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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

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:

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

where c is a fixed constant equal to the propagation speed of the wave and where ${\displaystyle \Delta =\nabla ^{2}}$ 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:

${\displaystyle v_{\mathrm {p} }={\frac {\omega }{k}}.}$

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:

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

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:

${\displaystyle \rho {\ddot {\mathbf {u}}}={\mathbf {f}}+(\lambda +2\mu )\nabla (\nabla \cdot {\mathbf {u}})-\mu \nabla \times (\nabla \times {\mathbf {u}})}$

where:

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

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 ${\displaystyle m}$ at the location ${\displaystyle x+h}$ are:

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

${\displaystyle F_{Hooke}=F_{x+2h}+F_{x}=k\left+k}$

The equation of motion for the weight at the location x+h is:

${\displaystyle m{\partial ^{2}u(x+h,t) \over \partial t^{2}}=k}$

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:

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

Taking the limit N ${\displaystyle \rightarrow \infty }$, h${\displaystyle \rightarrow 0}$ (and assuming smoothness) one gets:

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

(K ${\displaystyle L^{2}}$)/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 was derived by d'Alembert. The wave equation may be written in the factored form

${\displaystyle \left\leftu=0.\,}$

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

${\displaystyle u(x,t)=F(x-ct)+G(x+ct)\,}$

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:

${\displaystyle u(x,0)=f(x)\,}$

${\displaystyle u_{t}(x,0)=g(x)\,}$

The result is d'Alembert's formula:

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

In the classical sense if ${\displaystyle f(x)\in C^{k}}$ and ${\displaystyle g(x)\in C^{k-1}}$ then ${\displaystyle u(t,x)\in C^{k}}$. 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.

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

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

This equation may be rewritten as

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

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

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

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.

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

${\displaystyle r^{2}=(x-\xi )^{2}+(y-\eta )^{2}+(z-\zeta )^{2}.\,}$

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

${\displaystyle u(t,x,y,z)={\frac {1}{4\pi c}}\iiint \varphi (\xi ,\eta ,\zeta ){\frac {\delta (r-ct)}{r}}d\xi \,d\eta \,d\zeta ;\,}$

the denominator 4πc is a convenience.

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

${\displaystyle u(t,x,y,z)={\frac {t}{4\pi }}\iint _{S}\varphi (x+ct\alpha ,y+ct\beta ,z+ct\gamma )d\omega ,\,}$

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:

${\displaystyle u(t,x,y,z)=tM_{ct}.\,}$

It follows that

${\displaystyle u(0,x,y,z)=0,\quad u_{t}(0,x,y,z)=\phi (x,y,z).\,}$

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

${\displaystyle v(t,x,y,z)={\frac {\partial }{\partial t}}\left(tM_{ct}\right),\,}$

then

${\displaystyle v(0,x,y,z)=\psi (x,y,z),\quad v_{t}(0,x,y,z)=0.\,}$

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 Garding (1970, 1973).

Scalar wave equation in two space dimensions

In two space dimensions, the wave equation is

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

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

${\displaystyle u(0,x,y)=0,\quad u_{t}(0,x,y)=\phi (x,y),\,}$

then the three-dimensional solution formula becomes

${\displaystyle u(t,x,y)=tM_{ct}={\frac {t}{4\pi }}\iint _{S}\phi (x+ct\alpha ,\,y+ct\beta )d\omega ,\,}$

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:

${\displaystyle u(t,x,y)={\frac {1}{2\pi c}}\iint _{D}{\frac {\phi (x+\xi ,y+\eta )}{\sqrt {(ct)^{2}-\xi ^{2}-\eta ^{2}}}}d\xi \,d\eta .\,}$

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

${\displaystyle (x-\xi )^{2}+(y-\eta )^{2}=c^{2}t^{2},\,}$

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

${\displaystyle -u_{x}(t,0)+au(t,0)=0,\,}$

${\displaystyle u_{x}(t,L)+bu(t,L)=0,\,}$

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

${\displaystyle u(t,x)=T(t)v(x).\,}$

A consequence is that

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

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

${\displaystyle v''+\lambda v=0,\,}$

${\displaystyle -v'(0)+av(0)=0,\quad v'(L)+bv(L)=0.\,}$

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.

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

${\displaystyle {\frac {\partial u}{\partial n}}+au=0,\,}$

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

${\displaystyle u(0,x)=f(x),\quad u_{t}=g(x),\,}$

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

${\displaystyle \nabla \cdot \nabla v+\lambda v=0,\,}$

in D, and

${\displaystyle {\frac {\partial v}{\partial n}}+av=0,\,}$

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.

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

The inhomogenous wave equation in one dimension is the following:

${\displaystyle c^{2}u_{xx}(x,t)-u_{tt}(x,t)=s(x,t)}$

with initial conditions given by

${\displaystyle u(x,0)=f(x)}$

${\displaystyle u_{t}(x,0)=g(x).}$

The function ${\displaystyle s(x,t)}$ 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 ${\displaystyle (x_{i},t_{i})}$, the value of ${\displaystyle u(x_{i},t_{i})}$ depends only on the values of ${\displaystyle f(x_{i}+ct_{i})}$ and ${\displaystyle f(x_{i}-ct_{i})}$ and the values of the function ${\displaystyle g(x)}$ between ${\displaystyle (x_{i}-ct_{i})}$ and ${\displaystyle (x_{i}-ct_{i})}$. 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 ${\displaystyle c}$, 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 ${\displaystyle (x_{i},t_{i})}$ as ${\displaystyle R_{C}}$. Suppose we integrate the in-homogenous wave equation over this region.

${\displaystyle \int \int _{R_{C}}\left(c^{2}u_{xx}(x,t)-u_{tt}(x,t)\right)dxdt=\int \int _{R_{C}}s(x,t)dxdt.}$

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

${\displaystyle \int _{L_{0}+L_{1}+L_{2}}\left(-c^{2}u_{x}(x,t)dt-u_{t}(x,t)dx\right)=\int \int _{R_{C}}s(x,t)dxdt.}$

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

${\displaystyle \int _{x_{i}-ct_{i}}^{x_{i}+ct_{i}}-u_{t}(x,0)dx=-\int _{x_{i}-ct_{i}}^{x_{i}+ct_{i}}g(x)dx.}$

In the above, the term to be integrated with respect to time disappears because the time interval involved is zero, thus ${\displaystyle dt=0}$.

For the other two sides of the region, it is worth noting that ${\displaystyle x\pm ct}$ is a constant, namingly ${\displaystyle x_{i}\pm ct_{i}}$, where the sign is chosen appropriately. Using this, we can get the relation ${\displaystyle dx\pm cdt=0}$, again choosing the right sign:

${\displaystyle \int _{L_{1}}\left(-c^{2}u_{x}(x,t)dt-u_{t}(x,t)dx\right)\,}$

${\displaystyle =\int _{L_{1}}\left(cu_{x}(x,t)dx+cu_{t}(x,t)dt\right)\,}$

${\displaystyle =c\int _{L_{1}}du(x,t)=cu(x_{i},t_{i})-cf(x_{i}+ct_{i}).\,}$

And similarly for the final boundary segment:

${\displaystyle \int _{L_{2}}\left(-c^{2}u_{x}(x,t)dt-u_{t}(x,t)dx\right)}$

${\displaystyle =-\int _{L_{2}}\left(cu_{x}(x,t)dx+cu_{t}(x,t)dt\right)}$

${\displaystyle =-c\int _{L_{2}}du(x,t)=-\left(cf(x_{i}-ct_{i})-cu(x_{i},t_{i})\right)}$

${\displaystyle =cu(x_{i},t_{i})-cf(x_{i}-ct_{i}).\,}$

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

${\displaystyle -\int _{x_{i}-ct_{i}}^{x_{i}+ct_{i}}g(x)dx+cu(x_{i},t_{i})-cf(x_{i}+ct_{i})+cu(x_{i},t_{i})-cf(x_{i}-ct_{i})=\int \int _{R_{C}}s(x,t)dxdt}$

${\displaystyle 2cu(x_{i},t_{i})-\int _{x_{i}-ct_{i}}^{x_{i}+ct_{i}}g(x)dx-cf(x_{i}+ct_{i})-cf(x_{i}-ct_{i})=\int \int _{R_{C}}s(x,t)dxdt}$

${\displaystyle 2cu(x_{i},t_{i})=\int _{x_{i}-ct_{i}}^{x_{i}+ct_{i}}g(x)dx+cf(x_{i}+ct_{i})+cf(x_{i}-ct_{i})+\int \int _{R_{C}}s(x,t)dxdt}$

${\displaystyle u(x_{i},t_{i})={\frac {f(x_{i}+ct_{i})+f(x_{i}-ct_{i})}{2}}+{\frac {1}{2c}}\int _{x_{i}-ct_{i}}^{x_{i}+ct_{i}}g(x)dx+{\frac {1}{2c}}\int _{0}^{t_{i}}\int _{x_{i}-c\left(t_{i}-t\right)}^{x_{i}+c\left(t_{i}-t\right)}s(x,t)dxdt.\,}$

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 ${\displaystyle (x_{i},t_{i})}$ 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.

See also

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

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.

• Partial differential equations
• Equations
• Wave mechanics