[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki21\/one-way-wave-equation-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki21\/one-way-wave-equation-wikipedia\/","headline":"One-way wave equation – Wikipedia","name":"One-way wave equation – Wikipedia","description":"before-content-x4 From Wikipedia, the free encyclopedia after-content-x4 Differential equation important in physics A one-way wave equation is a first-order partial","datePublished":"2021-09-13","dateModified":"2021-09-13","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki21\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki21\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6f3ad777d0973931376adc6c0f20df29ccb5cbe7","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6f3ad777d0973931376adc6c0f20df29ccb5cbe7","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki21\/one-way-wave-equation-wikipedia\/","wordCount":9067,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4From Wikipedia, the free encyclopedia       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Differential equation important in physicsA one-way wave equation is a first-order partial differential equation describing one wave traveling in a direction defined by the vector wave velocity. It contrasts with the second-order two-way wave equation describing a standing wavefield resulting from superposition of two waves in opposite directions.[1][2][3] In the one-dimensional case, the one-way wave equation allows wave propagation to be calculated without the mathematical complication of solving a 2nd order differential equation. Due to the fact that in the last decades no 3D one-way wave equation could be found numerous approximation methods based on the 1D one-way wave equation are used for 3D seismic and other geophysical calculations, see also the section \u00a7\u00a0Three-dimensional case.[4][5][1][6]       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Table of ContentsOne-dimensional case[edit]Three-dimensional case[edit]Inhomogeneous media[edit]Further mechanical and electromagnetic waves[edit]See also[edit]References[edit]One-dimensional case[edit]The scalar second-order (two-way) wave equation describing a standing wavefield can be written as:\u22022s\u2202t2\u2212c2\u22022s\u2202x2=0,{displaystyle {frac {partial ^{2}s}{partial t^{2}}}-c^{2}{frac {partial ^{2}s}{partial x^{2}}}=0,}where        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4x{displaystyle x} is the coordinate, t{displaystyle t} is time, s=s(x,t){displaystyle s=s(x,t)} is the displacement, and c{displaystyle c} is the wave velocity.Due to the ambiguity in the direction of the wave velocity, c2=(+c)2=(\u2212c)2{displaystyle c^{2}=(+c)^{2}=(-c)^{2}}, the equation does not contain information about the wave direction and therefore has solutions propagating in both the forward (+x{displaystyle +x}) and backward (\u2212x{displaystyle -x}) directions. The general solution of the equation is the summation of the solutions in these two directions is:s(x,t)=s+(t\u2212x\/c)+s\u2212(t+x\/c){displaystyle s(x,t)=s_{+}(t-x\/c)+s_{-}(t+x\/c)}where s+{displaystyle s_{+}} and s\u2212{displaystyle s_{-}} are the displacement amplitudes of the waves running in +c{displaystyle +c} and \u2212c{displaystyle -c} direction.When a one-way wave problem is formulated, the wave propagation direction has to be (manually) selected by keeping one of the two terms in the general solution.Factoring the operator on the left side of the equation yields a pair of one-way wave equations, one with solutions that propagate forwards and the other with solutions that propagate backwards.[7][8][9](\u22022\u2202t2\u2212c2\u22022\u2202x2)s=(\u2202\u2202t\u2212c\u2202\u2202x)(\u2202\u2202t+c\u2202\u2202x)s=0,{displaystyle left({partial ^{2} over partial t^{2}}-c^{2}{partial ^{2} over partial x^{2}}right)s=left({partial  over partial t}-c{partial  over partial x}right)left({partial  over partial t}+c{partial  over partial x}right)s=0,}The forward- and backward-travelling waves are described respectively,\u2202s\u2202t\u2212c\u2202s\u2202x=0\u2202s\u2202t+c\u2202s\u2202x=0{displaystyle {begin{aligned}&{{frac {partial s}{partial t}}-c{frac {partial s}{partial x}}=0}\\[6pt]&{{frac {partial s}{partial t}}+c{frac {partial s}{partial x}}=0}end{aligned}}}The one-way wave equations can also be physically derived directly from specific acoustic impedance.In a longitudinal plane wave, the specific impedance determines the local proportionality of pressure p=p(x,t){displaystyle p=p(x,t)} and particle velocity v=v(x,t){displaystyle v=v(x,t)}:[10]pv=\u03c1c,{displaystyle {frac {p}{v}}=rho c,}with \u03c1{displaystyle rho } = density.The conversion of the impedance equation leads to:[3]v\u22121\u03c1cp=0{displaystyle v-{frac {1}{rho c}}p=0}(\u204e)A longitudinal plane wave of angular frequency \u03c9{displaystyle omega } has the displacement s=s(x,t){displaystyle s=s(x,t)}.The pressure p{displaystyle p} and the particle velocity v{displaystyle v} can be expressed in terms of the displacement s{displaystyle s} (E{displaystyle E}: Elastic Modulus)[11][better\u00a0source\u00a0needed]:p:=E\u2202s\u2202x{displaystyle p:=E{partial s over partial x}} for the 1D case this is in full analogy to stress \u03c3{displaystyle sigma } in mechanics: \u03c3=E\u03b5{displaystyle sigma =Evarepsilon }, with strain being defined as \u03b5=\u0394LL{displaystyle varepsilon ={frac {Delta L}{L}}} [12]v=\u2202s\u2202t{displaystyle v={partial s over partial t}}These relations inserted into the equation above (\u204e) yield:\u2202s\u2202t\u2212E\u03c1c\u2202s\u2202x=0{displaystyle {partial s over partial t}-{E over rho c}{partial s over partial x}=0}With the local wave velocity definition (speed of sound):c=E(x)\u03c1(x)\u21d4c=E\u03c1c{displaystyle c={sqrt {E(x) over rho (x)}}Leftrightarrow c={E over rho c}}directly(!) follows the 1st-order partial differential equation of the one-way wave equation:\u2202s\u2202t\u2212c\u2202s\u2202x=0{displaystyle {{frac {partial s}{partial t}}-c{frac {partial s}{partial x}}=0}}The wave velocity c{displaystyle c} can be set within this wave equation as +c{displaystyle +c} or \u2212c{displaystyle -c} according to the direction of wave propagation.For wave propagation in the direction of +c{displaystyle +c} the unique solution iss(x,t)=s+(t\u2212x\/c){displaystyle s(x,t)=s_{+}(t-x\/c)}and for wave propagation in the \u2212c{displaystyle -c} direction the respective solution is[13]s(x,t)=s\u2212(t+x\/c){displaystyle s(x,t)=s_{-}(t+x\/c)}There also exists a spherical one-way wave equation describing the wave propagation of a monopole sound source in spherical coordinates, i.e., in radial direction. By a modification of the radial nabla operator an inconsistency between spherical divergence and Laplace operators is solved and the resulting solution does not show Bessel functions (in contrast to the known solution of the conventional two-way approach).[6]Three-dimensional case[edit]The one-way equation and solution in the three-dimensional case was assumed to be similar way as for the one-dimensional case by a mathematical decomposition (factorization) of a 2nd order differential equation.[14] In fact, the 3D One-way wave equation can be derived from first principles: a) derivation from impedance theorem [3] and  b) derivation from a tensorial impulse flow equilibrium in a field point.[6]Inhomogeneous media[edit]For inhomogeneous media with location-dependent elasticity module E(x){displaystyle E(x)}, density \u03c1(x){displaystyle rho (x)} and wave velocity c(x){displaystyle c(x)} an analytical solution of the one-way wave equation can be derived by introduction of a new field variable.[9]Further mechanical and electromagnetic waves[edit]The method of PDE factorization can also be transferred to other 2nd or 4th order wave equations, e.g. transversal, and string, Moens\/Korteweg, bending, and electromagnetic wave equations and electromagnetic waves.[9]See also[edit]References[edit]^ a b Angus, D. A. (2014-03-01). “The One-Way Wave Equation: A Full-Waveform Tool for Modeling Seismic Body Wave Phenomena” (PDF). Surveys in Geophysics. 35 (2): 359\u2013393. Bibcode:2014SGeo…35..359A. doi:10.1007\/s10712-013-9250-2. ISSN\u00a01573-0956. S2CID\u00a0121469325.^ Trefethen, L N. “19. One-way wave equations” (PDF).^ a b c Bschorr, Oskar; Raida, Hans-Joachim (March 2020). “One-Way Wave Equation Derived from Impedance Theorem”. Acoustics. 2 (1): 164\u2013170. doi:10.3390\/acoustics2010012.^ Qiqiang, Yang (2012-01-01). “Forward Modeling of the One-Way Acoustic Wave Equation by the Hartley Method”. Procedia Environmental Sciences. 2011 International Conference of Environmental Science and Engineering. 12: 1116\u20131121. doi:10.1016\/j.proenv.2012.01.396. ISSN\u00a01878-0296.^ Zhang, Yu; Zhang, Guanquan; Bleistein, Norman (September 2003). “True amplitude wave equation migration arising from true amplitude one-way wave equations”. Inverse Problems. 19 (5): 1113\u20131138. Bibcode:2003InvPr..19.1113Z. doi:10.1088\/0266-5611\/19\/5\/307. ISSN\u00a00266-5611. S2CID\u00a0250860035.^ a b c Bschorr, Oskar; Raida, Hans-Joachim (March 2021). “Spherical One-Way Wave Equation”. Acoustics. 3 (2): 309\u2013315. doi:10.3390\/acoustics3020021.^ Baysal, Edip; Kosloff, Dan D.; Sherwood, J. W. C. (February 1984), “A two\u2010way nonreflecting wave equation”, Geophysics, vol.\u00a049, no.\u00a02, pp.\u00a0132\u2013141, Bibcode:1984Geop…49..132B, doi:10.1190\/1.1441644, ISSN\u00a00016-8033^ Angus, D. A. (2013-08-17), “The One-Way Wave Equation: A Full-Waveform Tool for Modeling Seismic Body Wave Phenomena” (PDF), Surveys in Geophysics, vol.\u00a035, no.\u00a02, pp.\u00a0359\u2013393, Bibcode:2014SGeo…35..359A, doi:10.1007\/s10712-013-9250-2, ISSN\u00a00169-3298, S2CID\u00a0121469325^ a b c Bschorr, Oskar; Raida, Hans-Joachim (December 2021). “Factorized One-Way Wave Equations”. Acoustics. 3 (4): 717\u2013722. doi:10.3390\/acoustics3040045.^ “Sound – Impedance”. Encyclopedia Britannica. Retrieved 2021-05-20.^ “elastic modulus”. Encyclopedia Britannica. Retrieved 2021-12-15.^ “Young’s modulus | Description, Example, & Facts”. Encyclopedia Britannica. Retrieved 2021-05-20.^ “Wave Equation–1-Dimensional”.^ The mathematics of PDEs and the wave equation https:\/\/mathtube.org\/sites\/default\/files\/lecture-notes\/Lamoureux_Michael.pdf       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki21\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki21\/one-way-wave-equation-wikipedia\/#breadcrumbitem","name":"One-way wave equation – Wikipedia"}}]}]