1 Sinusoidal waves
2 More general waveforms
- 2.1 Envelope waves
- 2.2 Wave packets
3 Interference and diffraction
4 Subwavelength
5 See also
6 References
7 External links

Wavelength

From Wikipedia, the free encyclopedia

For other uses, see Wavelength (disambiguation).

Wavelength of a sine wave, λ, can be measured between any two points with the same phase, such as between crests, or troughs, or corresponding zero crossings as shown.

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]

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]

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]

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

Refraction: when a plane wave encounters a medium in which it has a slower speed, the wavelength decreases, and the direction adjusts accordingly.

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 speed is the speed of light, about 3×10⁸ m/s. For sound waves in air, the speed of sound is 343 m/s (1238 km/h) (at room temperature and atmospheric pressure). As an example, the wavelength of a 100 MHz electromagnetic (radio) wave is about: 3×10⁸ m/s divided by 100×10⁶ 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). 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; 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.

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.

Standing waves

A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).

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 ( $2π$ times the reciprocal of wavelength) and angular frequency ω ( $2π$ 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:

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

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

Dispersion causes separation of colors when light is refracted by a prism.

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]

Wavelength is decreased in a medium with higher refractive index.

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

$\lambda = \frac{\lambda_0}{n},$

Various local wavelengths on a crest-to-crest basis in an ocean wave approaching shore.[11]

where λ₀ is the wavelength in vacuum, and n is the refractive index of the medium. 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.

In general, the refractive index is a function of wavelength. This variation of n with λ, called dispersion, causes different colors of light to be separated when light is refracted by a prism.

Nonuniform media

A sinusoidal wave in a nonuniform medium, with loss. As the wave slows down, the wavelength gets shorter and the amplitude increases; after a place of maximum response, the short wavelength is associated with a high loss and the wave dies out.

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

A wave on a line of atoms can be interpreted according to a variety of wavelengths.

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