Wavelength

For other uses, see Wavelength (disambiguation).

Wavelength of a sine wave

In physics, wavelength is the distance between repeating units of a waveform. It is commonly designated by the Greek letter lambda (λ). Examples of wave-like phenomena are light, water waves, and sound waves. Waves may have arbitrary shapes, but recur periodically in time or space or both. A wave that does not move in space but oscillates in time is called a standing wave.

Assuming the speed of the wave is fixed, wavelength is inversely proportional to frequency: waves with higher frequencies have shorter wavelengths, and lower frequencies have longer wavelengths.

Different types of wave have different properties that may vary with position. For example, in a sound wave the air pressure oscillates, while in light and other electromagnetic radiation the strength of the electric and the magnetic field vary. In a crystal lattice vibration, atomic positions vary with location in the crystal lattice at a fixed time, and vary in time at a fixed lattice position.

Wavelength is generally not the distance that particles travel during a period. For instance, in acoustics and water waves, the particle displacements during a period are only a small fraction of the wavelength, apart from extreme conditions like breaking waves and shock waves.

Sinusoidal waves

Linear wave media are typically described in terms of the characteristics of sinusoidal waves, since sinusoids are the shapes that propagate unchanged in such media, and since any wave in such media can be described as a superposition of sinusoidal waves.

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

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

where v is called the phase speed (magnitude of the phase velocity) of the wave and f is its 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) (in air at room temperature and atmospheric pressure).

For example, the wavelength for 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 (cf. refraction).

General media

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

$\lambda ={\frac {\lambda _{0}}{n}},$

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.

More general waveforms

See also: Wave

Wavelength of an irregular periodic waveform f(x − vt) = f(x + λ − vt) at a particular moment in time; the same wavelength marks the distance between similar positions in the waveform, for example, the distance between peaks and the distance between troughs.

The figure shows a wave in space in one dimension at a particular time. A wave with a fixed shape but moving in space is called a traveling wave. If the shape repeats itself in space, it is also a periodic wave. To an observer at a fixed location, the amplitude of a traveling periodic wave will appear to vary in time, and if its velocity is constant, will repeat itself with a certain period, say T. Assuming one dimension, the wave thus recurs with a frequency, say f, given by:

$f={\frac {1}{T}}\ .$

Every period, one wavelength of the wave passes the observer, showing the velocity of the wave, say v, to be related to the frequency by:

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

which implies the wavelength is related to frequency as:

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

Therefore, the results presented above for the sinusoidal wave apply to general waveforms as well.

Formal description

Traveling periodic wave; wavelength is the distance between green dots

In one dimension, a traveling wave that propagates without changing shape is described by the equation

$y=f(x-vt)\ ,$

where f is an arbitrary function of its argument, y = wave amplitude, v = wave speed or velocity, x = position in the wave, and t = time. If we define a particular value of the argument, say x_t given by:

$x_{t}=x-vt\ ,$

then a fixed value of x_t is located at a position x that travels in time with a speed v, which means the point in the wave with amplitude y = f (x_t) also travels in time with speed v.

To possess a wavelength the waveform must be periodic, which requires f(x) to be a periodic function. That is, f(x) is restricted to functions such that:

$f(x+\lambda -vt)=f(x-vt)=f(x-v(t+T))\ ,$

which recaptures the relation between wavelength and wave speed found intuitively above: λ = vT.

Connection to sinusoidal waves: components with many wavelengths

Superposition of sinusoidal wave basis functions (bottom) to form a sawtooth wave (top); the basis functions have wavelengths λ/n (n=integer) shorter than the wavelength λ of the sawtooth itself (except for n=1). All basis functions have nodes at the nodes of the sawtooth, but all but the fundamental have additional nodes. The oscillation about the sawtooth is called the Gibbs phenomenon.

Under rather general conditions, a function f(x) can be expressed as a sum of basis functions {φ_n(x)} in the form:

$f(x)=\sum _{n=1}^{\infty }c_{n}\varphi _{n}(x)\ ,$

known variously as Fourier series, Fourier-Bessel series, generalized Fourier series, and so forth, depending upon the basis used.

For a periodic function f with spatial periodicity λ, the basis functions satisfy φ_n(x + λ) = φ_n(x). This condition can be satisfied by basis functions that repeat more often in space than does f itself, and so have wavelengths shorter than the function f.

In particular, for such a periodic function f , the basis may be chosen as a set of sinusoidal functions, selected with wavelengths λ/n (n an integer) to ensure φ_n(x + λ) = φ_n(x). For a sine wave sin(kx) the implication is kλ = 2nπ (n an integer), or k = 2πn/λ, where k is called the wave vector and n is called the wavenumber. The wavelength of sin(kx) = sin(2πn x /λ) is λ/n. In this case, the basis function with wavelength λ is referred to as the fundamental and the other basis functions as harmonics. Many examples of such representations are found in books on Fourier series. For example, application to a number of sawtooth waves is presented by Puckette.

Whatever the basis functions, the wave becomes:

$y=f(x-vt)=\sum _{n=1}^{\infty }c_{n}\varphi _{n}(x-vt)\ .$

Thus, all the components must travel at the same rate to insure that the waveform remains unchanged as it moves. To turn this discussion upside down, because a general waveform may be viewed as a superposition of shorter wavelength basis functions, a requirement upon the physical medium propagating a traveling wave of fixed shape is that the medium must be capable of propagating disturbances of different wavelengths at the same wave speed. This requirement is met in many simple wave propagating mediums, but is not a general property of all media. More commonly, a medium has a non-linear dispersion relation connecting wave vector to frequency of oscillation, and the medium is dispersive, which means propagation of rigid waveforms is not possible in general, but requires very particular circumstances.

Such circumstances sometimes do occur in nonlinear media. For example, in large-amplitude ocean waves, due to properties of the nonlinear surface-wave medium, wave shapes can propagate unchanged. A related phenomenon is the cnoidal wave, a periodic traveling wave named because it is described by the Jacobian elliptic function of m-th order, usually denoted as cn (x; m). Another is the wave motion in an inviscid incompressible fluid, where the wave shape is given by:

$y=A\ \mathrm {sech} ^{2}[\beta (x-vt)]\ ,$

one of the early solutions to the Korteweg–de Vries equation of 1895. Here β is a constant related to height of the wave and depth of the water. The Korteweg-de Vries equation is an exactly solvable model, that is, a non-linear partial differential equation whose solutions can be exactly and precisely specified. The solutions include examples of solitons, traveling waves without a wavelength because they are not periodically recurring, but still capable of representation as sums of functions with definite wavelengths using the Fourier integral.

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.

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

De Broglie wavelength of particles

Louis de Broglie postulated that all particles with 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.

See also

• Amplitude
• Angular frequency
• Fraunhofer lines, spectral lines traditionally used as standard optical wavelength references
• Frequency
• Periodic function
• Wave vector
• Ion acoustic wave
• Waves in plasmas
• Sound

External links

• Conversion: Wavelength to Frequency and vice versa - Sound waves and radio waves
• Teaching resource for 14-16yrs on sound including wavelength
• The visible electromagnetic spectrum displayed in web colors with according wavelengths