Dark state

In physics, a wave vector (also spelled wavevector) is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave (inversely proportional to the wavelength), and its direction is ordinarily the direction of wave propagation (but not always, see below).

In the context of special relativity the wave vector can also be defined as a four-vector.

Definitions

DTZ's public sale group in Singapore auctions all forms of residential, workplace and retail properties, outlets, homes, lodges, boarding homes, industrial buildings and development websites. Auctions are at present held as soon as a month.

We will not only get you a property at a rock-backside price but also in an space that you've got longed for. You simply must chill out back after giving us the accountability. We will assure you 100% satisfaction. Since we now have been working in the Singapore actual property market for a very long time, we know the place you may get the best property at the right price. You will also be extremely benefited by choosing us, as we may even let you know about the precise time to invest in the Singapore actual property market.

The Hexacube is offering new ec launch singapore business property for sale Singapore investors want to contemplate. Residents of the realm will likely appreciate that they'll customize the business area that they wish to purchase as properly. This venture represents one of the crucial expansive buildings offered in Singapore up to now. Many investors will possible want to try how they will customise the property that they do determine to buy by means of here. This location has offered folks the prospect that they should understand extra about how this course of can work as well.

Singapore has been beckoning to traders ever since the value of properties in Singapore started sky rocketing just a few years again. Many businesses have their places of work in Singapore and prefer to own their own workplace area within the country once they decide to have a everlasting office. Rentals in Singapore in the corporate sector can make sense for some time until a business has discovered a agency footing. Finding Commercial Property Singapore takes a variety of time and effort but might be very rewarding in the long term.

is changing into a rising pattern among Singaporeans as the standard of living is increasing over time and more Singaporeans have abundance of capital to invest on properties. Investing in the personal properties in Singapore I would like to applaud you for arising with such a book which covers the secrets and techniques and tips of among the profitable Singapore property buyers. I believe many novice investors will profit quite a bit from studying and making use of some of the tips shared by the gurus." – Woo Chee Hoe Special bonus for consumers of Secrets of Singapore Property Gurus Actually, I can't consider one other resource on the market that teaches you all the points above about Singapore property at such a low value. Can you? Condominium For Sale (D09) – Yong An Park For Lease

In 12 months 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebrities of the property market. Models are snapped up in report time and at document breaking prices. Builders are having fun with overwhelming demand and patrons need more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp responsibility imposed on residential properties, buyers change their focus to commercial and industrial properties. I imagine every property purchasers need their property funding to understand in value.

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

Unfortunately, there are two common definitions of wave vector, which differ by a factor of 2π in their magnitudes. One definition is preferred in physics and related fields, while the other definition is preferred in crystallography and related fields. For this article, they will be called the "physics definition" and the "crystallography definition", respectively.

Physics definition

A perfect one-dimensional traveling wave follows the equation:

$\psi (x,t)=A\cos(kx-\omega t+\varphi )$

where:

• x is position,
• t is time,
• $\psi$ (a function of x and t) is the disturbance describing the wave (for example, for an ocean wave, $\psi$ would be the excess height of the water, or for a sound wave, $\psi$ would be the excess air pressure).
• A is the amplitude of the wave (the peak magnitude of the oscillation),
• $\varphi$ is a "phase offset" describing how two waves can be out of sync with each other,
• $\omega$ is the angular frequency of the wave, related to how quickly it oscillates at a given point,
• $k$ is the wavenumber (more specifically called angular wavenumber) of the wave, related to the wavelength by the equation $k=2\pi /\lambda$.

This wave travels in the +x direction with speed (more specifically, phase velocity) $\omega /k$.

Crystallography definition

In crystallography, the same waves are described using slightly different equations. In one and three dimensions respectively:

$\psi (x,t)=A\cos(2\pi (kx-\omega t)+\varphi )$

$\psi \left({\mathbf {r} },t\right)=A\cos \left(2\pi ({\mathbf {k} }\cdot {\mathbf {r} }-\nu t)+\varphi \right)$

The differences are:

• The frequency $\nu$ instead of angular frequency $\omega$ is used. They are related by $2\pi \nu =\omega$. This substitution is not important for this article, but reflects common practice in crystallography.
• The wavenumber k and wave vector k are defined in a different way. Here, $k=|{\mathbf {k} }|=1/\lambda$, while in the physics definition above, $k=|{\mathbf {k} }|=2\pi /\lambda$.

The direction of k is discussed below.

Direction of the wave vector

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. 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 wave fronts.

In a lossless isotropic medium such as air, any gas, any liquid, or some solids (such as glass), the direction of the wavevector is exactly the same as the direction of wave propagation. If the medium is lossy, the wave vector in general points in directions other than that of wave propagation. The condition for 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 lossy. In a homogeneous wave, the surfaces of constant phase are also surfaces of constant amplitude. In case of inhomogeneous waves, these two species of surfaces differ in orientation. Wave vector is always perpendicular to surfaces of constant phase.

However, 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.

In solid-state physics

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. 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 wave for further details.

In special relativity

A beam of coherent, monochromatic light can be characterized by the (null) wave 4-vector

$k^{\mu }=\left({\frac {\omega }{c}},{\vec {k}}\right)\,$

which, when written out explicitly in its contravariant and covariant forms is

$k^{\mu }=\left({\frac {\omega }{c}},k_{x},k_{y},k_{z}\right)\,$ and

$k_{\mu }=\left({\frac {\omega }{c}},-k_{x},-k_{y},-k_{z}\right).\,$

The null character of the wave 4-vector gives a relation between the frequency and the magnitude of the spatial part of the wave 4-vector:

$k^{\mu }k_{\mu }=\left({\frac {\omega }{c}}\right)^{2}-k_{x}^{2}-k_{y}^{2}-k_{z}^{2}\ =0$

The wave 4-vector is related to the four-momentum as follows:

$p^{\mu }=(E/c,p_{x},p_{y},p_{z})=(\hbar \omega /c,\hbar k_{x},\hbar k_{y},\hbar k_{z})=\hbar k^{\mu }$

Lorentz transformation

Taking the Lorentz transformation of the wave vector is one way to derive the relativistic Doppler effect. The Lorentz matrix is defined as

$\Lambda ={\begin{pmatrix}\gamma &-\beta \gamma &0&0\\-\beta \gamma &\gamma &0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}$

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 S^s and earth is in the observing frame, S^obs. Applying the lorentz transformation to the wave vector

$k_{s}^{\mu }=\Lambda _{\nu }^{\mu }k_{\mathrm {obs} }^{\nu }\,$

and choosing just to look at the $\mu =0$ component results in

$k_{s}^{0}=\Lambda _{0}^{0}k_{\mathrm {obs} }^{0}+\Lambda _{1}^{0}k_{\mathrm {obs} }^{1}+\Lambda _{2}^{0}k_{\mathrm {obs} }^{2}+\Lambda _{3}^{0}k_{\mathrm {obs} }^{3}\,$

${\frac {\omega _{s}}{c}}\,$	$=\gamma {\frac {\omega _{\mathrm {obs} }}{c}}-\beta \gamma k_{\mathrm {obs} }^{1}\,$
	$\quad =\gamma {\frac {\omega _{\mathrm {obs} }}{c}}-\beta \gamma {\frac {\omega _{\mathrm {obs} }}{c}}\cos \theta .\,$ where $\cos \theta \,$ is the direction cosine of $k^{1}$ wrt $k^{0},k^{1}=k^{0}\cos \theta .$

So

${\frac {\omega _{\mathrm {obs} }}{\omega _{s}}}={\frac {1}{\gamma (1-\beta \cos \theta )}}\,$

Source moving away

As an example, to apply this to a situation where the source is moving directly away from the observer ($\theta =\pi$), this becomes:

${\frac {\omega _{\mathrm {obs} }}{\omega _{s}}}={\frac {1}{\gamma (1+\beta )}}={\frac {\sqrt {1-\beta ^{2}}}{1+\beta }}={\frac {\sqrt {(1+\beta )(1-\beta )}}{1+\beta }}={\frac {\sqrt {1-\beta }}{\sqrt {1+\beta }}}\,$

Source moving towards

To apply this to a situation where the source is moving straight towards the observer ($\theta =0$), this becomes:

${\frac {\omega _{\mathrm {obs} }}{\omega _{s}}}={\frac {\sqrt {1+\beta }}{\sqrt {1-\beta }}}\,$

See also

• Plane wave expansion

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

Further reading

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534