En física , un vector de onda (también deletreado vector de onda ) es un vector que ayuda a describir una onda . Como cualquier vector, tiene una magnitud y una dirección , las cuales son importantes. Su magnitud es el número de onda o el número de onda angular de la onda (inversamente proporcional a la longitud de onda ), y su dirección es normalmente la dirección de propagación de la onda (pero no siempre, ver más abajo ).
En el contexto de la relatividad especial, el vector de onda también se puede definir como un cuatro-vector .
Definiciones
Hay dos definiciones comunes de vector de onda, que difieren en un factor de 2π en sus magnitudes. Se prefiere una definición en física y campos relacionados, mientras que la otra definición se prefiere en cristalografía y campos relacionados. [1] En este artículo, se denominarán "definición de física" y "definición de cristalografía", respectivamente.
En las dos definiciones siguientes, la magnitud del vector de onda está representada por ; la dirección del vector de onda se analiza en la siguiente sección.
Definición de física
Una onda viajera unidimensional perfecta sigue la ecuación:
dónde:
- x es la posición,
- t es el tiempo,
- (una función de x y t ) es la perturbación que describe la ola (por ejemplo, para una ola del océano ,sería el exceso de altura del agua, o para una onda de sonido ,sería el exceso de presión de aire ).
- A es la amplitud de la onda (la magnitud máxima de la oscilación),
- es un desfase que describe cómo dos ondas pueden estar desincronizadas entre sí,
- es la frecuencia angular temporal de la onda, que describe cuántas oscilaciones completa por unidad de tiempo y está relacionada con el período por la ecuación ,
- is the spatial angular frequency (wavenumber) of the wave, describing how many oscillations it completes per unit of space, and related to the wavelength by the equation .
is the magnitude of the wave vector. In this one-dimensional example, the direction of the wave vector is trivial: this wave travels in the +x direction with speed (more specifically, phase velocity) . In a multidimensional system, the scalar would be replaced by the vector dot product , representing the wave vector and the position vector, respectively.
Crystallography definition
In crystallography, the same waves are described using slightly different equations.[2] In one and three dimensions respectively:
The differences between the above two definitions are:
- The angular frequency is used in the physics definition, while the frequency is used in the crystallography definition. They are related by . This substitution is not important for this article, but reflects common practice in crystallography.
- The wavenumber and wave vector k are defined differently: in the physics definition above, , while in the crystallography definition below, .
The direction of k is discussed in the next section.
Dirección del vector de onda
The direction in which the wave vector points must be distinguished from the "direction of wave propagation". The "direction of wave propagation" is the direction of a wave's energy flow, and the direction that a small wave packet will move, i.e. the direction of the group velocity. For light waves, this is also the direction of the Poynting vector. On the other hand, the wave vector points in the direction of phase velocity. In other words, the wave vector points in the normal direction to the surfaces of constant phase, also called wavefronts.
In a lossless isotropic medium such as air, any gas, any liquid, amorphous solids (such as glass), and cubic crystals the direction of the wavevector is exactly the same as the direction of wave propagation. If the medium is anisotropic, the wave vector in general points in directions other than that of the wave propagation. The condition for the wave vector to point in the same direction in which the wave propagates is that the wave has to be homogeneous, which isn't necessarily satisfied when the medium is anisotropic. In a homogeneous wave, the surfaces of constant phase are also surfaces of constant amplitude. In case of heterogeneous waves, these two species of surfaces differ in orientation. The wave vector is always perpendicular to surfaces of constant phase.
For example, when a wave travels through an anisotropic medium, such as light waves through an asymmetric crystal or sound waves through a sedimentary rock, the wave vector may not point exactly in the direction of wave propagation.[3][4]
En física del estado sólido
In solid-state physics, the "wavevector" (also called k-vector) of an electron or hole in a crystal is the wavevector of its quantum-mechanical wavefunction. These electron waves are not ordinary sinusoidal waves, but they do have a kind of envelope function which is sinusoidal, and the wavevector is defined via that envelope wave, usually using the "physics definition". See Bloch's theorem for further details.[5]
En relatividad especial
A moving wave surface in special relativity may be regarded as a hypersurface (a 3D subspace) in spacetime, formed by all the events passed by the wave surface. A wavetrain (denoted by some variable X) can be regarded as a one-parameter family of such hypersurfaces in spacetime. This variable X is a scalar function of position in spacetime. The derivative of this scalar is a vector that characterizes the wave, the four-wavevector.[6]
The four-wavevector is a wave four-vector that is defined, in Minkowski coordinates, as:
where the angular frequency is the temporal component, and the wavenumber vector is the spatial component.
Alternately, the wavenumber can be written as the angular frequency divided by the phase-velocity , or in terms of inverse period and inverse wavelength .
When written out explicitly its contravariant and covariant forms are:
In general, the Lorentz scalar magnitude of the wave four-vector is:
The four-wavevector is null for massless (photonic) particles, where the rest mass
An example of a null four-wavevector would be a beam of coherent, monochromatic light, which has phase-velocity
- {for light-like/null}
which would have the following relation between the frequency and the magnitude of the spatial part of the four-wavevector:
- {for light-like/null}
The four-wavevector is related to the four-momentum as follows:
The four-wavevector is related to the four-frequency as follows:
The four-wavevector is related to the four-velocity as follows:
Lorentz transformation
Taking the Lorentz transformation of the four-wavevector is one way to derive the relativistic Doppler effect. The Lorentz matrix is defined as
In the situation where light is being emitted by a fast moving source and one would like to know the frequency of light detected in an earth (lab) frame, we would apply the Lorentz transformation as follows. Note that the source is in a frame Ss and earth is in the observing frame, Sobs. Applying the Lorentz transformation to the wave vector
and choosing just to look at the component results in
where is the direction cosine of wrt
So
Source moving away (redshift)
As an example, to apply this to a situation where the source is moving directly away from the observer (), this becomes:
Source moving towards (blueshift)
To apply this to a situation where the source is moving straight towards the observer (), this becomes:
Source moving tangentially (transverse Doppler effect)
To apply this to a situation where the source is moving transversely with respect to the observer (), this becomes:
Ver también
- Plane wave expansion
- Plane of incidence
Referencias
- ^ Physics definition example:Harris, Benenson, Stöcker (2002). Handbook of Physics. p. 288. ISBN 978-0-387-95269-7.CS1 maint: multiple names: authors list (link). Crystallography definition example: Vaĭnshteĭn (1994). Modern Crystallography. p. 259. ISBN 978-3-540-56558-1.
- ^ Vaĭnshteĭn, Boris Konstantinovich (1994). Modern Crystallography. p. 259. ISBN 978-3-540-56558-1.
- ^ Fowles, Grant (1968). Introduction to modern optics. Holt, Rinehart, and Winston. p. 177.
- ^ "This effect has been explained by Musgrave (1959) who has shown that the energy of an elastic wave in an anisotropic medium will not, in general, travel along the same path as the normal to the plane wavefront...", Sound waves in solids by Pollard, 1977. link
- ^ Donald H. Menzel (1960). "§10.5 Bloch wave". Fundamental Formulas of Physics, Volume 2 (Reprint of Prentice-Hall 1955 2nd ed.). Courier-Dover. p. 624. ISBN 978-0486605968.
- ^ Wolfgang Rindler (1991). "§24 Wave motion". Introduction to Special Relativity (2nd ed.). Oxford Science Publications. pp. 60–65. ISBN 978-0-19-853952-0.
Otras lecturas
- Brau, Charles A. (2004). Modern Problems in Classical Electrodynamics. Oxford University Press. ISBN 978-0-19-514665-3.