The wave equation is an important partial differential equation that describes 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.

Introduction

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The wave equation is the prototypical example of a hyperbolic partial differential equation. In its simplest form, the wave equation refers to a scalar quantity u that satisfies:

$\{\; \backslash partial^2\; u\; \backslash over\; \backslash partial\; t^2\; \}\; =\; c^2\; \backslash Delta\; u,$

where c is a fixed constant equal to the propagation speed of the wave and where $\backslash Delta\; =\; \backslash nabla^2$ is the Laplacian. For a sound wave in air at 20°C this 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:

$v\_\backslash mathrm\{p\}\; =\; \backslash frac\{\backslash omega\}\{k\}.$

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:

$\{\; \backslash partial^2\; u\; \backslash over\; \backslash partial\; t^2\; \}\; =\; c(u)^2\; \backslash Delta\; u$

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).

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:

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

where:

• $\backslash lambda$ and $\backslash mu$ are the so-called Lamé moduli describing the elastic properties of the medium,
• $\backslash rho$ is density,
• $\backslash bold\{f\}$ is the source function (driving force),
• and $\backslash bold\{u\}$ 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.

Variations of the wave equation are also found in quantum mechanics and general relativity.

Scalar wave equation in one space dimension

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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:

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:

$F\_\{Newton\}=m\; \backslash cdot\; a(t)=m\; \backslash cdot\; d\backslash xi\backslash ,d\backslash eta.\; \backslash ,$

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

$(x\; -\backslash xi)^2\; +\; (y\; -\; \backslash eta)^2\; =\; c^2\; t^2,\; \backslash ,$

but also on data that are interior to that cone.

Problems with boundaries

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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

$-u\_x(t,0)\; +\; a\; u(t,0)\; =\; 0,\; \backslash ,$

$u\_x(t,L)\; +\; b\; u(t,L)\; =\; 0,\backslash ,$

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

$u(t,x)\; =\; T(t)\; v(x).\backslash ,$

A consequence is that

$\backslash frac\{T$}{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

$v\text{'}\text{'}\; +\; \backslash lambda\; v=0,\; \backslash ,$

$-v\text{'}(0)\; +\; a\; v(0)\; =\; 0,\; \backslash quad\; v\text{'}(L)\; +\; b\; v(L)=0.\backslash ,$

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.

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

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

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

$u(0,x)\; =\; f(x),\; \backslash quad\; u\_t=g(x),\; \backslash ,$

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

$\backslash nabla\; \backslash cdot\; \backslash nabla\; v\; +\; \backslash lambda\; v\; =\; 0,\; \backslash ,$

in D, and

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

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.

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.

Inhomogenous wave equation in one dimension

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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.

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 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.

$\backslash int\; \backslash int\_\{R\_C\}\; \backslash left\; (\; c^2\; u\_\{x\; x\}(x,t)\; -\; u\_\{t\; t\}(x,t)\; \backslash right\; )\; dx\; dt\; =\; \backslash int\; \backslash int\_\{R\_C\}\; s(x,t)\; dx\; dt.$

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

$\backslash int\_\{\; L\_0\; +\; L\_1\; +\; L\_2\; \}\; \backslash left\; (\; -\; c^2\; u\_x(x,t)\; dt\; -\; u\_t(x,t)\; dx\; \backslash right\; )\; =\; \backslash int\; \backslash int\_\{R\_C\}\; s(x,t)\; dx\; dt.$

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

$\backslash int^\{x\_i\; +\; c\; t\_i\}\_\{x\_i\; -\; c\; t\_i\}\; -\; u\_t(x,0)\; dx\; =\; -\; \backslash int^\{x\_i\; +\; c\; t\_i\}\_\{x\_i\; -\; c\; t\_i\}\; g(x)\; dx.$

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 $x\; \backslash pm\; c\; t$ is a constant, namingly $x\_i\; \backslash pm\; c\; t\_i$, where the sign is chosen appropriately. Using this, we can get the relation $dx\; \backslash pm\; c\; dt\; =\; 0$, again choosing the right sign:

$\backslash int\_\{L\_1\}\; \backslash left\; (\; -\; c^2\; u\_x(x,t)\; dt\; -\; u\_t(x,t)\; dx\; \backslash right\; )\; \backslash ,$

$=\; \backslash int\_\{L\_1\}\; \backslash left\; (\; c\; u\_x(x,t)\; dx\; +\; c\; u\_t(x,t)\; dt\; \backslash right)\backslash ,$

$=\; c\; \backslash int\_\{L\_1\}\; d\; u(x,t)\; =\; c\; u(x\_i,t\_i)\; -\; c\; f(x\_i\; +\; c\; t\_i).\backslash ,$

And similarly for the final boundary segment:

$\backslash int\_\{L\_2\}\; \backslash left\; (\; -\; c^2\; u\_x(x,t)\; dt\; -\; u\_t(x,t)\; dx\; \backslash right\; )$

$=\; -\; \backslash int\_\{L\_2\}\; \backslash left\; (\; c\; u\_x(x,t)\; dx\; +\; c\; u\_t(x,t)\; dt\; \backslash right\; )$

$=\; -\; c\; \backslash int\_\{L\_2\}\; d\; u(x,t)\; =\; -\; \backslash left\; (\; c\; f(x\_i\; -\; c\; t\_i)\; -\; c\; u(x\_i,t\_i)\; \backslash right\; )$

$=\; c\; u(x\_i,t\_i)\; -\; c\; f(x\_i\; -\; c\; t\_i).\backslash ,$

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

$-\; \backslash int^\{x\_i\; +\; c\; t\_i\}\_\{x\_i\; -\; c\; t\_i\}\; g(x)\; dx\; +\; c\; u(x\_i,t\_i)\; -\; c\; f(x\_i\; +\; c\; t\_i)\; +\; c\; u(x\_i,t\_i)\; -\; c\; f(x\_i\; -\; c\; t\_i)\; =\; \backslash int\; \backslash int\_\{R\_C\}\; s(x,t)\; dx\; dt$

$2\; c\; u(x\_i,t\_i)\; -\; \backslash int^\{x\_i\; +\; c\; t\_i\}\_\{x\_i\; -\; c\; t\_i\}\; g(x)\; dx\; -\; c\; f(x\_i\; +\; c\; t\_i)\; -\; c\; f(x\_i\; -\; c\; t\_i)\; =\; \backslash int\; \backslash int\_\{R\_C\}\; s(x,t)\; dx\; dt$

$2\; c\; u(x\_i,t\_i)\; =\; \backslash int^\{x\_i\; +\; c\; t\_i\}\_\{x\_i\; -\; c\; t\_i\}\; g(x)\; dx\; +\; c\; f(x\_i\; +\; c\; t\_i)\; +\; c\; f(x\_i\; -\; c\; t\_i)\; +\; \backslash int\; \backslash int\_\{R\_C\}\; s(x,t)\; dx\; dt$

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

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

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In three dimensions, the wave equation, when written in elliptic cylindrical coordinates, may be solved by separation of variables, leading to the Mathieu differential equation.

See also

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• Helmholtz equation
• Acoustic wave equation
• Electromagnetic wave equation
• Motor variable
• Doppler effect
• Schrödinger equation

References

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• 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.

Partial differential equations | Equations | Wave mechanics

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