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[[Слика:Wave equation 1D fixed endpoints.gif|оквир|Имплус који се простире кроз жицу са фиксираним крајевима моделован преко таласне једначине]]
'''Таласна једначина''' је важна парцијална [[диференцијална једначина]] којом се описује простирање [[талас]]а. Таласи могу бити звучни, елетромагнетни, водени итд., али се сви простиру на истом принципу сажетом у таласну једначину. Таласна једначина се јавља и користи у акустици, [[електромагнетизам|електромагнетизму]], [[оптика|оптици]], динамици флуида. Најзначајнији допринос решењу проблема описивања осцилација и простирања таласа дали су [[Жан ле Рон Д'Аламбер|Жан дАламбер]], [[Леонард Ојлер]], [[Данијел Бернули]] и [[Жозе-Луј Лагранж]].
== Увод==
где је '''c''' (константна) брзина таласа а <math>\Delta = \nabla^2</math> је [[Лапласијан]]. За звучне таласе у ваздуху на 20°C износи око 343 m/s (видети [[брзина звука]]). За осцилујућу жицу брзина може да се мења у великом опсегу јер зависи од линеарне густине жице и њене затегнутости. Много реалистичнији модел једначине таласа узима у обзир [[дисперзија|дисперзију]], тј, зависност брзине таласа од његове фреквенције. Тада се ''c'' замењује [[фазна брзина|фазном брзином]]:
:<math>v_\mathrm{p} = \frac{\omega}{k}.</math>
<!--Another common correction is that, in realistic systems, the speed also can depend on the amplitude of the wave, leading to a nonlinear wave equation:-->
:<math>{ \partial^2 u \over \partial t^2 } = c(u)^2 \Delta u </math>
<!--Also note that a wave may be superimposed onto another movement (for instance sound propagation in a moving medium like a gas flow). In that case the scalar ''u'' will contain a [[Mach number|Mach factor]] (which is positive for the wave moving along the flow and negative for the reflected wave).-->
<!--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 [[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:
*<math>\lambda</math> and <math>\mu</math> are the so-called Lamé moduli describing the elastic properties of the medium,
*<math>\rho</math> is density,
*<math>\bold{f}</math> is the source function (driving force),
*and <math>\bold{u}</math> is displacement.
Note that in this equation, both force and displacement are [[vector (spatial)|vector]] quantities. Thus, this equation is sometimes known as the vector wave equation.-->
<!--Variations of the wave equation are also found in [[quantum mechanics]] and [[general relativity]].-->
==Скаларна таласна једначина у једној просторној димензији==
:<math>u(x,t) = \frac{f(x-ct) + f(x+ct)}{2} + \frac{1}{2c} \int_{x-ct}^{x+ct} g(s) ds</math>
<!--
In the classical sense if <math>f(x) \in C^k</math> and <math>g(x) \in C^{k-1}</math> then <math>u(t,x) \in C^k</math>. 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.
The basic wave equation is a [[linear differential equation]] which means that the amplitude of two waves interacting is simply the sum of the waves. This means also that a behavior of a wave can be analyzed by breaking up the wave into components. The [[Fourier transform]] breaks up a wave into sinusoidal components and is useful for analyzing the wave equation.-->
==Скаларна таласна једначина у три просторне димензије==
:<math> u(t,r) = \frac{1}{r} F(r-ct) + \frac{1}{r} G(r+ct), \,</math>
где су ''F'' и ''G'' произвољне функције. Сваки члан може да се тумачи као сферни талас који се шири или скупља (контрахује) брзином ''c''. Такве таласе ствара тачкасти извор. <!-- 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. Such waves exist only in cases of space with odd dimensions. Fortunately, we live in a world that has three space dimensions, so that we can communicate clearly with acoustic and electromagnetic waves.-->
===Решење општег проблема почетних вредности===
Ове формуле дају решење за проблем почетних вредности таласне једначине. Оне показују да решење у датој тачки ''Р'', за дато (''t'',''x'',''y'',''z'') зависи само од података на сфери радијуса ''ct'' пресеченој '''светлосним конусом''' нацртаним уназад из тачке ''Р''. Решење не зависи од података унутар те сфере.
<!--
Thus the interior of the sphere is a '''lacuna''' for the solution. This phenomenon is called '''Huygens' principle'''. It is true for odd numbers of space dimension, except for one dimension. It is not satisfied in even space dimensions. The phenomenon of lacunas has been extensively investigated in Atiyah, Bott and Garding (1970, 1973).-->
==Скаларна таласна једначина у две просторне димензије==
==Проблем с границама==
===Једна просторна димензија===
<!--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-->
:<math> -u_x(t,0) + a u(t,0) = 0, \,</math>
:<math> u_x(t,L) + b u(t,L) = 0,\,</math>
<!--
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-->
:<math> u(t,x) = T(t) v(x).\,</math>
<!--A consequence is that-->
:<math> \frac{T''}{c^2T} = \frac{v''}{v} = -\lambda. \,</math>
<!--The [[eigenvalue]] λ must be determined so that there is a non-trivial solution of the boundary-value problem
-->
:<math> v'' + \lambda v=0, \,</math>
:<math> -v'(0) + a v(0) = 0, \quad v'(L) + b v(L)=0.\,</math>
<!--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.-->
===Неколико просторних димензија===
<!--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 0<''t''. One the boundary of ''D'', the solution ''u'' shall satisfy-->
:<math> \frac{\part u}{\part n} + a u =0, \,</math>
<!--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=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>
на ''B''.
<!--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]].
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.-->
==Нехомогена таласна једначина у једној димензији==
<!--The inhomogenous 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 <math>s(x,t)</math> 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 the property of the wave equation that its solutions obey causality. That is, for any point <math>(x_i,t_i)</math>, the value of <math>u(x_i,t_i)</math> depends only on the values of <math>f(x_i + c t_i)</math> and <math>f(x_i - c t_i)</math> and the values of the function <math>g(x)</math> between <math>(x_i - c t_i)</math> and <math>(x_i - c t_i)</math>. 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 <math>c</math>, 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.
-->
<!--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 <math>(x_i,t_i)</math> as <math>R_C</math>. Suppose we integrate the in-homogenous wave equation over this region.-->
:<math>\int \int_{R_C} \left ( c^2 u_{x x}(x,t) - u_{t t}(x,t) \right ) dx dt = \int \int_{R_C} s(x,t) dx dt. </math>
<!--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 ) = \int \int_{R_C} s(x,t) dx dt. </math>
<!--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>
<!--In the above, the term to be integrated with respect to time disappears because the time interval involved is zero, thus <math> d t = 0 </math>.
-->
<!--For the other two sides of the region, it is worth noting that <math>x \pm c t</math> is a constant, namingly <math>x_i \pm c t_i</math>, where the sign is chosen appropriately. Using this, we can get the relation <math>dx \pm c dt = 0</math>, again choosing the right sign:
-->
:<math>\int_{L_1} \left ( - c^2 u_x(x,t) dt - u_t(x,t) dx \right ) \,</math>
:: <math>= \int_{L_1} \left ( c u_x(x,t) dx + c u_t(x,t) dt \right)\, </math>
::<math>= c \int_{L_1} d u(x,t) = c u(x_i,t_i) - c f(x_i + c t_i).\,</math>
<!--And similarly for the final boundary segment:
-->
:<math>\int_{L_2} \left ( - c^2 u_x(x,t) dt - u_t(x,t) dx \right ) </math>
::<math>= - \int_{L_2} \left ( c u_x(x,t) dx + c u_t(x,t) dt \right ) </math>
:: <math>= c u(x_i,t_i) - c f(x_i - c t_i).\,</math>
<!--Adding the three results together and putting them back in the original integral:
-->
:<math>- \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) = \int \int_{R_C} s(x,t) dx dt </math>
:<math>2 c u(x_i,t_i) - \int^{x_i + c t_i}_{x_i - c t_i} g(x) dx - c f(x_i + c t_i) - c f(x_i - c t_i) = \int \int_{R_C} s(x,t) dx dt </math>
:<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>
<!--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 <math>(x_i,t_i)</math> 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 homogenous wave equation in one dimension. The difference is in the third term, the integral over the source.
-->
==Други координатни системи==
<!--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]].
-->
==Види још==
*[[Једначина електромагнетних таласа]]
*[[Нехомогена једначина електромагнетних таласа]]
<!--*[[Motor variable]]
*[[Шредингерова једначина]]
-->*Математички аспект таласне једначине је дискутован на [http://tosio.math.toronto.edu/wiki/index.php/Main_Page Дисперзивне парцијалне диференцијалне једначине Wiki]. ▼<!--*[[Theoretical and experimental justification for the Schrödinger equation]]
▲-->*Математички аспект таласне једначине је дискутован на [http://tosio.math.toronto.edu/wiki/index.php/Main_Page Дисперзивне парцијалне диференцијалне једначине Wiki].
==Литература==
* 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.
* "[http://eqworld.ipmnet.ru/en/solutions/npde/npde-toc2.pdf Nonlinear Wave Equations]", ''EqWorld: The World of Mathematical Equations.''
* William C. Lane, "[http://35.9.69.219/home/modules/pdf_modules/m201.pdf <small>MISN-0-201</small> The Wave Equation and Its Solutions]", ''[http://www.physnet.org Project PHYSNET]''.
<!--[[Категорија:Hyperbolic partial differential equations]]
[[Категорија:Wave mechanics]]
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[[Категорија:Физика]]
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