Wave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves). It arises in fields like acoustics, electromagnetism, and fluid dynamics. A single wave propagating in a pre-defined direction can also be described with the one-way wave equation. 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. In 1746, d'Alembert discovered the one-dimensional wave equation, and within ten years Euler discovered the three-dimensional wave equation. Introduction. The (two-way) wave equation is a second-order partial differential equation describing standing wave field (superposition of two waves travelling in opposite directions). This article mostly focuses on the scalar wave equation describing waves in scalars by scalar functions of a time variable (a variable representing time) and one or more spatial variables (variables representing a position in a space under discussion) while there are vector wave equations describing waves in vectors  such as waves for electrical field, magnetic field, and magnetic vector potential and elastic waves. By comparison with vector wave equations, the scalar wave equation can be seen as a special case of the vector wave equations; in the Cartesian coordinate system, the scalar wave equation is the equation to be satisfied by each component (for each coordinate axis, such as the x-component for the x-axis) of a vector wave without sources of waves in the considered domain (i.e., a space and time). For example, in the Cartesian coordinate system, for formula_1 as the representation of an electric  vector field wave formula_2 in the absence of wave sources, each coordinate axis component formula_3 ("i" = "x", "y", or "z") must satisfy the scalar wave equation. Other scalar wave equation solutions are for physical quantities in scalars such as pressure in a liquid or gas, or the displacement, along some specific direction, of particles of a vibrating solid away from their resting (equilibrium) positions. The scalar wave equation is formula_4 where is a fixed non-negative real coefficient. Using the notations of Newtonian mechanics and vector calculus, the wave equation can be written more compactly as formula_5 where the  double dot denotes double time derivative of , is the nabla operator, and is the (spatial) Laplacian operator (not vector Laplacian): formula_6 An even more compact notation sometimes used in physics reads simply formula_7 where all operators are combined into the d'Alembert operator: formula_8 A solution of this (two-way) wave equation can be quite complicated, but it can be analyzed as a linear combination of simple solutions that are sinusoidal plane waves with various directions of propagation and wavelengths but all with the same propagation speed . This analysis is possible because the wave equation is linear and homogeneous; so that  any multiple of a solution is also a solution, and the sum of any two solutions is again a solution. This property is called the superposition principle in physics. The wave equation alone does not specify a physical solution; a unique solution is usually obtained by setting a problem with further conditions, such as initial conditions, which prescribe the amplitude and phase of the wave. Another important class of problems occurs in enclosed spaces specified by boundary conditions, for which the solutions represent standing waves, or harmonics, analogous to the harmonics of musical instruments. The two-way wave equation – describing  a standing wave field – is the simplest example of a second-order hyperbolic differential equation. It, and its modifications, play fundamental roles in continuum mechanics, quantum mechanics, plasma physics, general relativity, geophysics, and many other scientific and technical disciplines. In case that only the propagation of a single wave in a predefined direction is of interest, a first-order partial differential equation – one-way wave equation – can be considered. Wave equation in one space dimension. The wave equation in one space dimension can be written as follows: formula_9 This equation is typically described as having only one space dimension , because  the only other independent variable is the time . Nevertheless, the dependent variable may represent a second space dimension, if, for example, the displacement takes place in -direction, as in the case of a string that is located in the . Derivation of the wave equation. The wave equation in one space dimension can be derived in a variety of different physical settings. Most famously, it can be derived for the case of a string that is vibrating in a two-dimensional plane, with each of its elements being pulled in opposite directions by the force of tension. Another physical setting for  derivation of the wave equation in one space dimension utilizes Hooke's Law. In the theory of elasticity, Hooke's Law is an approximation for certain materials, stating that the amount by which a material body is deformed (the strain) is linearly related to the force causing the deformation (the stress). From Hooke's law. 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 interconnected with massless springs of length . The springs have a spring constant of : Here the dependent variable measures  the distance from the equilibrium of the mass situated at , so that essentially measures the magnitude of a disturbance (i.e. strain) that is traveling in an elastic material. The forces exerted on the mass at the location are: formula_10 The equation of motion for the weight at the location is given by equating these two forces: formula_11 where the time-dependence of has been made explicit. If the array of weights consists of weights spaced evenly over the length of total mass , and the total spring constant of the array we can write the above equation as: formula_12 Taking the  limit and assuming smoothness one gets: formula_13 which is from the definition of a second derivative. is the square of the propagation speed in this particular case. Stress pulse in a bar. In the case of a stress pulse propagating longitudinally through a bar, the bar acts much like an infinite number of springs in series and can be taken as an extension of the equation derived for Hooke's law. A uniform bar, i.e. of constant cross-section, made from a linear elastic material has a stiffness given by formula_14 where is the cross-sectional area and is the  Young's modulus of the material. The wave equation becomes formula_15 formula_16 where is the density of the material. The wave equation reduces to formula_17 The speed of a stress wave in a bar is therefore . General solution. Algebraic approach. 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: formula_18 changes the wave equation into formula_19 which leads to the general solution formula_20 or equivalently: formula_21 In other words, solutions of the 1D wave equation are sums of a right traveling function and a  left traveling function . "Traveling" means that the shape of these individual arbitrary functions with respect to stays constant, however the functions are translated left and right with time at the speed . This was derived by Jean le Rond d'Alembert. Another way to arrive at this result is to factor the wave equation into two one-way wave equations: formula_22 i.e. formula_23 As a result, if we define thus, formula_24 then formula_25 From this, must have the form , and from this the correct form of the full solution can be deduced. The usual second order wave equation is sometimes called the  two-way wave equation (superposition of two waves) to distinguish it from the first-order one-way wave equation] describing the wave propagation of a single wave in a pre-defined direction. For an initial value problem, the arbitrary functions and can be determined to satisfy initial conditions: formula_26 formula_27 The result is d'Alembert's formula : formula_28 In the classical sense if and then . However, the waveforms and 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 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. Plane wave eigenmodes. Another way to solve the one-dimensional wave equation is to first analyze its frequency eigenmodes . A so-called eigenmode is a solution that oscillates in time  with a well-defined "constant" angular frequency , so that the temporal part of the wave function takes the form , and the amplitude is a function of the spatial variable , giving a separation of variables for the wave function: formula_29 This produces an ordinary differential equation for the spatial part : formula_30 Therefore: formula_31 which is precisely an eigenvalue equation for , hence the name eigenmode. It has the well-known plane wave solutions formula_32 with wave number . The total wave function for this eigenmode is then the linear combination formula_33 where complex numbers , depend in general on any initial and boundary conditions  of the problem. Eigenmodes are useful in constructing a full solution to the wave equation, because each of them evolves in time trivially with the phase factor formula_34. so that a full solution can be decomposed into an eigenmode expansion formula_35 or in terms of the plane waves, formula_36 which is exactly in the same form as in the algebraic approach. Functions are known as the Fourier component and are determined by initial and boundary conditions. This is a so-called frequency-domain method, alternative to direct time-domain propagations, such as FDTD method, of the wave packet , which is  complete for representing waves in absence of time dilations. Completeness of the Fourier expansion for representing waves in the presence of time dilations has been challenged by chirp wave solutions allowing for time variation of . The chirp wave solutions seem particularly implied by very large but previously inexplicable radar residuals in the flyby anomaly , and differ from the sinusoidal solutions in being receivable at any distance only at proportionally shifted frequencies and time dilations, corresponding to past chirp states of the source. Vectorial wave equation in three space dimensions. The vectorial wave equation (from which the scalar wave equation can  be directly derived) can be obtained by applying a force equilibrium to an infinitesimal volume element. In a homogeneous continuum (cartesian coordinate formula_37) with a constant modulus of elasticity formula_38 [Pa] a vectorial, elastic deflection formula_39 [m] causes the stress tensor formula_40 [Pa]. The local equilibrium of a) the tension force formula_41 [N/mformula_42] due to deflection formula_43 [m] and b) the inertial force formula_44 formula_45 [N/mformula_42] caused by the local acceleration formula_47 [m/sformula_48] can be written as formula_49 By merging density formula_44 [kg/m3] and elasticity module formula_38 the sound velocity formula_52 [m/s] results (material law ). After insertion follows the well-known governing wave equation for a homogeneous medium: formula_53 The above vectorial partial differential equation of the 2nd order delivers two mutually independent solutions. From the quadratic velocity term formula_54 can be seen that there are two waves travelling in opposite directions formula_55 and formula_56 are possible, hence results the designation “Two-way wave equation”. The "Two-Way wave equation" can also be factorized: formula_57 and the vectorial 1st order One-way wave equation with waves travelling in a pre-defined propagation direction formula_58 results: formula_59 Scalar wave equation in three space dimensions. | thumb|right|Swiss  mathematician and physicist Leonhard Euler (b. 1707) discovered the wave equation in three space dimensions. A solution of the initial-value problem for the wave equation in three space dimensions can be obtained from the corresponding solution for a spherical wave. The result can then be also used to obtain the same solution in two space dimensions. Spherical waves. The wave equation can be solved using the technique of separation of variables . To obtain a solution with constant frequencies, let us first Fourier-transform the wave equation in time as formula_60 So we get, formula_61 This is the Helmholtz equation  and can be solved using separation of variables. If spherical coordinates are used to describe a problem, then the solution to the angular part of the Helmholtz equation is given by spherical harmonics and the radial equation now becomes formula_62 Here and the complete solution is now given by formula_63 where and are the Bessel function#Spherical Hankel functions|spherical Hankel functions . Example. To gain a better understanding of the nature of these spherical waves, let us go back and look at the case when . In this case, there is no angular dependence and the amplitude depends only on the radial distance  i.e. . In this case, the wave equation reduces to formula_64 This equation can be rewritten as formula_65 where the quantity satisfies the one-dimensional wave equation. Therefore, there are solutions in the form formula_66 where and are general solutions to the one-dimensional wave equation, and can be interpreted as respectively an outgoing or incoming spherical wave. The outgoing wave can be generated by a point source , and they make possible sharp signals whose form is altered only by a decrease in amplitude as increases (see an illustration of a spherical wave on the top right). Such waves exist only  in cases of space with odd dimensions. For physical examples of non-spherical wave solutions to the 3D wave equation that do possess angular dependence, see dipole radiation . Monochromatic spherical wave. |thumb|Cut-away of spherical wavefronts, with a wavelength of 10 units, propagating from a point source. Although the word "monochromatic" is not exactly accurate since it refers to light or electromagnetic radiation with well-defined frequency, the spirit is to discover the eigenmode of the wave equation in three dimensions. Following the derivation in the previous section on Wave Equation#Plane wave eigenmodes|Plane wave eigenmodes , if we again restrict our  solutions to spherical waves that oscillate in time with well-defined "constant" angular frequency , then the transformed function has simply plane wave solutions, formula_67 or formula_68 From this we can observe that the peak intensity of the spherical wave oscillation, characterized as the squared wave amplitude formula_69 then Problems with boundaries. One space dimension. Reflection and Transmission at the boundary of two mediums. For an incident wave traveling from one medium (where the wave speed is ) to another medium (where the wave speed is ), one part of the wave will transmit into the second medium, while another part reflects back  into the other direction and stays in the first medium. The amplitude of the transmitted wave and the reflected wave can be calculated by using the continuity condition at the boundary. Consider the component of the incident wave with an angular frequency of , which has the waveformformula_72At t=0, the incident reaches the boundary between the two mediums at x=0. Therefore, the corresponding reflected wave and the transmitted wave will have the waveformsformula_73The continuity condition at the boundary isformula_74This gives the equationsformula_75And we have the reflectivity and transmissivityformula_76When , the reflected wave has a reflection phase change of 180°, since . The  energy conservation can be verified by formula_77The above discussion holds true for any component, regardless of its angular frequency of . The limiting case of corresponds to a "fixed end" that doesn't move, whereas the limiting case of corresponds to a "free end". The Sturm–Liouville formulation. A flexible string that is stretched between two points and satisfies the wave equation for and . On the boundary points, may satisfy a variety of boundary conditions. A general form that is appropriate for applications is formula_78 formula_79 where and are non-negative. The case where u is required to vanish at an  endpoint (i.e. "fixed end") is the limit of this condition when the respective or approaches infinity. The method of separation of variables consists in looking for solutions of this problem in the special form formula_80 A consequence is that formula_81 The eigenvalue must be determined so that there is a non-trivial solution of the boundary-value problem formula_82 formula_83 This is a special case of the general problem of Sturm–Liouville theory . If and are positive, the eigenvalues are all positive, and the solutions are trigonometric functions. A solution that satisfies square-integrable initial conditions for and can be  obtained from expansion of these functions in the appropriate trigonometric series. Investigation by numerical methods. Approximating the continuous string with a finite number of equidistant mass points one gets the following physical model: |frame|Figure 1: Three consecutive mass points of the discrete model for a string If each mass point has the mass , the tension of the string is , the separation between the mass points is and are the offset of these 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  is and the vertical component of the force towards point is Taking the sum of these two forces and dividing with the mass one gets for the vertical motion: As the mass density is formula_84 this can be written The wave equation is obtained by letting in which case takes the form where is continuous function of two variables, takes the form and formula_85 for formula_86 with formula_89</ref> with all formula_90. The blue curve is the state at time formula_91 i.e. after a time that corresponds to the time a wave that is moving with the nominal wave velocity  would need for one fourth of the length of the string. Figure 3 displays the shape of the string at the times formula_92. The wave travels in direction right with the speed without being actively constraint by the boundary conditions at the two extremes of the string. The shape of the wave is constant, i.e. the curve is indeed of the form . Figure 4 displays the shape of the string at the times formula_93. 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 formula_94 when the direction of motion is reversed. The red, green and blue curves are the states at the times formula_95 while the 3 black curves correspond to the states at times formula_96 with the wave starting to move back towards left. Figure 6 and figure 7 finally display the shape of the string at the times formula_97 and formula_98. 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. Several space dimensions. 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. The one-dimensional initial-boundary value theory may be extended to an arbitrary number of space dimensions. Consider a domain in -dimensional space, with boundary . Then the wave equation is to be satisfied if is in and . On the boundary of , the solution shall satisfy formula_99 where is the unit outward normal to , and is a non-negative function defined on . The case where  vanishes on is a limiting case for approaching infinity. The initial conditions are formula_100 where and are defined in . This problem may be solved by expanding and in the eigenfunctions of the Laplacian in , which satisfy the boundary conditions. Thus the eigenfunction satisfies formula_101 in , and formula_102 on . In the case of two space dimensions, the eigenfunctions may be interpreted as the modes of vibration of a drumhead stretched over the boundary . If is a circle, then these eigenfunctions have an angular component that is a trigonometric function of the polar angle , 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. Inhomogeneous wave equation in one dimension. The inhomogeneous wave equation in one dimension is the following: formula_103 with initial conditions given by formula_26 formula_105 The function 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 , the value of depends only on the values of and and the values of the function between and . 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 , 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 as . Suppose we integrate the inhomogeneous wave equation over this region. formula_106 To simplify  this greatly, we can use Green's theorem to simplify the left side to get the following: formula_107 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 formula_108 In the above, the term to be integrated with respect to time disappears because the time interval involved is zero, thus . For the other two sides of the region, it is worth noting that is a constant, namely , where the sign is chosen appropriately. Using this, we can get the relation , again choosing the  right sign: formula_109 And similarly for the final boundary segment: formula_110 Adding the three results together and putting them back in the original integral: formula_111 Solving for we arrive at formula_112 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 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. Wave equation for inhomogeneous media, three-dimensional case. For one-way wave propagation, i.e. wave are travelling in a pre-defined wave direction (formula_55 or formula_56) in inhomogeneous media, wave propagation can also be calculated with a tensorial one-way wave equation (resulting from factorization of the vectorial two way wave equation) and an analytical solution can be derived. Other coordinate systems. 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 . Further generalizations. Elastic waves. The elastic wave equation  (also known as the Linear elasticity#Elastodynamics in terms of displacements|Navier–Cauchy 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: formula_115 where: By using the elastic wave equation can be rewritten into the more common form of the Navier–Cauchy equation . Note that in the elastic wave equation, both force and displacement are vector (geometry)|vector quantities. Thus, this equation is sometimes known as the vector wave equation. As an aid to understanding, the reader will observe that if and are set to zero, this becomes (effectively) Maxwell's equation for the propagation of the electric field , which has only transverse waves. Dispersion relation. In Dispersion (optics)|dispersive wave phenomena, the speed of wave propagation varies with the wavelength of the wave, which is reflected by a dispersion relation formula_116 where is the angular frequency and is the wavevector describing plane wave solutions . For light waves, the dispersion relation is , but in general, the constant speed gets replaced by a variable phase velocity : formula_117 External links. Category:Equations of physics Category:Hyperbolic partial differential equations Category:Wave mechanics