In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), and its direction is perpendicular to the wavefront. In isotropic media, this is also the direction of wave propagation.
A closely related vector is the angular wave vector (or angular wavevector), with a typical unit being radian per metre. The wave vector and angular wave vector are related by a fixed constant of proportionality, 2π radians per cycle.[a]
It is common in several fields of physics to refer to the angular wave vector simply as the wave vector, in contrast to, for example, crystallography.[1][2] It is also common to use the symbol k for whichever is in use.
In the context of special relativity, wave vector can refer to a four-vector, in which the (angular) wave vector and (angular) frequency are combined.
The terms wave vector and angular wave vector have distinct meanings. Here, the wave vector is denoted by ν ν --> ~ ~ --> {\displaystyle {\tilde {\boldsymbol {\nu }}}} and the wavenumber by ν ν --> ~ ~ --> = | ν ν --> ~ ~ --> | {\displaystyle {\tilde {\nu }}=\left|{\tilde {\boldsymbol {\nu }}}\right|} . The angular wave vector is denoted by k and the angular wavenumber by k = |k|. These are related by k = 2 π π --> ν ν --> ~ ~ --> {\displaystyle \mathbf {k} =2\pi {\tilde {\boldsymbol {\nu }}}} .
A sinusoidal traveling wave follows the equation
where:
The equivalent equation using the wave vector and frequency is[3]
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 in vacuum, 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 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 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.[4][5]
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.[6]
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.[7]
The four-wavevector is a wave four-vector that is defined, in Minkowski coordinates, as:
where the angular frequency ω ω --> c {\displaystyle {\tfrac {\omega }{c}}} is the temporal component, and the wavenumber vector k → → --> {\displaystyle {\vec {k}}} is the spatial component.
Alternately, the wavenumber k can be written as the angular frequency ω divided by the phase-velocity vp, or in terms of inverse period T and inverse wavelength λ.
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 m o = 0 {\displaystyle m_{o}=0}
An example of a null four-wavevector would be a beam of coherent, monochromatic light, which has phase-velocity v p = c {\displaystyle v_{p}=c}
which would have the following relation between the frequency and the magnitude of the spatial part of the four-wavevector:
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:
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 μ μ --> = 0 {\displaystyle \mu =0} component results in
where cos --> θ θ --> {\displaystyle \cos \theta } is the direction cosine of k 1 {\displaystyle k^{1}} with respect to k 0 , k 1 = k 0 cos --> θ θ --> . {\displaystyle k^{0},k^{1}=k^{0}\cos \theta .}
So
As an example, to apply this to a situation where the source is moving directly away from the observer ( θ θ --> = π π --> {\displaystyle \theta =\pi } ), this becomes:
To apply this to a situation where the source is moving straight towards the observer (θ = 0), this becomes:
To apply this to a situation where the source is moving transversely with respect to the observer (θ = π/2), this becomes:
{{cite book}}
Questa voce o sezione sugli argomenti film d'azione e film thriller non cita le fonti necessarie o quelle presenti sono insufficienti. Puoi migliorare questa voce aggiungendo citazioni da fonti attendibili secondo le linee guida sull'uso delle fonti. Segui i suggerimenti dei progetti di riferimento 1, 2. Kill Bill: Volume 1La Sposa (Uma Thurman) mentre si prepara ad affrontare gli 88 FolliLingua originaleinglese, giapponese, francese Paese di produzioneStati Uniti d'America Anno2003 Durata1…
Lambang Jawa BaratDetailDigunakan sejak1961PerisaiHijau, kujang Putih berhulu Merah, dikelilingi padi 17 butir dan kapas 8 kuntum tiga daun kuning warna alam, muncul pada jajaran pegunungan Hitam. Di bawah gambar bendungan, di sebelah kiri gelombang biru 3 dan putih 2 (total 5), serta di sebelah kanan kotak-kotak biru dan putih 6 × 5Mottogemah ripah, répéh rapih tulisan Hitam pita Kuning. Prangko Lambang Provinsi Jawa Barat Lambang Jawa Barat terdiri dari sebuah perisai berbentuk bulat telur …
American politician (born 1954) For the baseball player, see Hank Johnson (baseball). For others with a similar name, see Henry Johnson (disambiguation). Hank JohnsonMember of the U.S. House of Representativesfrom Georgia's 4th districtIncumbentAssumed office January 3, 2007Preceded byCynthia McKinney Personal detailsBornHenry Calvin Johnson Jr. (1954-10-02) October 2, 1954 (age 69)Washington, D.C., U.S.Political partyDemocraticSpouseMereda DavisChildren2Residence(s)Lithonia…
1989 live album by Cecil Taylor & Han BenninkSpots, Circles, and FantasyLive album by Cecil Taylor & Han BenninkReleased1989RecordedJuly 10, 1988GenreFree jazzLabelFMPCecil Taylor chronology Pleistozaen Mit Wasser(1989) Spots, Circles, and Fantasy(1989) Legba Crossing(1989) Spots, Circles, and Fantasy is a live album featuring a performance by Cecil Taylor and Han Bennink recorded in Berlin on July 10, 1988 as part of month-long series of concerts by Taylor and released on the FM…
Xianbei-led dynasty of China Bei Zhou redirects here. For the historical prefecture, see Bei Prefecture. Zhou周557–581KyrgyzsCHAM-PA576CHENLAFIRST TURKIC KHAGANATESASANIANEMPIREALCHONHUNSCHALU-KYASLATERGUPTASNORTH.ZHOUNORTH.QIZHANGZHUNGCHENBYZANTINEEMPIREAVARKHAGANATETUYUHUNKhitansPaleo-SiberiansTungusGOGU-RYEOTOCHA-RIANS ◁ ▷ The Northern Zhou ( ) and main contemporary polities in Asia c. 576 Northern Zhou territories in light blueCapitalChang'anGovernmentMonarchyEmperor …
Artikel ini sebatang kara, artinya tidak ada artikel lain yang memiliki pranala balik ke halaman ini.Bantulah menambah pranala ke artikel ini dari artikel yang berhubungan atau coba peralatan pencari pranala.Tag ini diberikan pada Januari 2023. Artikel ini tidak memiliki referensi atau sumber tepercaya sehingga isinya tidak bisa dipastikan. Tolong bantu perbaiki artikel ini dengan menambahkan referensi yang layak. Tulisan tanpa sumber dapat dipertanyakan dan dihapus sewaktu-waktu.Cari sumber:…
Relationship between Buddhism and Hinduism Hinduism and Buddhism redirects here. For the book, see The Religion of India: The Sociology of Hinduism and Buddhism. Hinduismand other religions Indian religions Buddhism Jainism Sikhism Abrahamic religions Baháʼí Faith Christianity Islam Judaism Hinduism and... Abortion Apostasy Cattle Blasphemy LGBT Science Theosophy Hinduism portalvtePart of a series onHinduism Hindus History Timeline Origins History Indus Valley Civilisation Historica…
Canoeing governing body in Europe European Canoe AssociationAbbreviationECAFormation1993; 30 years ago (1993)TypeSports federationHeadquartersZagrebMembership 45 member countriesPresidentJean ZoungranaWebsitewww.canoe-europe.org/ The European Canoe Association (ECA) is the umbrella organization for canoeing sport in Europe. It was founded in Rome, Italy, on December 11, 1993.[1] The organization has 45 member countries.[2] The association was recognized by the I…
هذه المقالة يتيمة إذ تصل إليها مقالات أخرى قليلة جدًا. فضلًا، ساعد بإضافة وصلة إليها في مقالات متعلقة بها. (سبتمبر 2018) دينيس جرين (بالإنجليزية: Dennis Allan Green) معلومات شخصية الميلاد 26 مايو 1931[1] سيدني[1] الوفاة 5 سبتمبر 2018 (87 سنة) [2] سيدني[1] سبب الوفاة…
Este artículo se refiere o está relacionado con un evento de salud pública reciente o actualmente en curso. La información de este artículo puede cambiar frecuentemente. Por favor, no agregues datos especulativos y recuerda colocar referencias a fuentes fiables para dar más detalles. Pandemia de COVID-19 en Mongolia Parte de la Pandemia de COVID-19 Agente patógenoPatógeno SARS-CoV-2Tipo de patógeno VirusEnfermedad causada COVID-19Datos históricosInicio 10 de marzo de 2020 (3 años,…
Never Let You Go: Shindemo HanasanaiNever Let You Go ~死んでも離さない~Sampul edisi regulerLagu oleh 2AMdari album VoiceSisi-BWaratte Agerarenakute GomenDirilis11 Januari 2012 (2012-01-11)FormatCD single, Unduhan digitalDirekam2011GenrePop, dance-pop, J-popDurasi3:16LabelAriola Japan Never Let You Go: Shindemo Hanasanai (Never Let You Go ~死んでも離さない~) adalah singel berbahasa Jepang pertama dari boy band asal Korea Selatan 2AM. Album ini dirilis pada tanggal 11 …
Rhode Island LGBT court case 1980 Fricke v. LynchCourtUnited States District Court for the District of Rhode IslandFull case nameAaron Fricke v. Richard B. Lynch, in his official capacity as Principal of Cumberland High School DecidedMay 28, 1980Docket nos.Civ. A. No. 80-214Citation(s)491 F. Supp. 381Court membershipJudge(s) sittingRaymond James PettineKeywordsLGBT rights Fricke v. Lynch, 491 F. Supp. 381 (D.R.I. 1980), was a decision in the United States District Court for the District of Rhode…
Untuk tempat lain yang bernama sama, lihat Kandangan. KandanganNagoriGapura selamat datang di Nagori KandanganNegara IndonesiaProvinsiSumatera UtaraKabupatenSimalungunKecamatanPematang BandarKode pos21186Luas... km²Jumlah penduduk... jiwaKepadatan... jiwa/km² Kandangan merupakan salah satu nagori yang ada di kecamatan Pematang Bandar, kabupaten Simalungun, provinsi Sumatera Utara, Indonesia. Kantor Pangulu Nagori Kandangan lbsKecamatan Pematang Bandar, Kabupaten Simalungun, Sumatera Utara…
1-PentanolNombre IUPAC 1-Pentanol, Pentan-1-olGeneralOtros nombres n-pentanol, alcohol amílico normalFórmula semidesarrollada CH3CH2CH2CH2CH2OHFórmula estructural Ver imagenFórmula molecular C5H12OIdentificadoresNúmero CAS 71-41-0[1]Número RTECS SB9800000ChEBI 44884ChEMBL CHEMBL14568ChemSpider 6040PubChem 6276UNII M9L931X26YKEGG C16834 InChIInChI=InChI=1S/C5H12O/c1-2-3-4-5-6/h6H,2-5H2,1H3Key: AMQJEAYHLZJPGS-UHFFFAOYSA-N Propiedades físicasApariencia IncoloroDensidad 814,4…
Recep Tayyip Erdoğan for PresidentCampaigned for2018 Turkish presidential electionCandidateRecep Tayyip ErdoğanIncumbent President of Turkey (2014–present)Prime Minister of Turkey (2003–2014)Mayor of Istanbul (1994–1998)AffiliationPeople's AllianceStatusAnnounced28 April 2018Official nominee3 May 2018 Recep Tayyip Erdoğan, incumbent President of Turkey since 2014, was officially nominated as the presidential candidate of the Justice and Development Party (AKP) on 3 May 2018.[1] …
Mosaik Romawi (220–250 M) dari El Djem, Tunisia (Afrika Romawi), dengan tulisan Latin Diam! Biarkan banteng-banteng itu tidur (Silentiu[m] dormiant tauri) dan perbincangan antara lima orang yang berpesta (kemungkinan gladiator):- Kita akan telanjang ([N]os nudi [f]iemus)- Kita datang untuk minum (Bibere venimus)- Kalian semua banyak omong (Ia[m] multu[m] loquimini)- Kita mungkin akan dipanggil (Avocemur)- Kita minum tiga [gelas] (Nos tres tenemus)[1] Bahasa-bahasa di Kekaisaran Romawi …
Grégory Coupet Informasi pribadiTanggal lahir 31 Desember 1972 (umur 50)Tempat lahir Le Puy-en-Velay, PrancisTinggi 1,81 m (5 ft 11 in)Posisi bermain Penjaga gawangKarier junior Olympique Le PuyKarier senior*Tahun Tim Tampil (Gol)1993–1997 Saint-Étienne 88 (0)1997–2008 Lyon 518 (0)2008–2009 Atlético Madrid 6 (0)2009–2011 Paris Saint-Germain 31 (0)Total 643 (0)Tim nasional‡2001–2008 Prancis 34 (0) * Penampilan dan gol di klub senior hanya dihitung dari liga dome…
Depression in the textile industry of North West England An 1862 newspaper illustration showing people queueing for food and coal tickets at a district Provident Society office The Lancashire Cotton Famine, also known as the Cotton Famine or the Cotton Panic (1861–65), was a depression in the textile industry of North West England, brought about by overproduction in a time of contracting world markets. It coincided with the interruption of baled cotton imports caused by the American Civil War …
2012 video game 2012 video gameFarming Simulator 2012The icon for the 3DS releaseDeveloper(s)Giants Software (3DS)Astragon (iOS)Publisher(s)EU: Excalibur Publishing (3DS)NA: Focus Home Interactive (3DS)WW: Astragon (iOS)WW: Giants Software (Android)SeriesFarming SimulatorPlatform(s)Nintendo 3DS, iOS, AndroidReleaseNintendo 3DSEU: March 30, 2012NA: June 23, 2013iOSJuly 30, 2012AndroidJanuary 9, 2013Genre(s)SimulationMode(s)Single-player Farming Simulator 2012 (Farming Simulator 3D for the U.S. 3D…
American gallerist, art collector and photographer Gloria KirbyBorn19 April 1928Nice, FranceDied2017Tangier, MoroccoOccupation(s)Galerist, art collector and photographerKnown forArt collector Gloria Price Kirby, (Nice, France, April 19, 1928 – Tangier, Morocco, 2017) was an American gallerist, art collector and photographer, she founded the Vandrés Gallery in Madrid. Biography Gloria Kirby was the granddaughter of Fred Morgan Kirby, the founder of the F. M. Kirby & Co. 5 & 10…
Lokasi Pengunjung: 3.12.149.139