NationMaster - Encyclopedia: Wave equation

The wave equation is an important partial differential equation that describes the propagation of a variety of waves, such as sound waves, light waves and water waves. It arises in fields such as acoustics, 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. In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... This article is about waves in the most general scientific sense. ... Sound is a disturbance of mechanical energy that propagates through matter as a longitudinal wave, and therefore is a mechanical wave. ... It has been suggested that this article or section be merged with Electromagnetic radiation. ... Impact of a drop of water creating circular capillary waves. ... Acoustics is a branch of physics and is the study of sound (mechanical waves in gases, liquids, and solids). ... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge and magnetic charge, and is in turn affected by the presence and motion of those particles. ... Fluid dynamics is the sub-discipline of fluid mechanics dealing with fluids (liquids and gases) in motion. ... A musical instrument is a device constructed or modified with the purpose of making music. ... Jean le Rond dAlembert, pastel by Maurice Quentin de La Tour Jean le Rond dAlembert (November 16, 1717 â€“ October 29, 1783) was a French mathematician, mechanician, physicist and philosopher. ... Leonhard Euler (pronounced Oiler; IPA ) (April 15, 1707 â€“ September 7, 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... Daniel Bernoulli Daniel Bernoulli (Groningen, February 8, 1700 â€“ Basel, March 17, 1782) was a Dutch-born mathematician who spent much of his life in Basel, Switzerland. ... Joseph Louis Lagrange (January 25, 1736 – April 10, 1813) was an Italian mathematician and astronomer who later lived in France and Prussia. ...

1 Introduction
2 Scalar wave equation in one space dimension
- 2.1 Derivation of the wave equation
- 2.2 Solution of the initial value problem
3 Scalar wave equation in three space dimensions
- 3.1 Spherical waves
- 3.2 Solution of a general initial-value problem
4 Scalar wave equation in two space dimensions
5 Problems with boundaries
- 5.1 One space dimension
- 5.2 Several space dimensions
6 Inhomogenous wave equation in one dimension
7 Other coordinate systems
8 See also
9 References

Introduction

The wave equation is the prototypical example of a hyperbolic partial differential equation. In its simplest form, the wave equation refers to a scalar function u that satisfies: A hyperbolic partial differential equation is usually a second-order partial differential equation of the form with . ... In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector. ...

${ partial^2 u over partial t^2 } = c^2 nabla^2 u,$

where $nabla^2$ is the Laplacian and where c is a fixed constant equal to the propagation speed of the wave. For a sound wave in air at 20°C this constant is about 343 m/s (see speed of sound). For the vibration of a string the speed can vary widely, depending upon the linear density of the string and the tension on it. For a spiral spring (a slinky) it can be as slow as a meter per second. More realistic differential equations for waves allow for the speed of wave propagation to vary with the frequency of the wave, a phenomenon known as dispersion. In such a case, c must be replaced by the phase velocity: In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ... In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ... The speed of sound is a term used to describe the speed of sound waves passing through an elastic medium. ... A vibration in a string is a wave. ... Metal Slinky Rainbow-colored plastic Slinky A Slinky, or Lazy-Spring, is a coil-shaped toy invented by mechanical engineer Richard James in Philadelphia, Pennsylvania. ... Dispersion can mean any of several things: A phenomenon that causes the separation of a wave into components of varying frequency. ... The phase velocity of a wave is the rate at which the phase of the wave propagates in space. ...

$v_mathrm{p} = frac{omega}{k}.$

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:

${ partial^2 u over partial t^2 } = c(u)^2 nabla^2 u$

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 factor (which is positive for the wave moving along the flow and negative for the reflected wave). An F/A-18 Hornet breaking the sound barrier. ...

The elastic wave equation in three dimensions describes the propagation of waves in an isotropic homogeneous 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: Isotropic means independent of direction. Isotropic radiation has the same intensity regardless of the direction of measurement, and an isotropic field exerts the same action regardless of how the test particle is oriented. ... Elasticity is a branch of physics which studies the properties of elastic materials. ... A seismic wave is a wave that travels through the Earth, often as the result of an earthquake or explosion. ... Adjectives: Terrestrial, Terran, Telluric, Tellurian, Earthly Atmosphere Surface pressure: 101. ... Ultrasound is sound with a frequency greater than the upper limit of human hearing, approximately 20 kilohertz. ...

$rho{ ddot{bold{u}}} = bold{f} + ( lambda + 2mu )nabla(nabla cdot bold{u}) - munabla times (nabla times bold{u})$

where:

$λ$ and $μ$ are the so-called Lamé parameters describing the elastic properties of the medium,
$ρ$ is density,
$bold{f}$ is the source function (driving force),
and $bold{u}$ is displacement.

Note that in this equation, both force and displacement are vector quantities. Thus, this equation is sometimes known as the vector wave equation. In linear elasticity, the LamÃ© parameters are the two parameters which in homogenous, isotropic materials satisfy the equation where is the stress and the strain tensor. ... In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ...

Variations of the wave equation are also found in quantum mechanics and general relativity. Fig. ... General relativity (GR) [also called the general theory of relativity (GTR) and general relativity theory (GRT)] is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ...

Scalar wave equation in one space dimension

Derivation of the wave equation

The wave equation in the one dimensional case can be derived in the following way: Imagine an array of little weights of mass m interconnected with springs (or slinkies) of length h . The springs have a stiffness of k: A Slinky is a coil-shaped toy, invented by Naval engineer Richard James and his wife, Betty James. ... Stiffness is the resistance of an elastic body to deflection or deformation by an applied force. ...

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: Image File history File links File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...

$F_{mathit{Newton}}=m cdot a(t)=m cdot {{partial^2 over partial t^2}u(x+h,t)}$

$F_mathit{Hooke} = F_{x+2h} + F_x = k left [ {u(x+2h,t) - u(x+h,t)} right ] + k[u(x,t) - u(x+h,t)]$

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

$m{partial^2u(x+h,t) over partial t^2}= k[u(x+2h,t)-u(x+h,t)-u(x+h,t)+u(x,t)]$

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 = N h of total mass M = N m, and the total stiffness of the array K = k/N we can write the above equation as: Stiffness is the resistance of an elastic body to deflection or deformation by an applied force. ...

${partial^2u(x+h,t) over partial t^2}={KL^2 [over M}{u(x+2h,t)-2u(x+h,t)+u(x,t)] over h^2}$

Taking the limit $Nrightarrow infty,hrightarrow 0$ (and assuming smoothness) one gets:

${partial^2 u(x,t) over partial t^2}={KL^2 over M}{ partial^2 u(x,t) over partial x^2 }$

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

Solution of the initial value problem

The general solution to the one dimensional scalar wave equation

${ partial^2 u over partial t^2 } = c^2 { partial^2 u over partial x^2 }$

was derived by d'Alembert. The wave equation may be written in the factor form Jean le Rond dAlembert, pastel by Maurice Quentin de La Tour Jean le Rond dAlembert (November 16, 1717 â€“ October 29, 1783) was a French mathematician, mechanician, physicist and philosopher. ...

$left[ frac{part}{part t} - cfrac{part}{part x}right] left[ frac{part}{part t} + cfrac{part}{part x}right] u = 0.,$

Consequently, if F and G are arbitrary functions, then any sum of the form

$u(x,t) = F(x-ct) + G(x+ct) ,$

will satisfy the wave equation. The two terms are traveling waves: any point on the wave form given by a specific argument for F or G will move with velocity c in either the forward or backwards direction: forwards for F and backwards for G. These functions can be determined to satisfy arbitrary initial conditions:

$u(x,0)=f(x) ,$

$u_t(x,0)=g(x) ,$

The result is d'Alembert's formula: In mathematics, and specifically partial differential equations, dÂ´Alemberts formula is the general solution to the one-dimensional wave equation: . It is named after the mathematician Jean le Rond dAlembert. ...

$u(x,t) = frac{f(x-ct) + f(x+ct)}{2} + frac{1}{2c} int_{x-ct}^{x+ct} g(s) ds$

In the classical sense if $f(x) in C^k$ and $g(x) in C^{k-1}$ then $u(t,x) in C^k$ . 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. In mathematics, a linear differential equation is a differential equation of the form Ly = f, where the differential operator L is a linear operator, y is the unknown function, and the right hand side f is a given function. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ...

Scalar wave equation in three space dimensions

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.

Spherical waves

The wave equation is unchanged under rotations of the spatial coordinates, and therefore one may expect to find solutions that depend only on the radial distance from a given point. Such solutions must satisfy

$u_{tt} - c^2 left( u_{rr} + frac{2}{r} u_r right) =0. ,$

This equation may be rewritten as

$(ru)_{tt} -c^2 (ru)_{rr}=0; ,$

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

$u(t,r) = frac{1}{r} F(r-ct) + frac{1}{r} G(r+ct), ,$

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. 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. Look up point source in Wiktionary, the free dictionary. ...

Solution of a general initial-value problem

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

$r^2 = (x-&# 0;^2 + (y-eta)^2 + (z-zeta)^2. ,$

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

$u(t,x,y,z) = frac{1}{4pi c} iiint varphi(&# 0;eta,zeta) frac{delta(r-ct)}{r} d&# 0;,deta,dzeta; ,$

the denominator 4πc is a convenience.

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

$u(t,x,y,z) = frac{t}{4pi} iint_S varphi(x +ctalpha, y +ctbeta, z+ctgamma) domega, ,$

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:

$u(t,x,y,z) = t M_{ct}[phi]. ,$

It follows that

$u(0,x,y,z) = 0, quad u_t(0,x,y,z) = phi(x,y,z). ,$

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

$v(t,x,y,z) = frac{part}{part t} left( t M_{ct}[psi] right), ,$

then

$v(0,x,y,z) = psi(x,y,z), quad v_t(0,x,y,z) = 0. ,$

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 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 Gårding (1970, 1973). Sir Michael Francis Atiyah, OM, FRS (born 22 April 1929) is a mathematician who was born in London. ... Raoul Bott (Harvard University News Office) Raoul Bott, FRS (born September 24, 1923, died December 20, 2005) was a mathematician known for numerous basic contributions to geometry in its broad sense. ... Lars GÃ¥rding (born 1919) is a Swedish mathematician. ...

Scalar wave equation in two space dimensions

In two space dimensions, the wave equation is

$u_{tt} = c^2 left( u_{xx} + u_{yy} right). ,$

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

$u(0,x,y)=0, quad u_t(0,x,y) = phi(x,y), ,$

then the three-dimensional solution formula becomes

$u(t,x,y) = tM_{ct}[phi] = frac{t}{4pi} iint_S phi(x + ctalpha,, y + ctbeta) domega,,$

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:

$u(t,x,y) = frac{1}{2pi c} iint_D frac{phi(x+&# 0; y +eta)}{sqrt{(ct)^2 - &# 0;2 - eta^2}} d&# 0;,deta. ,$

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

$(x -&# 0;^2 + (y - eta)^2 = c^2 t^2, ,$

but also on data that are interior to that cone.

Problems with boundaries

One space dimension

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

$-u_x(t,0) + a u(t,0) = 0, ,$

$u_x(t,L) + b u(t,L) = 0,,$

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 In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to re-write an equation so that each of two variables occurs on a different side of the equation. ...

$u(t,x) = T(t) v(x).,$

A consequence is that

$frac{T''}{c^2T} = frac{v''}{v} = -lambda. ,$

The eigenvalue λ must be determined so that there is a non-trivial solution of the boundary-value problem In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...

$v'' + lambda v=0, ,$

$-v'(0) + a v(0) = 0, quad v'(L) + b v(L)=0.,$

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_t can be obtained from expansion of these functions in the appropriate trigonometric series. In mathematics and its applications, a classical Sturm-Liouville equation, named after Jacques Charles FranÃ§ois Sturm (1803-1855) and Joseph Liouville (1809-1882), is a real second-order linear differential equation of the form where the functions p(x), q(x), and w(x) are specified at the outset...

Several space dimensions

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 $t > 0$ . One the boundary of D, the solution u shall satisfy

$frac{part u}{part n} + a u =0, ,$

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

$u(0,x) = f(x), quad u_t=g(x), ,$

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

$nabla cdot nabla v + lambda v = 0, ,$

in D, and

$frac{part v}{part n} + a v =0, ,$

on 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. In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: for an arbitrary real or complex number Î±. The most common and important special case is where Î± is an integer n, then Î± is referred to... The Helmholtz equation, named for Hermann von Helmholtz, is the following elliptic partial differential equation: The Helmholtz equation often arises in the study of physical problems involving partial differential equations (PDEs) in both space and time. ...

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 functions of half-integer order. In mathematics, the spherical harmonics are the angular portion of an orthogonal set of solutions to Laplaces equation represented in a system of spherical coordinates. ... In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: for an arbitrary real or complex number Î±. The most common and important special case is where Î± is an integer n, then Î± is referred to...

Inhomogenous wave equation in one dimension

The inhomogenous wave equation in one dimension is the following:

c 2 u x x (x, t) - u t t (x, t) = s (x, t)

with initial conditions given by

u (x,0) = f (x)

u t (x,0) = g (x).

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. The Lorenz gauge (or Lorenz gauge condition) was published by the Danish physicist Ludwig Lorenz. ... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge and magnetic charge, and is in turn affected by the presence and motion of those particles. ...

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 $(x i, t i)$ , the value of $u (x i, t i)$ depends only on the values of $f (x i + c t i)$ and $f (x i - c t i)$ and the values of the function $g (x)$ between $(x i - c t i)$ and $(x i - c t i)$ . 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. In mathematics, and specifically partial differential equations, dÂ´Alemberts formula is the general solution to the one-dimensional wave equation: . It is named after the mathematician Jean le Rond dAlembert. ...

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 $(x i, t i)$ as $R C$ . Suppose we integrate the in-homogenous wave equation over this region.

$iint limits_{R_C} left ( c^2 u_{x x}(x,t) - u_{t t}(x,t) right ) dx dt = iint limits_{R_C} s(x,t) dx dt.$

To simplify this greatly, we can use Green's theorem to simplify the left side to get the following: In physics and mathematics, Greens theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. Greens theorem was named after British scientist George Green and is a special two-dimensional case of...

$int_{ L_0 + L_1 + L_2 } left ( - c^2 u_x(x,t) dt - u_t(x,t) dx right ) = iint limits_{R_C} s(x,t) dx dt.$

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

$int^{x_i + c t_i}_{x_i - c t_i} - u_t(x,0) dx = - int^{x_i + c t_i}_{x_i - c t_i} g(x) dx.$

In the above, the term to be integrated with respect to time disappears because the time interval involved is zero, thus $d t = 0$ .

For the other two sides of the region, it is worth noting that $x pm c t$ is a constant, namingly $x_i pm c t_i$ , where the sign is chosen appropriately. Using this, we can get the relation $dx pm c dt = 0$ , again choosing the right sign:

$int_{L_1} left ( - c^2 u_x(x,t) dt - u_t(x,t) dx right ) ,$

$= int_{L_1} left ( c u_x(x,t) dx + c u_t(x,t) dt right),$

$= c int_{L_1} d u(x,t) = c u(x_i,t_i) - c f(x_i + c t_i).,$

And similarly for the final boundary segment:

$int_{L_2} left ( - c^2 u_x(x,t) dt - u_t(x,t) dx right )$

$= - int_{L_2} left ( c u_x(x,t) dx + c u_t(x,t) dt right )$

$= - c int_{L_2} d u(x,t) = - left ( c f(x_i - c t_i) - c u(x_i,t_i) right )$

$= c u(x_i,t_i) - c f(x_i - c t_i).,$

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

$- int^{x_i + c t_i}_{x_i - c t_i} g(x) dx + c u(x_i,t_i) - c f(x_i + c t_i) + c u(x_i,t_i) - c f(x_i - c t_i) = iint limits_{R_C} s(x,t) dx dt$

$2 c u(x_i,t_i) - int^{x_i + c t_i}_{x_i - c t_i} g(x) dx - c f(x_i + c t_i) - c f(x_i - c t_i) = iint limits_{R_C} s(x,t) dx dt$

$2 c u(x_i,t_i) = int^{x_i + c t_i}_{x_i - c t_i} g(x) dx + c f(x_i + c t_i) + c f(x_i - c t_i) + iint limits_{R_C} s(x,t) dx dt$

$u(x_i,t_i) = frac{f(x_i + c t_i) + f(x_i - c t_i)}{2} + frac{1}{2 c}int^{x_i + c t_i}_{x_i - c t_i} g(x) dx + frac{1}{2 c}int^{t_i}_0 int^{x_i + c left ( t_i - t right )}_{x_i - c left ( t_i - t right )} s(x,t) dx dt. ,$

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 $(x i, t i)$ 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.

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. Elliptic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional elliptic coordinate system in the perpendicular -direction. ... In mathematics, the Mathieu functions are certain special functions useful for treating a variety of interesting problems in applied mathematics, including vibrating elliptical drumheads, the phenomenon of parametric resonance in forced oscillators, exact plane wave solutions in general relativity. ...

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.
"Linear Wave Equations", EqWorld: The World of Mathematical Equations.
"Nonlinear Wave Equations", EqWorld: The World of Mathematical Equations.
William C. Lane, "MISN-0-201 The Wave Equation and Its Solutions", Project PHYSNET.
Relativistic wave equations with fractional derivatives and pseudodifferential operators, by Petr Zavada, Journal of Applied Mathematics, vol. 2, no. 4, pp. 163-197, 2002. doi:10.1155/S1110757X02110102 (available online or as the arXiv preprint)

Categories: Fundamental physics concepts | Hyperbolic partial differential equations | Wave mechanics

Results from FactBites:

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