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

${\displaystyle \mathrm{\lambda <!--\; \lambda \; -->}=\frac{v}{f},}$

where v is called the phase speed (magnitude of the phase velocity) of the wave and f is the wave's frequency. In a dispersive medium, the phase speed itself depends upon the frequency of the wave, making the relationship between wavelength and frequency nonlinear.

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. The wavelength of visible light ranges from deep red, roughly 700 nm, to violet, roughly 400 nm (for other examples, see electromagnetic spectrum).

For sound waves in air, the speed of sound is 343 m/s (at room temperature and atmospheric pressure). The wavelengths of sound frequencies audible to the human ear (20 Hz–20 kHz) are thus between approximately 17 m and 17 mm, respectively. Somewhat higher frequencies are used by bats so they can resolve targets smaller than 17 mm. Wavelengths in audible sound are much longer than those in visible light.

Standing wavesedit

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 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 boundary conditions) determining which wavelengths are allowed. For example, for an electromagnetic wave, if the box has ideal metal walls, the condition for nodes at the walls results because the metal walls cannot support a tangential electric field, forcing the wave to have zero amplitude at the wall.

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 speed of light can be determined from observation of standing waves in a metal box containing an ideal vacuum.

Mathematical representationedit

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

${\displaystyle y(x,t)=A\cos <!--\; \; -->\left(2\mathrm{\pi <!--\; \pi \; -->}\left(\frac{x}{\mathrm{\lambda <!--\; \lambda \; -->}}-<!--\; -\; -->ft\right)\right)=A\cos <!--\; \; -->\left(\frac{2\mathrm{\pi <!--\; \pi \; -->}}{\mathrm{\lambda <!--\; \lambda \; -->}}(x-<!--\; -\; -->vt)\right)}$

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 wavenumber k (2π times the reciprocal of wavelength) and angular frequency ω (2π times the frequency) as:

${\displaystyle y(x,t)=A\cos <!--\; \; -->\left(kx-<!--\; -\; -->\mathrm{\omega <!--\; \omega \; -->}t\right)=A\cos <!--\; \; -->\left(k(x-<!--\; -\; -->vt)\right)}$

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

${\displaystyle k=\frac{2\mathrm{\pi <!--\; \pi \; -->}}{\mathrm{\lambda <!--\; \lambda \; -->}}=\frac{2\mathrm{\pi <!--\; \pi \; -->}f}{v}=\frac{\mathrm{\omega <!--\; \omega \; -->}}{v},}$

or

${\displaystyle \mathrm{\lambda <!--\; \lambda \; -->}=\frac{2\mathrm{\pi <!--\; \pi \; -->}}{k}=\frac{2\mathrm{\pi <!--\; \pi \; -->}v}{\mathrm{\omega <!--\; \omega \; -->}}=\frac{v}{f}.}$

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

${\displaystyle A{e}^{i\left(kx-<!--\; -\; -->\mathrm{\omega <!--\; \omega \; -->}t\right)}.}$

General mediaedit

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 vacuum, 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.

This change in speed upon entering a medium causes refraction, or a change in direction of waves that encounter the interface between media at an angle. For electromagnetic waves, this change in the angle of propagation is governed by Snell's law.

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.

For electromagnetic waves the speed in a medium is governed by its refractive index according to

${\displaystyle v=\frac{c}{n({\mathrm{\lambda <!--\; \lambda \; -->}}_0)},}$

where c is the speed of light in vacuum and n(λ₀) is the refractive index of the medium at wavelength λ₀, where the latter is measured in vacuum rather than in the medium. The corresponding wavelength in the medium is

${\displaystyle \mathrm{\lambda <!--\; \lambda \; -->}=\frac{{\mathrm{\lambda <!--\; \lambda \; -->}}_0}{n({\mathrm{\lambda <!--\; \lambda \; -->}}_0)}.}$

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 specified medium.

The variation in speed of light with wavelength is known as dispersion, and is also responsible for the familiar phenomenon in which light is separated into component colors by a prism. 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 refract at different angles. The mathematical relationship that describes how the speed of light within a medium varies with wavelength is known as a dispersion relation.

Nonuniform mediaedit

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.

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 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 with a high loss and the wave dies out.

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. 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.

Crystalsedit

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. 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.

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.