Wave equation 1

Wave equationThe wave equation is an important second-order linear partial differential equation for the description of waves – asthey occur in physics – such as sound waves, light waves and water waves. It arises in fields like acoustics,electromagnetics, and fluid dynamics. Historically, the problem of a vibrating string such as that of a musicalinstrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-LouisLagrange.[1][2][3][4]

A pulse traveling through a string with fixed endpoints as modeledby the wave equation.

Spherical waves coming from a point source.

Introduction

Wave equations are examples of hyperbolic partialdifferential equations, but there are many variations.

In its simplest form, the wave equation concerns a timevariable t, one or more spatial variables x1, x2, …, xn,and a scalar function u = u (x1, x2, …, xn; t), whosevalues could model the displacement of a wave. Thewave equation for u is

where ∇2 is the (spatial) Laplacian and where c is afixed constant.

Solutions of this equation that are initially zero outsidesome restricted region propagate out from the region ata fixed speed in all spatial directions, as do physicalwaves from a localized disturbance; the constant c isidentified with the propagation speed of the wave. Thisequation is linear, as the sum of any two solutions isagain a solution: in physics this property is called thesuperposition principle.

The equation alone does not specify a solution; aunique solution is usually obtained by setting a problemwith further conditions, such as initial conditions, which prescribe the value and velocity of the wave. Anotherimportant class of problems specifies boundary conditions, for which the solutions represent standing waves, orharmonics, analogous to the harmonics of musical instruments.

To model dispersive wave phenomena, those in which the speed of wave propagation varies with the frequency ofthe wave, the constant c is replaced by the phase velocity:

The elastic wave equation in three dimensions describes the propagation of waves in an isotropic homogeneouselastic medium. Most solid materials are elastic, so this equation describes such phenomena as seismic waves in theEarth and ultrasonic waves used to detect flaws in materials. While linear, this equation has a more complex formthan the equations given above, as it must account for both longitudinal and transverse motion:

where:• λ and μ are the so-called Lamé parameters describing the elastic properties of the medium,

Wave equation 2

•• 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 quantities. Thus, this equation is sometimes knownas the vector wave equation.Variations of the wave equation are also found in quantum mechanics, plasma physics and general relativity.

Scalar wave equation in one space dimension

Derivation of the wave equation

From Hooke's law

The wave equation in the one dimensional case can be derived from Hooke's law in the following way: Imagine anarray of little weights of mass m interconnected with massless springs of length h . The springs have a springconstant of k:

Here u(x) measures the distance from the equilibrium of the mass situated at x. The forces exerted on the mass m atthe location x+h are:

The equation of motion for the weight at the location x+h is given by equating these two forces:

where the time-dependence of u(x) has been made explicit.If the array of weights consists of N weights spaced evenly over the length L = Nh of total mass M = Nm, and thetotal spring constant of the array K = k/N we can write the above equation as:

Taking the limit N → ∞, h → 0 and assuming smoothness one gets:

(KL2)/M is the square of the propagation speed in this particular case.

Wave equation 3

General solutionThe one dimensional wave equation is unusual for a partial differential equation in that a relatively simple generalsolution may be found. Defining new variables:[5]

changes the wave equation into

which leads to the general solution

or equivalently:

In other words, solutions of the 1D wave equation are sums of a right traveling function F and a left travelingfunction 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 Rondd'Alembert.[6]

Another way to arrive at this result is to note that the wave equation may be "factored":

and therefore:

These last two equations are advection equations, one left traveling and one right, both with constant speed c.For an initial value problem, the arbitrary functions F and G can be determined to satisfy initial conditions:

The result is d'Alembert's formula:

In the classical sense if f(x) ∈ Ck and g(x) ∈ Ck−1 then u(t, x) ∈ Ck. However, the waveforms F and G may also begeneralized functions, such as the delta-function. In that case, the solution may be interpreted as an impulse thattravels to the right or the left.The basic wave equation is a linear differential equation and so it will adhere to the superposition principle. Thismeans that the net displacement caused by two or more waves is the sum of the displacements which would havebeen caused by each wave individually. In addition, the behavior of a wave can be analyzed by breaking up the waveinto components, e.g. the Fourier transform breaks up a wave into sinusoidal components.

Wave equation 4

Scalar wave equation in three space dimensionsThe solution of the initial-value problem for the wave equation in three space dimensions can be obtained from thesolution for a spherical wave. This result can then be used to obtain the solution in two space dimensions.

Spherical waves

Cut-away of spherical wavefronts, with awavelength of 10 units, propagating from a point

source.

The wave equation is unchanged under rotations of the spatialcoordinates, because the Laplacian operator is invariant under rotation,and therefore one may expect to find solutions that depend only on theradial distance from a given point. Such solutions must satisfy

This equation may be rewritten as

the quantity ru satisfies the one-dimensional wave equation. Thereforethere are solutions in the form

where F and G are arbitrary functions. Each term may be interpreted as a spherical wave that expands or contractswith velocity c. Such waves are generated by a point source, and they make possible sharp signals whose form isaltered only by a decrease in amplitude as r increases (see an illustration of a spherical wave on the top right). Suchwaves exist only in cases of space with odd dimensions.

Monochromatic spherical wave

A point source is vibrating at a single frequency f with phase = 0 at t = 0 with a peak-to-peak magnitude of 2a. Aspherical wave is propagated from the point. The phase of the propagated wave changes as kr where r is the distancetravelled from the source. The magnitude falls off as 1/r since the energy falls off as r−2. The amplitude of thespherical wave at r is therefore given by:[7]

Solution of a general initial-value problemThe wave equation is linear in u and it is left unaltered by translations in space and time. Therefore we can generate agreat variety of solutions by translating and summing spherical waves. Let φ(ξ,η,ζ) be an arbitrary function of threeindependent variables, and let the spherical wave form F be a delta-function: that is, let F be a weak limit ofcontinuous functions whose integral is unity, but whose support (the region where the function is non-zero) shrinksto 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

If u is a superposition of such waves with weighting function φ, then

the denominator 4πc is a convenience.

Wave equation 5

From the definition of the delta-function, u may also be written as

where α, β, and γ are coordinates on the unit sphere S, and ω is the area element on S. This result has theinterpretation that u(t,x) is t times the mean value of φ on a sphere of radius ct centered at x:

It follows that

The mean value is an even function of t, and hence if

then

These formulas provide the solution for the initial-value problem for the wave equation. They show that the solutionat 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 lightcone drawn backwards from P. It does not depend upon data on the interior of this sphere. Thus the interior of thesphere is a lacuna for the solution. This phenomenon is called Huygens' principle. It is true for odd numbers ofspace dimension, where for one dimension the integration is performed over the boundary of an interval with respectto the Dirac measure. It is not satisfied in even space dimensions. The phenomenon of lacunas has been extensivelyinvestigated in Atiyah, Bott and Gårding (1970, 1973).

Scalar wave equation in two space dimensionsIn two space dimensions, the wave equation is

We can use the three-dimensional theory to solve this problem if we regard u as a function in three dimensions that isindependent of the third dimension. If

then the three-dimensional solution formula becomes

where α and β are the first two coordinates on the unit sphere, and dω is the area element on the sphere. This integralmay be rewritten as an integral over the disc D with center (x,y) and radius ct:

It is apparent that the solution at (t,x,y) depends not only on the data on the light cone where

but also on data that are interior to that cone.

Wave equation 6

Scalar wave equation in general dimension and Kirchhoff's formulaeWe want to find solutions to utt−Δu = 0 for u : Rn × (0, ∞) → R with u(x, 0) = g(x) and ut(x, 0) = h(x). See Evans formore details.

Odd dimensions

Assume n ≥ 3 is an odd integer and g ∈ Cm+1(Rn), h ∈ Cm(Rn) for m = (n+1)/2. Let and let

thenu ∈ C2(Rn × [0, ∞))utt−Δu = 0 in Rn × (0, ∞)

Even dimensions

Assume n ≥ 2 is an even integer and g ∈ Cm+1(Rn), h ∈ Cm(Rn), for m = (n+2)/2. Let and let

thenu ∈ C2(Rn × [0, ∞))utt−Δu = 0 in Rn × (0, ∞)

Problems with boundaries

One space dimension

The Sturm-Liouville formulation

A flexible string that is stretched between two points x = 0 and x = L satisfies the wave equation for t > 0 and0 < x < L. On the boundary points, u may satisfy a variety of boundary conditions. A general form that is appropriatefor applications is

where a and b are non-negative. The case where u is required to vanish at an endpoint is the limit of this conditionwhen the respective a or b approaches infinity. The method of separation of variables consists in looking forsolutions of this problem in the special form

A consequence is that

Wave equation 7

The eigenvalue λ must be determined so that there is a non-trivial solution of the boundary-value problem

This is a special case of the general problem of Sturm–Liouville theory. If a and b are positive, the eigenvalues areall positive, and the solutions are trigonometric functions. A solution that satisfies square-integrable initial conditionsfor u and ut 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 physicalmodel:

Figure 1: Three consecutive mass points of thediscrete model for a string

If each mass point has the mass m, the tension of the string is f, theseparation between the mass points is Δx and ui, i = 1, ..., n are theoffset of these n points from their equilibrium points (i.e. their positionon a straight line between the two attachment points of the string) thevertical component of the force towards point i+1 is

(1)

and the vertical component of the force towards point i−1 is

(2)

Taking the sum of these two forces and dividing with the mass m one gets for the vertical motion:

(3)

As the mass density is

this can be written

(4)

The wave equation is obtained by letting Δx → 0 in which case ui(t) takes the form u(x, t) where u(x, t) is continuous

function of two variables, takes the form and

But the discrete formulation (3) of the equation of state with a finite number of mass point is just the suitable one fora numerical propagation of the string motion. The boundary condition

Wave equation 8

where L is the length of the string takes in the discrete formulation the form that for the outermost points u1 and unthe equation of motion are

(5)

and

(6)

while for 1 < i < n

(7)

where

If the string is approximated with 100 discrete mass points one gets the 100 coupled second order differentialequations (5), (6) and (7) or equivalently 200 coupled first order differential equations.Propagating these up to the times

using an 8-th order multistep method the 6 states displayed in figure 2 are found:

Figure 2: The string at 6 consecutive epochs, thefirst (red) corresponding to the initial time with

the string in rest

Figure 3: The string at 6 consecutive epochs

Figure 4: The string at 6 consecutive epochs

The red curve is the initial state at time zero at which the string is "letfree" in a predefined shape [8] with all . The blue curve is the

state at time , 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. The wave travels in direction right with

the speed without being actively constraint by the boundary

conditions at the two extrems of the string. The shape of the wave isconstant, i.e. the curve is indeed of the form f(x−ct).

Figure 4 displays the shape of the string at the times. The constraint on the right extreme

starts to interfere with the motion preventing the wave to raise the endof the string.

Figure 5 displays the shape of the string at the times when the direction

of motion is

Wave equation 9

Figure 5: The string at 6 consecutive epochs

Figure 6: The string at 6 consecutive epochs

Figure 7: The string at 6 consecutive epochs

reversed. The red, green and blue curves are the states at the times

while the 3 black curves correspond to the states at times with the wave

starting to move back towards left.Figure 6 and figure 7 finally display the shape of the string at the times

and .

The wave now travels towards left and the constraints at the end pointsare not active any more. When finally the other extreme of the stringthe direction will again be reversed in a way similar to what isdisplayed in figure 6

Several space dimensions

A solution of the wave equation in two dimensions with azero-displacement boundary condition along the entire outer

edge.

The one-dimensional initial-boundary value theory may beextended to an arbitrary number of space dimensions.Consider a domain D in m-dimensional x space, withboundary B. Then the wave equation is to be satisfied if x is inD and t > 0. On the boundary of D, the solution u shall satisfy

where n is the unit outward normal to B, and a is anon-negative function defined on B. The case where uvanishes on B is a limiting case for a approaching infinity.The initial conditions are

where f and g are defined in D. This problem may be solved by expanding f and g in the eigenfunctions of theLaplacian in D, which satisfy the boundary conditions. Thus the eigenfunction v satisfies

in D, and

on B.

Wave equation 10

In the case of two space dimensions, the eigenfunctions may be interpreted as the modes of vibration of a drumheadstretched over the boundary B. If B is a circle, then these eigenfunctions have an angular component that is atrigonometric function of the polar angle θ, multiplied by a Bessel function (of integer order) of the radialcomponent. Further details are in Helmholtz equation.If the boundary is a sphere in three space dimensions, the angular components of the eigenfunctions are sphericalharmonics, and the radial components are Bessel functions of half-integer order.

Inhomogeneous wave equation in one dimensionThe inhomogeneous wave equation in one dimension is the following:

with initial conditions given by

The function s(x, t) is often called the source function because in practice it describes the effects of the sources ofwaves on the medium carrying them. Physical examples of source functions include the force driving a wave on astring, 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 theproperty of the wave equation that its solutions obey causality. That is, for any point (xi, ti), the value of u(xi, ti)depends only on the values of f(xi+cti) and f(xi−cti) and the values of the function g(x) between (xi−cti) and (xi+cti).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 agiven 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, theonly area that needs to be considered is the area encompassing all the points that could causally affect the point beingconsidered. Denote the area that casually affects point (xi, ti) as RC. Suppose we integrate the inhomogeneous waveequation over this region.

To simplify this greatly, we can use Green's theorem to simplify the left side to get the following:

The left side is now the sum of three line integrals along the bounds of the causality region. These turn out to befairly easy to compute

In the above, the term to be integrated with respect to time disappears because the time interval involved is zero, thusdt = 0.For the other two sides of the region, it is worth noting that x±ct is a constant, namingly xi±cti, where the sign ischosen appropriately. Using this, we can get the relation dx±cdt = 0, again choosing the right sign:

Wave equation 11

And similarly for the final boundary segment:

Adding the three results together and putting them back in the original integral:

Solving for u(xi, ti) we arrive at

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 (xi, ti) compatible with the wave equation, it is clear that thefirst two terms are simply d'Alembert's formula, as stated above as the solution of the homogeneous wave equationin one dimension. The difference is in the third term, the integral over the source.

Other coordinate systemsIn three dimensions, the wave equation, when written in elliptic cylindrical coordinates, may be solved by separationof variables, leading to the Mathieu differential equation.

Notes[1] (http:/ / homes. chass. utoronto. ca/ ~cfraser/ vibration. pdf) (retrieved 13 Nov 2012).[2] Gerard F Wheeler. The Vibrating String Controversy, (http:/ / www. scribd. com/ doc/ 32298888/

The-Vibrating-String-Controversy-Am-J-Phys-1987-v55-n1-p33-37) (retrieved 13 Nov 2012). Am. J. Phys., 1987, v55, n1, p33-37.[3] For a special collection of the 9 groundbreaking papers by the three authors, see First Appearance of the wave equation: D'Alembert,

Leonhard Euler, Daniel Bernoulli. - the controversy about vibrating strings (http:/ / www. lynge. com/ item. php?bookid=38975&s_currency=EUR& c_sourcepage=) (retrieved 13 Nov 2012). Herman HJ Lynge and Son.

[4] For de Lagrange's contributions to the acoustic wave equation, can consult Acoustics: An Introduction to Its Physical Principles andApplications (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 paper on the wave equation& f=false) Allan D.Pierce, Acoustical Soc of America, 1989; page 18.(retrieved 9 Dec 2012)

[6] D'Alembert (1747) "Recherches sur la courbe que forme une corde tenduë mise en vibration" (http:/ / books. google. com/books?id=lJQDAAAAMAAJ& pg=PA214#v=onepage& q& f=false) (Researches on the curve that a tense cord forms [when] set intovibration), Histoire de l'académie royale des sciences et belles lettres de Berlin, vol. 3, pages 214-219.

• See also: D'Alembert (1747) "Suite des recherches sur la courbe que forme une corde tenduë mise en vibration" (http:/ / books. google.com/ books?id=lJQDAAAAMAAJ& pg=PA220#v=onepage& q& f=false) (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.

• See also: D'Alembert (1750) "Addition au mémoire sur la courbe que forme une corde tenduë mise en vibration," (http:/ / books. google.

com/ books?id=m5UDAAAAMAAJ& pg=PA355#v=onepage& q& f=false) Histoire de l'académie royale des sciences et belles lettres de

Wave equation 12

Berlin, vol. 6, pages 355-360.[7][7] RS Longhurst, Geometrical and Physical Optics, 1967, Longmans, Norwich[8] The initial state for "Investigation by numerical methods" is set with quadratic splines as follows:

UNIQ-math-0-0c56a222eb4da01b-QINU for UNIQ-math-1-0c56a222eb4da01b-QINUUNIQ-math-2-0c56a222eb4da01b-QINU for UNIQ-math-3-0c56a222eb4da01b-QINU

for

with

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.• " Linear Wave Equations (http:/ / eqworld. ipmnet. ru/ en/ solutions/ lpde/ wave-toc. pdf)", EqWorld: The World

of Mathematical Equations.• " Nonlinear Wave Equations (http:/ / eqworld. ipmnet. ru/ en/ solutions/ npde/ npde-toc2. pdf)", EqWorld: The

World of Mathematical Equations.• William C. Lane, " MISN-0-201 The Wave Equation and Its Solutions (http:/ / physnet2. pa. msu. edu/ home/

modules/ pdf_modules/ m201. pdf)", Project PHYSNET (http:/ / www. physnet. org).

External links• Francis Redfern. "Kinematic Derivation of the Wave Equation" (http:/ / prism. texarkanacollege. edu/

physicsjournal/ wave-eq. html). Physics Journal. — a step-by-step derivation suitable for an introductoryapproach to the subject.

• Nonlinear Wave Equations (http:/ / demonstrations. wolfram. com/ NonlinearWaveEquations/ ) by StephenWolfram and Rob Knapp, Nonlinear Wave Equation Explorer (http:/ / demonstrations. wolfram. com/NonlinearWaveEquationExplorer/ ) by Stephen Wolfram, and Wolfram Demonstrations Project.

• Mathematical aspects of wave equations are discussed on the Dispersive PDE Wiki (http:/ / tosio. math. toronto.edu/ wiki/ index. php/ Main_Page).

• Graham W Griffiths and William E. Schiesser (2009). Linear and nonlinear waves (http:/ / www. scholarpedia.org/ article/ Linear_and_nonlinear_waves). Scholarpedia (http:/ / www. scholarpedia. org/ ), 4(7):4308.doi:10.4249/scholarpedia.4308 (http:/ / dx. doi. org/ 10. 4249/ scholarpedia. 4308)

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