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| {{distinguish|Wave function}}
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| The '''wave equation''' is an important second-order linear [[partial differential equation]] for the description of [[wave]]s – as they occur in [[physics]] – such as [[sound]] waves, [[light]] waves and [[water]] waves. It arises in fields like [[acoustics]], [[electromagnetism|electromagnetics]], and [[fluid dynamics]]. Historically, the problem of a vibrating string such as that of a [[musical instrument]] was studied by [[Jean le Rond d'Alembert]], [[Leonhard Euler]], [[Daniel Bernoulli]], and [[Joseph-Louis Lagrange]].<ref>[http://homes.chass.utoronto.ca/~cfraser/vibration.pdf {{cite journal|last1= Cannon |first1=John T.|last2=Dostrovsky|first2=Sigalia|title=The evolution of dynamics, vibration theory from 1687 to 1742|year=1981|volume= 6|series=Studies in the History of Mathematics and Physical Sciences|ISBN= 0-3879-0626-6|publisher=Springer-Verlag|location=New York|pages=ix + 184 pp.}}] {{cite journal|last= GRAY|first=JW|title=BOOK REVIEWS |journal=BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY |date=July 1983 |volume= 9| issue = 1}} (retrieved 13 Nov 2012).</ref><ref>Gerard F Wheeler. [http://www.scribd.com/doc/32298888/The-Vibrating-String-Controversy-Am-J-Phys-1987-v55-n1-p33-37 The Vibrating String Controversy,] (retrieved 13 Nov 2012). Am. J. Phys., 1987, v55, n1, p33-37.</ref><ref>For a special collection of the 9 groundbreaking papers by the three authors, see [http://www.lynge.com/item.php?bookid=38975&s_currency=EUR&c_sourcepage= First Appearance of the wave equation: D'Alembert, Leonhard Euler, Daniel Bernoulli. - the controversy about vibrating strings] (retrieved 13 Nov 2012). Herman HJ Lynge and Son.</ref><ref>For de Lagrange's contributions to the acoustic wave equation, can consult [http://books.google.co.uk/books?id=D8GqhULfKfAC&pg=PA18&lpg=PA18&dq=lagrange+paper+on+the+wave+equation&source=bl&ots=E-RPop_GGD&sig=aJ41g1nlDTDKUqvw9OAXFjjutV4&hl=en&sa=X&ei=KCPEUMaOCI2V0QXz5YC4DQ&ved=0CDQQ6AEwAQ#v=onepage&q=lagrange%20paper%20on%20the%20wave%20equation&f=false Acoustics: An Introduction to Its Physical Principles and Applications] Allan D. Pierce, Acoustical Soc of America, 1989; page 18.(retrieved 9 Dec 2012)</ref>
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| [[Image:Wave equation 1D fixed endpoints.gif|right|thumb|250px|A [[pulse (physics)|pulse]] traveling through a string with fixed endpoints as modeled by the wave equation.]]
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| [[Image:Spherical wave2.gif|frame|right|Spherical waves coming from a point source.]]
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| == Introduction ==
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| Wave equations are examples of [[hyperbolic partial differential equation]]s, but there are many variations.
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| In its simplest form, the wave equation concerns a time variable {{math|<var >t</var >}}, one or more spatial variables {{math|''x''<sub>1</sub>, ''x''<sub>2</sub>, …, ''x<sub>n</sub>''}}, and a [[scalar (mathematics)|scalar]] function {{math|''u'' {{=}} ''u'' (''x''<sub>1</sub>, ''x''<sub>2</sub>, …, ''x<sub>n</sub>''; ''t'')}}, whose values could model the [[Displacement (vector)|displacement]] of a wave. The wave equation for {{math|''u''}} is
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| :<math>{ \partial^2 u \over \partial t^2 } = c^2 \nabla^2 u </math>
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| where ∇<sup>2</sup> is the (spatial) [[Laplace operator|Laplacian]] and where ''c'' is a fixed [[coefficient|constant]].
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| Solutions of this equation that are initially zero outside some restricted region propagate out from the region at a fixed speed in all spatial directions, as do physical waves from a localized disturbance; the constant ''c'' is identified with the propagation speed of the wave. This equation is linear, as the sum of any two solutions is again a solution: in physics this property is called the [[superposition principle]].
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| The equation alone does not specify a solution; a unique solution is usually obtained by setting a problem with further conditions, such as [[initial conditions]], which prescribe the value and velocity of the wave. Another important class of problems specifies [[boundary conditions]], for which the solutions represent [[standing waves]], or [[harmonics]], analogous to the harmonics of musical instruments.
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| Variations of the wave equation are also found in [[elastic (solid mechanics)|elastic]], [[quantum mechanics]], [[plasma physics]] and [[general relativity]].
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| ==Scalar wave equation in one space dimension==
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| ===Derivation of the wave equation===
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| ====From Hooke's law====
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| The wave equation in the one dimensional case can be derived from [[Hooke's law]] in the following way: Imagine an array of little weights of mass ''m'' interconnected with massless springs of length ''h'' . The springs have a [[stiffness|spring constant]] of ''k'':
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| :[[Image:array of masses.svg|300px]]
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| Here ''u(x)'' measures the distance from the equilibrium of the mass situated at ''x''. The forces exerted on the mass ''m'' at the location ''x''+''h'' are:
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| :<math>F_{\mathit{Newton}}=m \cdot a(t)=m \cdot {{\partial^2 \over \partial t^2}u(x+h,t)}</math>
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| :<math>F_\mathit{Hooke} = F_{x+2h} - F_x = k \left [ {u(x+2h,t) - u(x+h,t)} \right ] - k[u(x+h,t) - u(x,t)]</math>
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| The equation of motion for the weight at the location ''x+h'' is given by equating these two forces:
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| :<math>m{\partial^2\over \partial t^2} u(x+h,t) = k[u(x+2h,t)-u(x+h,t)-u(x+h,t)+u(x,t)]</math>
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| where the time-dependence of ''u''(''x'') has been made explicit.
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| If the array of weights consists of ''N'' weights spaced evenly over the length ''L'' = ''Nh'' of total mass ''M'' = ''Nm'', and the total [[stiffness|spring constant]] of the array ''K'' = ''k''/''N'' we can write the above equation as:
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| :<math>{\partial^2 \over \partial t^2} u(x+h,t)={KL^2 \over M}{u(x+2h,t)-2u(x+h,t)+u(x,t) \over h^2}</math>
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| Taking the limit ''N'' → ∞, ''h'' → 0 and assuming smoothness one gets:
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| :<math> {\partial^2 u(x,t) \over \partial t^2}={KL^2 \over M}{ \partial^2 u(x,t) \over \partial x^2 } </math>
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| (''KL<sup>2</sup>)''/''M'' is the square of the propagation speed in this particular case.
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| [[File:1d wave equation animation.gif|thumbnail|1-d standing wave as a superposition of two waves traveling in opposite directions]]
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| ===General solution===
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| The one dimensional wave equation is unusual for a [[partial differential equation]] in that a relatively simple general solution may be found. Defining new variables:<ref>{{cite web | url = http://mathworld.wolfram.com/dAlembertsSolution.html | title = d'Alembert's Solution | author = [[Eric W. Weisstein]]| publisher = [[MathWorld]] | accessdate = 2009-01-21 }}</ref>
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| :<math>\xi = x - c t \quad ; \quad \eta = x + c t</math>
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| changes the wave equation into
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| :<math>\frac{\partial^2 u}{\partial \xi \partial \eta} = 0</math>
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| which leads to the general solution
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| :<math>u(\xi, \eta) = F(\xi) + G(\eta)</math>
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| or equivalently:
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| :<math>u(x, t) = F(x - c t) + G(x + c t)</math>
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| In other words, solutions of the 1D wave equation are sums of a right traveling function ''F'' and a left traveling function ''G''. "Traveling" means that the shape of these individual arbitrary functions with respect to ''x'' stays constant, however the functions are translated left and right with time at the speed ''c''. This was derived by [[Jean le Rond d'Alembert]].<ref>D'Alembert (1747) [http://books.google.com/books?id=lJQDAAAAMAAJ&pg=PA214#v=onepage&q&f=false "Recherches sur la courbe que forme une corde tenduë mise en vibration"] (Researches on the curve that a tense cord forms [when] set into vibration), ''Histoire de l'académie royale des sciences et belles lettres de Berlin'', vol. 3, pages 214-219.
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| * See also: D'Alembert (1747) [http://books.google.com/books?id=lJQDAAAAMAAJ&pg=PA220#v=onepage&q&f=false "Suite des recherches sur la courbe que forme une corde tenduë mise en vibration"] (Further researches on the curve that a tense cord forms [when] set into vibration), ''Histoire de l'académie royale des sciences et belles lettres de Berlin'', vol. 3, pages 220-249.
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| *See also: D'Alembert (1750) [http://books.google.com/books?id=m5UDAAAAMAAJ&pg=PA355#v=onepage&q&f=false "Addition au mémoire sur la courbe que forme une corde tenduë mise en vibration,"] ''Histoire de l'académie royale des sciences et belles lettres de Berlin'', vol. 6, pages 355-360.</ref>
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| Another way to arrive at this result is to note that the wave equation may be "factored":
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| :<math>\left[\frac{\part}{\part t} - c\frac{\part}{\part x}\right] \left[ \frac{\part}{\part t} + c\frac{\part}{\part x}\right] u = 0</math>
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| and therefore:
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| :<math>\frac{\part u}{\part t} - c\frac{\part u}{\part x} = 0 \qquad \mbox{and} \qquad \frac{\part u}{\part t} + c\frac{\part u}{\part x} = 0</math>
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| These last two equations are [[advection equation]]s, one left traveling and one right, both with constant speed ''c''.
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| For an initial value problem, the arbitrary functions ''F'' and ''G'' can be determined to satisfy initial conditions:
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| :<math>u(x,0)=f(x) \,</math>
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| :<math>u_t(x,0)=g(x) \,</math>
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| The result is [[d'Alembert's formula]]:
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| :<math>u(x,t) = \frac{f(x-ct) + f(x+ct)}{2} + \frac{1}{2c} \int_{x-ct}^{x+ct} g(s) ds</math>
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| In the classical sense if ''f''(''x'') ∈ ''C<sup>k</sup>'' and ''g''(''x'') ∈ ''C''<sup>''k''−1</sup> then ''u''(''t'', ''x'') ∈ ''C<sup>k</sup>''. However, the waveforms ''F'' and ''G'' may also be generalized functions, such as the delta-function. In that case, the solution may be interpreted as an impulse that travels to the right or the left.
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| The basic wave equation is a [[linear differential equation]] and so it will adhere to the [[superposition principle]]. This means that the net displacement caused by two or more waves is the sum of the displacements which would have been caused by each wave individually. In addition, the behavior of a wave can be analyzed by breaking up the wave into components, e.g. the [[Fourier transform]] breaks up a wave into sinusoidal components.
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| ==Scalar wave equation in three space dimensions==
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| The solution of the initial-value problem for the wave equation in three space dimensions can be obtained from the solution for a spherical wave. This result can then be used to obtain the solution in two space dimensions.
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| ===Spherical waves===
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| [[File:Spherical Wave.gif|thumb|Cut-away of spherical wavefronts, with a wavelength of 10 units, propagating from a point source.]]
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| The wave equation is unchanged under rotations of the spatial coordinates, because the [[Laplacian]] operator is invariant under rotation, and therefore one may expect to find solutions that depend only on the radial distance from a given point. Such solutions must satisfy
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| :<math> u_{tt} - c^2 \left( u_{rr} + \frac{2}{r} u_r \right) =0. \,</math>
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| This equation may be rewritten as
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| :<math> (ru)_{tt} -c^2 (ru)_{rr}=0; \,</math>
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| the quantity ''ru'' satisfies the one-dimensional wave equation. Therefore there are solutions in the form
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| :<math> u(t,r) = \frac{1}{r} F(r-ct) + \frac{1}{r} G(r+ct), \,</math>
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| where ''F'' and ''G'' are arbitrary functions. Each term may be interpreted as a spherical wave that expands or contracts with velocity ''c''. Such waves are generated by a [[point source]], and they make possible sharp signals whose form is altered only by a decrease in amplitude as ''r'' increases (see an illustration of a spherical wave on the top right). Such waves exist only in cases of space with odd dimensions.
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| ====Monochromatic spherical wave====
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| A point source is vibrating at a single [[frequency]] ''f'' with phase = 0 at ''t'' = 0 with a peak-to-peak magnitude of 2''a''. A spherical wave is propagated from the point. The phase of the propagated wave changes as ''kr'' where ''r'' is the distance travelled from the source. The magnitude falls off as 1/''r'' since the energy falls off as ''r''<sup>−2</sup>. The amplitude of the spherical wave at ''r'' is therefore given by:<ref>RS Longhurst, Geometrical and Physical Optics, 1967, Longmans, Norwich</ref>
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| <math>u(t,r)= Re \left[ \frac{a}{r} e^{i \left( \omega t - kr \right)} \right] </math>
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| ===Solution of a general initial-value problem===
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| The wave equation is linear in ''u'' and it is left unaltered by translations in space and time. Therefore we can generate a great variety of solutions by translating and summing spherical waves. Let φ(ξ,η,ζ) be an arbitrary function of three independent variables, and let the spherical wave form ''F'' be a delta-function: that is, let ''F'' be a weak limit of continuous functions whose integral is unity, but whose support (the region where the function is non-zero) shrinks to the origin. Let a family of spherical waves have center at (ξ,η,ζ), and let ''r'' be the radial distance from that point. Thus
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| :<math> r^2 = (x-\xi)^2 + (y-\eta)^2 + (z-\zeta)^2. \,</math>
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| If ''u'' is a superposition of such waves with weighting function φ, then
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| :<math> u(t,x,y,z) = \frac{1}{4\pi c} \iiint \varphi(\xi,\eta,\zeta) \frac{\delta(r-ct)}{r} d\xi\,d\eta\,d\zeta; \,</math>
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| the denominator 4πc is a convenience.
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| From the definition of the delta-function, ''u'' may also be written as
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| :<math> u(t,x,y,z) = \frac{t}{4\pi} \iint_S \varphi(x +ct\alpha, y +ct\beta, z+ct\gamma) d\omega, \,</math>
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| where α, β, and γ are coordinates on the unit sphere ''S'', and ω is the area element on ''S''. This result has the interpretation that ''u''(''t'',''x'') is ''t'' times the mean value of φ on a sphere of radius ''ct'' centered at ''x'':
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| :<math> u(t,x,y,z) = t M_{ct}[\phi]. \,</math>
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| It follows that
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| :<math> u(0,x,y,z) = 0, \quad u_t(0,x,y,z) = \phi(x,y,z). \,</math>
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| The mean value is an even function of ''t'', and hence if
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| :<math> v(t,x,y,z) = \frac{\part}{\part t} \left( t M_{ct}[\psi] \right), \,</math>
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| then
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| :<math> v(0,x,y,z) = \psi(x,y,z), \quad v_t(0,x,y,z) = 0. \,</math>
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| These formulas provide the solution for the initial-value problem for the wave equation. They show that the solution at a given point ''P'', given (''t'',''x'',''y'',''z'') depends only on the data on the sphere of radius ''ct'' that is intersected by the '''light cone''' drawn backwards from ''P''. It does ''not'' depend upon data on the interior of this sphere. Thus the interior of the sphere is a [[Petrovsky lacuna|lacuna]] for the solution. This phenomenon is called '''[[Huygens' principle]]'''. It is true for odd numbers of space dimension, where for one dimension the integration is performed over the boundary of an interval with respect to the Dirac measure. It is not satisfied in even space dimensions. The phenomenon of lacunas has been extensively investigated in [[Michael Atiyah|Atiyah]], [[Raoul Bott|Bott]] and [[Lars Gårding|Gårding]] (1970, 1973).
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| ==Scalar wave equation in two space dimensions==
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| In two space dimensions, the wave equation is
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| :<math> u_{tt} = c^2 \left( u_{xx} + u_{yy} \right). \,</math>
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| We can use the three-dimensional theory to solve this problem if we regard ''u'' as a function in three dimensions that is independent of the third dimension. If
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| :<math> u(0,x,y)=0, \quad u_t(0,x,y) = \phi(x,y), \,</math>
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| then the three-dimensional solution formula becomes
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| :<math> u(t,x,y) = tM_{ct}[\phi] = \frac{t}{4\pi} \iint_S \phi(x + ct\alpha,\, y + ct\beta) d\omega,\,</math>
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| where ''α'' and ''β'' are the first two coordinates on the unit sphere, and ''dω'' is the area element on the sphere. This integral may be rewritten as an integral over the disc ''D'' with center (''x'',''y'') and radius ''ct'':
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| :<math> u(t,x,y) = \frac{1}{2\pi c} \iint_D \frac{\phi(x+\xi, y +\eta)}{\sqrt{(ct)^2 - \xi^2 - \eta^2}} d\xi\,d\eta. \,</math>
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| It is apparent that the solution at (''t'',''x'',''y'') depends not only on the data on the light cone where
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| :<math> (x -\xi)^2 + (y - \eta)^2 = c^2 t^2, \,</math>
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| but also on data that are interior to that cone.
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| ==Scalar wave equation in general dimension and Kirchhoff's formulae==
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| We want to find solutions to ''u<sub>tt</sub>''−Δ''u'' = 0 for ''u'' : '''R'''<sup>''n''</sup> × (0, ∞) → '''R''' with ''u''(''x'', 0) = ''g''(''x'') and ''u<sub>t</sub>''(''x'', 0) = ''h''(''x''). See Evans for more details.
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| ===Odd dimensions===
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| Assume ''n'' ≥ 3 is an odd integer and ''g'' ∈ ''C''<sup>''m''+1</sup>('''R'''<sup>''n''</sup>), ''h'' ∈ ''C<sup>m</sup>''('''R'''<sup>''n''</sup>) for ''m'' = (''n''+1)/2. Let <math>\gamma_n = 1\cdot 3 \cdot 5 \cdot .. \cdot (n-2)</math> and let
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| :<math>u(x,t) = \frac{1}{\gamma_n}\left [\partial_t \left (\frac{1}{t} \partial_t \right )^{\frac{n-3}{2}} \left (t^{n-2} \int^{\text{average}}_{\partial B_t(x)} g dS \right ) + \left (\frac{1}{t}\partial_t \right )^{\frac{n-3}{2}} \left (t^{n-2} \int^{\text{average}}_{\partial B_t(x)} h dS \right ) \right]</math>
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| then
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| :''u'' ∈ ''C''<sup>2</sup>('''R'''<sup>''n''</sup> × [0, ∞))
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| :''u<sub>tt</sub>''−Δ''u'' = 0 in '''R'''<sup>''n''</sup> × (0, ∞)
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| :<math>\begin{align}
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| \lim_{(x,t)\to (x^0,0)} u(x,t) &= g(x^0) \\
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| \lim_{(x,t)\to (x^0,0)} u_t(x,t) &= h(x^0)
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| \end{align}</math>
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| ===Even dimensions===
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| Assume ''n'' ≥ 2 is an even integer and ''g'' ∈ ''C''<sup>''m''+1</sup>('''R'''<sup>''n''</sup>), ''h'' ∈ ''C<sup>m</sup>''('''R'''<sup>''n''</sup>), for ''m'' = (''n''+2)/2. Let <math>\gamma_n = 2 \cdot 4 \cdot .. \cdot n</math> and let
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| :<math>u(x,t) = \frac{1}{\gamma_n} \left [\partial_t \left (\frac{1}{t} \partial_t \right )^{\frac{n-2}{2}} \left (t^n \int^{\text{average}}_{B_t(x)} \frac{g}{(t^2 - |y - x|^2)^{\frac{1}{2}}} dy \right ) + \left (\frac{1}{t} \partial_t \right )^{\frac{n-2}{2}} \left (t^n \int^{\text{average}}_{B_t(x)} \frac{h}{(t^2 - |y-x|^2)^{\frac{1}{2}}} dy \right ) \right ] </math>
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| then
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| :''u'' ∈ ''C''<sup>2</sup>('''R'''<sup>''n''</sup> × [0, ∞))
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| :''u<sub>tt</sub>''−Δ''u'' = 0 in '''R'''<sup>''n''</sup> × (0, ∞)
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| :<math>\begin{align}
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| \lim_{(x,t)\to (x^0,0)} u(x,t) &= g(x^0)\\
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| \lim_{(x,t)\to (x^0,0)} u_t(x,t) &= h(x^0)
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| \end{align}</math>
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| ==Problems with boundaries==
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| ===One space dimension===
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| ==== The Sturm-Liouville formulation ====
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| A flexible string that is stretched between two points ''x'' = 0 and ''x'' = ''L'' satisfies the wave equation for ''t'' > 0 and 0 < ''x'' < ''L''. On the boundary points, ''u'' may satisfy a variety of boundary conditions. A general form that is appropriate for applications is
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| :<math> -u_x(t,0) + a u(t,0) = 0, \,</math>
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| :<math> u_x(t,L) + b u(t,L) = 0,\,</math>
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| where ''a'' and ''b'' are non-negative. The case where u is required to vanish at an endpoint is the limit of this condition when the respective ''a'' or ''b'' approaches infinity. The method of [[separation of variables]] consists in looking for solutions of this problem in the special form
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| :<math> u(t,x) = T(t) v(x).\,</math>
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| A consequence is that
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| :<math> \frac{T''}{c^2T} = \frac{v''}{v} = -\lambda. \,</math>
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| The [[eigenvalue]] λ must be determined so that there is a non-trivial solution of the boundary-value problem
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| :<math> v'' + \lambda v=0, \,</math> | |
| :<math> -v'(0) + a v(0) = 0, \quad v'(L) + b v(L)=0.\,</math> | |
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| This is a special case of the general problem of [[Sturm–Liouville theory]]. If ''a'' and ''b'' are positive, the eigenvalues are all positive, and the solutions are trigonometric functions. A solution that satisfies square-integrable initial conditions for ''u'' and ''u<sub>t</sub>'' can be obtained from expansion of these functions in the appropriate trigonometric series.
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| ==== Investigation by numerical methods ====
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| Approximating the continuous string with a finite number of equidistant mass points one gets the following physical model:
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| [[File:String wave 0.svg|frame|Figure 1: Three consecutive mass points of the discrete model for a string]]
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| If each mass point has the mass ''m'', the tension of the string is ''f'', the separation between the mass points is Δ''x'' and ''u<sub>i</sub>'', ''i'' = 1, ..., ''n'' are the offset of these ''n'' points from their equilibrium points (i.e. their position on a straight line between the two attachment points of the string) the vertical component of the force towards point ''i''+1 is
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| # {{NumBlk|:|<math>\frac{u_{i+1}-u_i}{\Delta x}\ f</math>|{{EquationRef|1}}}}
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| and the vertical component of the force towards point ''i''−1 is
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| # {{NumBlk|:|<math>\frac{u_{i-1}-u_i}{\Delta x}\ f</math>|{{EquationRef|2}}}}
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| Taking the sum of these two forces and dividing with the mass ''m'' one gets for the vertical motion:
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| # {{NumBlk|:|<math>\ddot u_i=\left(\frac{f}{m\ \Delta x} \right) \left(u_{i+1} + u_{i-1}\ -\ 2u_i\right)</math>|{{EquationRef|3}}}}
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| As the mass density is
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| :<math>\rho = \frac{m}{\Delta x}</math> | |
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| this can be written
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| # {{NumBlk|:|<math>\ddot u_i=\left(\frac{f}{\rho\ {\Delta x}^2} \right) \left(u_{i+1} + u_{i-1}\ -\ 2u_i\right)</math>|{{EquationRef|4}}}}
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| The wave equation is obtained by letting Δ''x'' → 0 in which case ''u<sub>i</sub>''(''t'') takes the form ''u''(''x'', ''t'') where ''u''(''x'', ''t'') is continuous function of two variables, <math>\ddot u_i</math> takes the form <math>\partial^2 u \over \partial t^2</math> and
| |
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| :<math>\frac{u_{i+1} + u_{i-1}\ -\ 2u_i}{{\Delta x}^2} \rightarrow \frac{\partial^2 u }{\partial x^2}</math>
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| | |
| But the discrete formulation ({{EquationNote|3}}) of the equation of state with a finite number of mass point is just the suitable one for a [[Numerical ordinary differential equations|numerical propagation]] of the string motion. The boundary condition
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| :<math>u(0,t) = u(L,t) = 0</math>
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| | |
| where ''L'' is the length of the string takes in the discrete formulation the form that for the outermost points ''u''<sub>1</sub> and ''u<sub>n</sub>'' the equation of motion are
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| # {{NumBlk|:|<math>\ddot u_1={\left(\frac{c}{\Delta x} \right)}^2 \left(u_2 \ -\ 2u_1\right)</math>|{{EquationRef|5}}}}
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| and
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| # {{NumBlk|:|<math>\ddot u_n={\left(\frac{c}{\Delta x} \right)}^2 \left(u_{n-1} \ -\ 2u_n\right)</math>|{{EquationRef|6}}}}
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| | |
| while for 1 < ''i'' < ''n''
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| # {{NumBlk|:|<math>\ddot u_i={\left(\frac{c}{\Delta x} \right)}^2 \left(u_{i+1} + u_{i-1}\ -\ 2u_i\right)</math>|{{EquationRef|7}}}}
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| | |
| where <math> c=\sqrt{\frac{f}{\rho}} </math>
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| If the string is approximated with 100 discrete mass points one gets the 100 coupled second order differential equations ({{EquationNote|5}}), ({{EquationNote|6}}) and ({{EquationNote|7}}) or equivalently 200 coupled first order differential equations.
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| Propagating these up to the times
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| :<math>\frac{L}{c}\ k\ 0.05\ \ k=0,\cdots ,5 </math>
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| | |
| using an 8-th order [[Multistep methods|multistep method]] the 6 states displayed in figure 2 are found:
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| [[File:String wave 1.svg|frame|Figure 2: The string at 6 consecutive epochs, the first (red) corresponding to the initial time with the string in rest]]
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| [[File:String wave 2.svg|frame|Figure 3: The string at 6 consecutive epochs]]
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| [[File:String wave 3.svg|frame|Figure 4: The string at 6 consecutive epochs]]
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| [[File:String wave 4.svg|frame|Figure 5: The string at 6 consecutive epochs]]
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| [[File:String wave 5.svg|frame|Figure 6: The string at 6 consecutive epochs]]
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| [[File:String wave 6.svg|frame|Figure 7: The string at 6 consecutive epochs]]
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| The red curve is the initial state at time zero at which the string is "let free" in a predefined shape <ref>The initial state for "Investigation by numerical methods" is set with quadratic [[Spline (mathematics)|splines]] as follows:
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| :<math>u(0,x)= u_0\ \left(1-\left(\frac{x-x_1}{x_1}\right)^2\right)</math> for <math>0 \le x \le x_2</math>
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| :<math>u(0,x)= u_0\ \left({\frac{x-x_3}{x_1}}\right)^2</math> for <math>x_2 \le x \le x_3</math>
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| :<math>u(0,x)= 0</math> for <math>x_3 \le x \le L</math>
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| with <math>x_1= \frac{1}{10}\ L\ ,\ x_2=x_1+\sqrt{\frac{1}{2}}\ x_1\ ,\ x_3=x_2+\sqrt{\frac{1}{2}}\ x_1</math></ref> with all <math>\dot u_i=0</math>. The blue curve is the state at time <math>\frac{L}{c}\ 0.25</math>, i.e. after a time that corresponds to the time a wave that is moving with the nominal wave velocity <math> c=\sqrt{\frac{f}{\rho}} </math> would need for one fourth of the length of the string.
| |
| | |
| Figure 3 displays the shape of the string at the times <math>\frac{L}{c}\ k\ 0.05\ \ k=6,\cdots ,11</math>. The wave travels in direction right with the speed <math> c=\sqrt{\frac{f}{\rho}} </math> without being actively constraint by the boundary conditions at the two extrems of the string. The shape of the wave is constant, i.e. the curve is indeed of the form ''f''(''x''−''ct'').
| |
| | |
| Figure 4 displays the shape of the string at the times <math>\frac{L}{c}\ k\ 0.05\ \ k=12,\cdots ,17</math>. The constraint on the right extreme starts to interfere with the motion preventing the wave to raise the end of the string.
| |
| | |
| Figure 5 displays the shape of the string at the times <math>\frac{L}{c}\ k\ 0.05\ \ k=18,\cdots ,23</math> when the direction of motion is reversed. The red, green and blue curves are the states at the times <math>\frac{L}{c}\ k\ 0.05\ \ k=18,\cdots ,20</math> while the 3 black curves correspond to the states at times <math>\frac{L}{c}\ k\ 0.05\ \ k=21,\cdots ,23</math> with the wave starting to move back towards left.
| |
| | |
| Figure 6 and figure 7 finally display the shape of the string at the times <math>\frac{L}{c}\ k\ 0.05\ \ k=24,\cdots ,29</math> and <math>\frac{L}{c}\ k\ 0.05\ \ k=30,\cdots ,35</math>. The wave now travels towards left and the constraints at the end points are not active any more. When finally the other extreme of the string the direction will again be reversed in a way similar to what is displayed in figure 6
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| | |
| ===Several space dimensions===
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| [[Image:Drum vibration mode12.gif|right|thumb|220px|A solution of the wave equation in two dimensions with a zero-displacement boundary condition along the entire outer edge.]]
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| The one-dimensional initial-boundary value theory may be extended to an arbitrary number of space dimensions. Consider a domain ''D'' in ''m''-dimensional ''x'' space, with boundary ''B''. Then the wave equation is to be satisfied if ''x'' is in ''D'' and ''t'' > 0. On the boundary of ''D'', the solution ''u'' shall satisfy
| |
| | |
| :<math> \frac{\part u}{\part n} + a u =0, \,</math>
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| | |
| where ''n'' is the unit outward normal to ''B'', and ''a'' is a non-negative function defined on ''B''. The case where ''u'' vanishes on ''B'' is a limiting case for ''a'' approaching infinity. The initial conditions are
| |
| | |
| :<math> u(0,x) = f(x), \quad u_t(0,x)=g(x), \,</math>
| |
| | |
| where ''f'' and ''g'' are defined in ''D''. This problem may be solved by expanding ''f'' and ''g'' in the eigenfunctions of the Laplacian in ''D'', which satisfy the boundary conditions. Thus the eigenfunction ''v'' satisfies
| |
| | |
| :<math> \nabla \cdot \nabla v + \lambda v = 0, \,</math>
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| | |
| in ''D'', and
| |
| | |
| :<math> \frac{\part v}{\part n} + a v =0, \,</math>
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| | |
| on ''B''.
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| | |
| In the case of two space dimensions, the eigenfunctions may be interpreted as the modes of vibration of a drumhead stretched over the boundary ''B''. If ''B'' is a circle, then these eigenfunctions have an angular component that is a trigonometric function of the polar angle θ, multiplied by a [[Bessel function]] (of integer order) of the radial component. Further details are in [[Helmholtz equation]].
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| | |
| If the boundary is a sphere in three space dimensions, the angular components of the eigenfunctions are [[spherical harmonics]], and the radial components are [[Bessel function]]s of half-integer order.
| |
| | |
| ==Inhomogeneous wave equation in one dimension==
| |
| | |
| The inhomogeneous wave equation in one dimension is the following:
| |
| | |
| :<math>c^2 u_{x x}(x,t) - u_{t t}(x,t) = s(x,t) \,</math>
| |
| with initial conditions given by
| |
| | |
| :<math>u(x,0)=f(x) \,</math> | |
| :<math>u_t(x,0)=g(x) \,</math> | |
| | |
| The function ''s''(''x'', ''t'') is often called the source function because in practice it describes the effects of the sources of waves on the medium carrying them. Physical examples of source functions include the force driving a wave on a string, or the charge or current density in the [[Lorenz gauge]] of [[electromagnetism]].
| |
| | |
| One method to solve the initial value problem (with the initial values as posed above) is to take advantage of a special property of the wave equation in an odd number of space dimensions, namely that its solutions respect causality. That is, for any point (''x<sub>i</sub>'', ''t<sub>i</sub>''), the value of ''u''(''x<sub>i</sub>'', ''t<sub>i</sub>'') depends only on the values of ''f''(''x<sub>i</sub>''+''ct<sub>i</sub>'') and ''f''(''x<sub>i</sub>''−''ct<sub>i</sub>'') and the values of the function ''g''(''x'') between (''x<sub>i</sub>''−''ct<sub>i</sub>'') and (''x<sub>i</sub>''+''ct<sub>i</sub>''). This can be seen in [[d'Alembert's formula]], stated above, where these quantities are the only ones that show up in it. Physically, if the maximum propagation speed is ''c'', then no part of the wave that can't propagate to a given point by a given time can affect the amplitude at the same point and time.
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| | |
| In terms of finding a solution, this causality property means that for any given point on the line being considered, the only area that needs to be considered is the area encompassing all the points that could causally affect the point being considered. Denote the area that casually affects point (''x<sub>i</sub>'', ''t<sub>i</sub>'') as ''R<sub>C</sub>''. Suppose we integrate the inhomogeneous wave equation over this region.
| |
| | |
| :<math>\iint \limits_{R_C} \left ( c^2 u_{x x}(x,t) - u_{t t}(x,t) \right ) dx dt = \iint \limits_{R_C} s(x,t) dx dt. </math>
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| | |
| To simplify this greatly, we can use [[Green's theorem]] to simplify the left side to get the following:
| |
| | |
| :<math>\int_{ L_0 + L_1 + L_2 } \left ( - c^2 u_x(x,t) dt - u_t(x,t) dx \right ) = \iint \limits_{R_C} s(x,t) dx dt. </math>
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| | |
| The left side is now the sum of three line integrals along the bounds of the causality region. These turn out to be fairly easy to compute
| |
| | |
| :<math>\int^{x_i + c t_i}_{x_i - c t_i} - u_t(x,0) dx = - \int^{x_i + c t_i}_{x_i - c t_i} g(x) dx.</math>
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| | |
| In the above, the term to be integrated with respect to time disappears because the time interval involved is zero, thus ''dt'' = 0.
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| | |
| For the other two sides of the region, it is worth noting that ''x''±''ct'' is a constant, namingly ''x<sub>i</sub>''±''ct<sub>i</sub>'', where the sign is chosen appropriately. Using this, we can get the relation ''dx''±''cdt'' = 0, again choosing the right sign:
| |
| | |
| :<math>\begin{align}
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| \int_{L_1} \left ( - c^2 u_x(x,t) dt - u_t(x,t) dx \right ) &= \int_{L_1} \left ( c u_x(x,t) dx + c u_t(x,t) dt \right)\\
| |
| &= c \int_{L_1} d u(x,t) \\
| |
| &= c u(x_i,t_i) - c f(x_i + c t_i).
| |
| \end{align}</math>
| |
| | |
| And similarly for the final boundary segment:
| |
| | |
| :<math>\begin{align}
| |
| \int_{L_2} \left ( - c^2 u_x(x,t) dt - u_t(x,t) dx \right ) &= - \int_{L_2} \left ( c u_x(x,t) dx + c u_t(x,t) dt \right )\\
| |
| &= - c \int_{L_2} d u(x,t) \\
| |
| &= c u(x_i,t_i) - c f(x_i - c t_i).
| |
| \end{align}</math>
| |
| | |
| Adding the three results together and putting them back in the original integral:
| |
| | |
| :<math>\begin{align}
| |
| \iint_{R_C} s(x,t) dx dt &= - \int^{x_i + c t_i}_{x_i - c t_i} g(x) dx + c u(x_i,t_i) - c f(x_i + c t_i) + c u(x_i,t_i) - c f(x_i - c t_i) \\
| |
| &= 2 c u(x_i,t_i) - c f(x_i + c t_i) - c f(x_i - c t_i) - \int^{x_i + c t_i}_{x_i - c t_i} g(x) dx
| |
| \end{align}</math>
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| | |
| Solving for ''u''(''x<sub>i</sub>'', ''t<sub>i</sub>'') we arrive at
| |
| | |
| :<math>u(x_i,t_i) = \frac{f(x_i + c t_i) + f(x_i - c t_i)}{2} + \frac{1}{2 c}\int^{x_i + c t_i}_{x_i - c t_i} g(x) dx + \frac{1}{2 c}\int^{t_i}_0 \int^{x_i + c \left ( t_i - t \right )}_{x_i - c \left ( t_i - t \right )} s(x,t) dx dt.</math>
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| | |
| In the last equation of the sequence, the bounds of the integral over the source function have been made explicit. Looking at this solution, which is valid for all choices (''x<sub>i</sub>'', ''t<sub>i</sub>'') compatible with the wave equation, it is clear that the first two terms are simply d'Alembert's formula, as stated above as the solution of the homogeneous wave equation in one dimension. The difference is in the third term, the integral over the source.
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| | |
| ==Other coordinate systems==
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| In three dimensions, the wave equation, when written in [[elliptic cylindrical coordinates]], may be solved by separation of variables, leading to the [[Mathieu differential equation]].
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| | |
| ==Further generalizations==
| |
| | |
| To model [[Dispersion (optics)|dispersive]] wave phenomena, those in which the speed of wave propagation varies with the frequency of the wave, the constant ''c'' is replaced by the [[phase velocity]]:
| |
| | |
| :<math>v_\mathrm{p} = \frac{\omega}{k}.</math>
| |
| | |
| The elastic wave equation in three dimensions describes the propagation of waves in an [[isotropic]] [[wiktionary:Homogeneous|homogeneous]] [[elastic (solid mechanics)|elastic]] medium. Most solid materials are elastic, so this equation describes such phenomena as [[seismic waves]] in the [[Earth]] and [[Ultrasound|ultrasonic]] waves used to detect flaws in materials. While linear, this equation has a more complex form than the equations given above, as it must account for both longitudinal and transverse motion:
| |
| | |
| :<math>\rho{ \ddot{\bold{u}}} = \bold{f} + ( \lambda + 2\mu )\nabla(\nabla \cdot \bold{u}) - \mu\nabla \times (\nabla \times \bold{u})</math>
| |
| | |
| where:
| |
| | |
| *λ and μ are the so-called [[Lamé parameters]] describing the elastic properties of the medium,
| |
| *ρ is the density,
| |
| *'''f''' is the source function (driving force),
| |
| *and '''u''' is the displacement vector.
| |
| Note that in this equation, both force and displacement are [[vector (geometry)|vector]] quantities. Thus, this equation is sometimes known as the vector wave equation.
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| | |
| | |
| ==See also==
| |
| *[[Acoustic wave equation]]
| |
| *[[Acoustic attenuation]]
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| *[[Electromagnetic wave equation]]
| |
| *[[Helmholtz equation]]
| |
| *[[Inhomogeneous electromagnetic wave equation]]
| |
| *[[Laplace operator]]
| |
| *[[Schrödinger equation]]
| |
| *[[Standing wave]]
| |
| *[[Vibrations of a circular drum]]
| |
| *[[Bateman transform]]
| |
| *[[Maxwell's equations]]
| |
| *[[Wheeler-Feynman absorber theory]]
| |
| | |
| ==Notes==
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| * M. F. Atiyah, R. Bott, L. Garding, "Lacunas for hyperbolic differential operators with constant coefficients I", ''Acta Math.'', '''124''' (1970), 109–189.
| |
| * M.F. Atiyah, R. Bott, and L. Garding, "Lacunas for hyperbolic differential operators with constant coefficients II", ''Acta Math.'', '''131''' (1973), 145–206.
| |
| * R. Courant, D. Hilbert, ''Methods of Mathematical Physics, vol II''. Interscience (Wiley) New York, 1962.
| |
| * L. Evans, "Partial Differential Equations". American Mathematical Society Providence, 1998.
| |
| * "[http://eqworld.ipmnet.ru/en/solutions/lpde/wave-toc.pdf Linear Wave Equations]", ''EqWorld: The World of Mathematical Equations.''
| |
| * "[http://eqworld.ipmnet.ru/en/solutions/npde/npde-toc2.pdf Nonlinear Wave Equations]", ''EqWorld: The World of Mathematical Equations.''
| |
| * William C. Lane, "[http://www.physnet.org/modules/pdf_modules/m201.pdf <small>MISN-0-201</small> The Wave Equation and Its Solutions]", ''[http://www.physnet.org Project PHYSNET]''.
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| | |
| == External links ==
| |
| * {{cite web |url = http://prism.texarkanacollege.edu/physicsjournal/wave-eq.html |title = Kinematic Derivation of the Wave Equation |author = Francis Redfern |work = Physics Journal }} — a step-by-step derivation suitable for an introductory approach to the subject.
| |
| * [http://demonstrations.wolfram.com/NonlinearWaveEquations/ Nonlinear Wave Equations] by [[Stephen Wolfram]] and Rob Knapp, [http://demonstrations.wolfram.com/NonlinearWaveEquationExplorer/ Nonlinear Wave Equation Explorer] by [[Stephen Wolfram]], and [[Wolfram Demonstrations Project]].
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| * Mathematical aspects of wave equations are discussed on the [http://tosio.math.toronto.edu/wiki/index.php/Main_Page Dispersive PDE Wiki].
| |
| * Graham W Griffiths and William E. Schiesser (2009). [http://www.scholarpedia.org/article/Linear_and_nonlinear_waves Linear and nonlinear waves]. [http://www.scholarpedia.org/ Scholarpedia], 4(7):4308. [http://dx.doi.org/10.4249/scholarpedia.4308 doi:10.4249/scholarpedia.4308]
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| [[Category:Concepts in physics]]
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| [[Category:Hyperbolic partial differential equations]]
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| [[Category:Wave mechanics]]
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