (in the x direction), frequency f and wavelength λ as:

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 )

where y is the value of the wave at any position x and time t, and A is the amplitude of the wave. They are also commonly expressed in terms of (radian) wavenumber k (2 \pi times the reciprocal of wavelength) and angular frequency ω (2 \pi times the frequency) as:

y (x, \ t) = A \cos \left( kx - \omega t \right) = A \cos \left(k(x - v t) \right)

in which wavelength and wavenumber are related to velocity and frequency as:

k = \frac{2 \pi}{\lambda} = \frac{2 \pi f}{v} = \frac{\omega}{v}

or

\lambda = \frac{2 \pi}{k} = \frac{2 \pi v}{\omega} = \frac{v}{f}.

The relationship between ω and λ (or k) is called a dispersion relation. This is not generally a simple inverse relation because the wave velocity itself typically varies with frequency.[10]

In the second form given above, the phase (kx − ωt) is often generalized to (k•r − ωt), by replacing the wavenumber k with a wave vector that specifies the direction and wavenumber of a plane wave in 3-space, parameterized by position vector r. In that case, the wavenumber k, the magnitude of k, is still in the same relationship with wavelength as shown above, with v 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.

Generalizations to sinusoids of other phases, and to complex exponentials, are also common; see plane wave. The typical convention of using the cosine phase instead of the sine phase when describing a wave is based on the fact that the cosine is the real part of the complex exponential in the wave

A e^{ i \left( kx - \omega t \right)}.

General media

The speed of a wave depends upon the medium in which it propagates. In particular, the speed of light in most media is lower than in vacuum, which means that the same frequency will correspond to a shorter wavelength in the medium than in vacuum. The wavelength in the medium is

\lambda = \frac{\lambda_0}{n(\lambda_0)},

where λ0 is the wavelength in vacuum, and n(λ0) is the refractive index of the medium, which varies with wavelength. This variation, called dispersion, causes different colors of light to be separated when light is refracted by a prism.

When wavelengths of electromagnetic radiation are quoted, the vacuum wavelength is usually 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 specified medium.

Nonuniform media

Wavelength can be a useful concept even if the wave is not periodic in space. For example, in an ocean wave approaching shore, shown in the figure, the incoming wave undulates with a varying local 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.[11]

Waves that are sinusoidal in time but propagate through a medium whose properties vary with position (an inhomogeneous medium) may propagate at a velocity that varies with position, and as a result may not be sinusoidal in space. The analysis of differential equations of such systems is often done approximately, using the WKB method (also known as the Liouville–Green method). The method integrates phase through space using a local wavenumber, which can be interpreted as indicating a "local wavelength" of the solution as a function of time and space.[12][13] 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 conservation of energy in the wave.

Crystals

Waves in crystalline solids are not continuous, because they are composed of vibrations of discrete particles arranged in a regular lattice. This produces aliasing because the same vibration can be considered to have a variety of different wavelengths, as shown in the figure.[14] 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 provide a description of all possible waves in a crystalline medium corresponds to the wave vectors confined to the Brillouin zone.[15]

This indeterminacy in wavelength in solids is important in the analysis of wave phenomena such as energy bands and lattice vibrations. It is mathematically equivalent to the aliasing of a signal that is sampled at discrete intervals.

More general waveforms

A wave moving in space is called a traveling wave. If the shape repeats itself, it is also a periodic wave.[16] In the special case of uniform and dispersionless media (see Dispersion relation), at a fixed moment in time, a snapshot of the wave shows a repeating form in space, with characteristics such as peaks and troughs repeating at equal intervals. To an observer at a fixed location the amplitude appears to vary in time, and repeats itself with a certain period, for example T. If the spatial period of this wave is referred to as its wavelength, then during every period, one wavelength of the wave passes the observer. In dispersion and uniform media, the wave propagates with unchanging shape and the velocity in the medium is uniform, so this period implies the wavelength is:

\lambda = vT \ .

This duality of space and time is expressed mathematically by the fact that, in such special media, the wave's behavior does not depend independently on position x and time t, but rather on the combination of position and time x − vt. The wave's amplitude u is then expressed as u(x − vt) and in the case of a periodic function u with period λ, that is, u(x + λ − vt) = u(x − vt), the periodicity of u in space means that a snapshot of the wave at a given time t finds the wave varying periodically in space with period λ. In a similar fashion, this periodicity of u implies a periodicity in time as well: u(x − v(t + T)) = u(x − vt) using the relation vT = λ described above, so an observation of the wave at a fixed location x finds the wave undulating periodically in time with period T = λ/v.[16]

Traveling waves with non-sinusoidal wave shapes can occur in linear dispersionless media such as free space, but also may arise in nonlinear media under certain circumstances. For example, large-amplitude ocean waves with certain shapes can propagate unchanged, because of properties of the nonlinear surface-wave medium.[17] An example is the cnoidal wave, a periodic traveling wave named because it is described by the Jacobi elliptic function of m-th order, usually denoted as cn (x; m).[18]

Envelope waves

The term wavelength is also sometimes applied to the envelopes of waves, such as the traveling sinusoidal envelope patterns that result from the interference of two sinusoidal waves close in frequency; such envelope characterizations are used in illustrating the derivation of group velocity, the speed at which slow envelope variations propagate.[19]

Wave packets

Main article: Wave packet

Localized wave packets, "bursts" of wave action where each wave packet travels as a unit, find application in many fields of physics; the notion of a wavelength also may be applied to these wave packets.[21] The wave packet has an envelope that describes the overall amplitude of the wave; within the envelope, the distance between adjacent peaks or troughs is sometimes called a local wavelength.[22][23] Using Fourier analysis, wave packets can be analyzed into infinite sums (or integrals) of sinusoidal waves of different wavenumbers or wavelengths.[24]

Louis de Broglie postulated that all particles with a specific value of momentum have a wavelength

\lambda = \frac{h}{p},

where h is Planck's constant, and p is the momentum of the particle. This hypothesis was at the basis of quantum mechanics. Nowadays, this wavelength is called the de Broglie wavelength. For example, the electrons in a CRT display have a De Broglie wavelength of about 10−13 m. To prevent the wave function for such a particle being spread over all space, De Broglie proposed using wave packets to represent particles that are localized in space.[25] The spread of wavenumbers of sinusoids that add up to such a wave packet corresponds to an uncertainty in the particle's momentum, one aspect of the Heisenberg uncertainty principle.[24]

Interference and diffraction

Double-slit interference

Main article: Interference (wave propagation)

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 interferometer. A simple example is an experiment due to Young where light is passed through two slits.[26] 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, s is large compared to the slit separation d) then the paths are nearly parallel, and the path difference is simply d sin θ. Accordingly the condition for constructive interference is:[27]

d \sin \theta = m \lambda \ ,

where m is an integer, and for destructive interference is:

d \sin \theta = (m %2B 1/2 )\lambda \ .

Thus, if the wavelength of the light is known, the slit separation can be determined from the interference pattern or fringes, and vice versa.

It should be noted that the effect of interference is to redistribute the light, so the energy contained in the light is not altered, just where it shows up.[28]

Single-slit diffraction

Main articles: Diffraction and Diffraction formalism

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 image on the screen. This distribution of wave energy is called diffraction.

Two types of diffraction are distinguished, depending upon the separation between the source and the screen: Fraunhofer diffraction or far-field diffraction at large separations and Fresnel diffraction or near-field diffraction at close separations.

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 (Huygen's wavelets). 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.

In the Fraunhofer diffraction pattern sufficiently far from a single slit, within a small-angle approximation, the intensity spread S is related to position x via a squared sinc function:[29]

S(u) = \mathrm{sinc}^2(u) = \left( \frac {\sin \pi u}{\pi u} \right) ^2 \�;  with  u = \frac {x L}{\lambda R} \ ,

where L is the slit width, R is the distance of the pattern (on the screen) from the slit, and λ is the wavelength of light used. The function S has zeros where u is a non-zero integer, where are at x values at a separation proportion to wavelength.

Diffraction-limited resolution

Main articles: Angular resolution and Diffraction-limited system

Diffraction is the fundamental limitation on the resolving power of optical instruments, such as telescopes (including radiotelescopes) and microscopes.[30] For a circular aperture, the diffraction-limited image spot is known as an Airy disk; the distance x in the single-slit diffraction formula is replaced by radial distance r and the sine is replaced by 2J1, where J1 is a first order Bessel function.[31]

The resolvable spatial size of objects viewed through a microscope is limited according to the Rayleigh criterion, 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 numerical aperture:[32]

r_{Airy} = 1.22 \frac {\lambda}{2\mathrm{NA}} \ ,

where the numerical aperture is defined as \mathrm{NA} = n \sin \theta\; for θ being the half-angle of the cone of rays accepted by the microscope objective.

The angular size of the central bright portion (radius to first null of the Airy disk) of the image diffracted by a circular aperture, a measure most commonly used for telescopes and cameras, is:[33]

\delta = 1.22 \frac {\lambda}{D} \ ,

where λ is the wavelength of the waves that are focused for imaging, D the entrance pupil diameter of the imaging system, in the same units, and the angular resolution δ is in radians.

As with other diffraction patterns, the pattern scales in proportion to wavelength, so shorter wavelengths can lead to higher resolution.

Subwavelength

The term subwavelength 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 subwavelength-diameter optical fibre means an optical fibre whose diameter is less than the wavelength of light propagating through it.

A subwavelength particle is a particle smaller than the wavelength of light with which it interacts (see Rayleigh scattering). Subwavelength apertures are holes smaller than the wavelength of light propagating through them. Such structures have applications in extraordinary optical transmission, and zero-mode waveguides, among other areas of photonics.

Subwavelength may also refer to a phenomenon involving subwavelength objects; for example, subwavelength imaging.

Angular wavelength

A quantity related to the wavelength is the angular wavelength (also known as reduced wavelength), usually symbolized by ƛ (lambda-bar). It is equal to the "regular" wavelength "reduced" by a factor of 2π (ƛ = λ/2π). It is usually encountered in quantum mechanics, where it is used in combination with the reduced Planck constant (symbol ħ , h-bar) and the angular frequency (symbol ω) or angular wavenumber (symbol k).

See also

References

  1. ^ a b Hecht, Eugene (1987). Optics (2nd ed.). Addison Wesley. pp. 15–16. ISBN 0-201-11609-X. 
  2. ^ Raymond A. Serway, John W. Jewett. Principles of physics (4th ed.). Cengage Learning. pp. 404, 440. ISBN 053449143X. http://books.google.com/books?id=1DZz341Pp50C&pg=PA404. 
  3. ^ A. A. Sonin (1995). The surface physics of liquid crystals. Taylor & Francis. p. 17. ISBN 2881249957. 
  4. ^ Brian Hilton Flowers (2000). "§21.2 Periodic functions". An introduction to numerical methods in C++ (2nd ed.). Cambridge University Press. p. 473. ISBN 0198506937. http://books.google.com/books?id=weYj75E_t6MC&pg=RA1-PA473. 
  5. ^ Keqian Zhang and Dejie Li (2007). Electromagnetic Theory for Microwaves and Optoelectronics. Springer,. p. 533. ISBN 9783540742951. http://books.google.com/books?id=3Da7MvRZTlAC&pg=PA533&dq=wavelength+modulated-wave+envelope. 
  6. ^ Theo Koupelis and Karl F. Kuhn (2007). In Quest of the Universe. Jones & Bartlett Publishers. ISBN 0763743879. http://books.google.com/books?id=WwKjznJ9Kq0C&pg=PA102&dq=wavelength+lambda+light+sound+frequency+wave+speed. 
  7. ^ Paul R Pinet (2008). Invitation to Oceanography (5th ed.). Jones & Bartlett Publishers. p. 237. ISBN 0763759937. http://books.google.com/books?id=6TCm8Xy-sLUC&pg=PA237. 
  8. ^ David C. Cassidy, Gerald James Holton, Floyd James Rutherford (2002). Understanding physics. Birkhäuser. pp. 339 ff. ISBN 0387987568. http://books.google.com/books?id=rpQo7f9F1xUC&pg=PA340. 
  9. ^ John Avison (1999). The World of Physics. Nelson Thornes. p. 460. ISBN 9780174387336. http://books.google.com/books?id=DojwZzKAvN8C&pg=PA460&dq=%22standing+wave%22+wavelength. 
  10. ^ John A. Adam (2003). Mathematics in nature. Princeton University Press. p. 148. ISBN 0691114293. http://books.google.com/books?id=2gO2sBp4ipQC&pg=PA148. "The relation between the frequency of a wave and its wavelength λ ... is referred to as a dispersion relation." 
  11. ^ a b Paul R Pinet. op. cit.. p. 242. ISBN 0763759937. http://books.google.com/books?id=6TCm8Xy-sLUC&pg=PA242. 
  12. ^ Bishwanath Chakraborty. Principles of Plasma Mechanics. New Age International. p. 454. ISBN 9788122414462. http://books.google.com/books?id=_MIdEiKqdawC&pg=PA454&dq=wkb+local-wavelength. 
  13. ^ Jeffrey A. Hogan and Joseph D. Lakey (2005). Time-frequency and time-scale methods: adaptive decompositions, uncertainty principles, and sampling. Birkhäuser. p. 348. ISBN 9780817642761. http://books.google.com/books?id=YOf0SRzxz3gC&pg=PA348&dq=wkb+local-wavelength. 
  14. ^ See Figure 4.20 in A. Putnis (1992). Introduction to mineral sciences. Cambridge University Press. p. 97. ISBN 0521429471. http://books.google.com/books?id=yMGzmOqYescC&pg=PA97.  and Figure 2.3 in Martin T. Dove (1993). Introduction to lattice dynamics (4th ed.). Cambridge University Press. p. 22. ISBN 0521392934. http://books.google.com/books?id=vM50l2Vf7HgC&pg=PA22. 
  15. ^ Manijeh Razeghi (2006). Fundamentals of solid state engineering (2nd ed.). Birkhäuser. pp. 165 ff. ISBN 0387281525. http://books.google.com/books?id=6x07E9PSzr8C&pg=PA165. 
  16. ^ a b Alexander McPherson (2009). "Waves and their properties". Introduction to Macromolecular Crystallography (2 ed.). Wiley. p. 77. ISBN 0470185902. http://books.google.com/books?id=o7sXm2GSr9IC&pg=PA77. 
  17. ^ Ken'ichi Okamoto (2001). Global environment remote sensing. IOS Press. p. 263. ISBN 9781586031015. http://books.google.com/books?id=tXQy5JdQyZoC&pg=PA263&dq=wave-length++non-sinusoidal. 
  18. ^ Roger Grimshaw (2007). "Solitary waves propagating over variable topography". In Anjan Kundu. Tsunami and Nonlinear Waves. Springer. pp. 52 ff. ISBN 3540712550. http://books.google.com/books?id=2Dtgq-1CGWIC&pg=PA52. 
  19. ^ Mark W. Denny (1995). Air and Water: The Biology and Physics of Life's Media. Princeton University Press. p. 289. ISBN 9780691025186. http://books.google.com/books?id=XjNS6v7q130C&pg=PA289&dq=envelope+wavelength+group-velocity. 
  20. ^ A. T. Fromhold (1991). "Wave packet solutions". Quantum Mechanics for Applied Physics and Engineering (Reprint of Academic Press 1981 ed.). Courier Dover Publications. pp. 59 ff. ISBN 0486667413. http://books.google.com/books?id=3SOwc6npkIwC&pg=PA59. "(p. 61) ... the individual waves move more slowly than the packet and therefore pass back through the packet as it advances" 
  21. ^ Paul A. LaViolette (2003). Subquantum Kinetics: A Systems Approach to Physics and Cosmology. Starlane Publications. p. 80. ISBN 9780964202559. http://books.google.com/books?id=8HQJAvA1EqkC&pg=PA80&dq=wave-packet-wavelength+de-Broglie-wavelength. 
  22. ^ Peter R. Holland (1995). The Quantum Theory of Motion: An Account of the de Broglie–Bohm Causal Interpretation of Quantum Mechanics. Cambridge University Press. p. 160. ISBN 9780521485432. http://books.google.com/books?id=BsEfVBzToRMC&pg=PA160&dq=wave-packet+local-wavelength. 
  23. ^ Jeffery Cooper (1998). Introduction to partial differential equations with MATLAB. Springer. p. 272. ISBN 0817639675. http://books.google.com/books?id=l0g2BcxOJVIC&pg=PA272. "The local wavelength λ of a dispersing wave is twice the distance between two successive zeros. ... the local wave length and the local wave number k are related by k = 2π / λ." 
  24. ^ a b See, for example, Figs. 2.8–2.10 in Joy Manners (2000). "Heisenberg's uncertainty principle". Quantum Physics: An Introduction. CRC Press. pp. 53–56. ISBN 9780750307208. http://books.google.com/books?id=LkDQV7PNJOMC&pg=PA54&dq=wave-packet+wavelengths. 
  25. ^ Ming Chiang Li (1980). "Electron Interference". In L. Marton and Claire Marton. Advances in Electronics and Electron Physics. 53. Academic Press. p. 271. ISBN 0120146533. http://books.google.com/books?id=g5q6tZRwUu4C&pg=PA271. 
  26. ^ Greenfield Sluder and David E. Wolf (2007). "IV. Young's Experiment: Two-Slit Interference". Digital microscopy (3rd ed.). Academic Press. p. 15. ISBN 0123740258. http://books.google.com/books?id=H--zxc_N-jMC&pg=PA15. 
  27. ^ Halliday, Resnick, Walker (2008). "§35-4 Young's interference experiment". Fundamentals of Physics (Extended 8th ed.). Wiley-India. p. 965. ISBN 8126514426. http://books.google.com/books?id=RVCE4EUjDCgC&pg=PT965. 
  28. ^ Douglas B. Murphy (2002). Fundamentals of light microscopy and electronic imaging. Wiley/IEEE. p. 64. ISBN 047123429X. http://books.google.com/books?id=UFgdjxTULJMC&pg=PA64. 
  29. ^ John C. Stover (1995). Optical scattering: measurement and analysis (2nd ed.). SPIE Press. p. 64. ISBN 9780819419347. http://books.google.com/books?id=ot0tjJL72uUC&pg=PA65&dq=single-slit+diffraction+sinc-function. 
  30. ^ Graham Saxby (2002). "Diffraction limitation". The science of imaging. CRC Press. p. 57. ISBN 075030734X. http://books.google.com/books?id=e5mC5TXlBw8C&pg=PA57. 
  31. ^ Grant R. Fowles (1989). Introduction to Modern Optics. Courier Dover Publications. pp. 117–120. ISBN 9780486659572. http://books.google.com/books?id=SL1n9TuJ5YMC&pg=PA119&dq=Airy-disk+Bessel+slit+diffraction+sin. 
  32. ^ James B. Pawley (1995). Handbook of biological confocal microscopy (2nd ed.). Springer. p. 112. ISBN 9780306448263. http://books.google.com/books?id=16Ft5k8RC-AC&pg=PA112. 
  33. ^ Ray N. Wilson (2004). Reflecting Telescope Optics I: Basic Design Theory and Its Historical Development. Springer. p. 302. ISBN 9783540401063. http://books.google.com/books?id=PuN7l2A2uzQC&pg=PA302&dq=telescope+diffraction-limited+resolution+sinc. 

External links

← higher frequencies       longer wavelengths →

Gamma rays · X-rays · Ultraviolet · Visible · Infrared · Terahertz radiation · Microwave · Radio
Visible (optical)
Violet · Blue · Green · Yellow · Orange · Red
Microwaves
W band · V band · Q band · Ka band · K band · Ku band · X band · S band · C band · L band
Radio
EHF · SHF · UHF · VHF · HF · MF · LF · VLF · ULF · SLF · ELF
Wavelength types