Wave Equation

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:

:

{ \partial^2 u \over \partial t^2 } = c^2 
abla^2u,

where c is a fixed Constant equal to the propagation speed of the wave. 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_\mathrm{p} = rac{\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 
abla^2u

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:

:

ho{ \ddot \bold{u}} = \bold{f} + ( \lambda + 2\mu )
abla(
abla \cdot \bold{u}) - \mu
abla 	imes (
abla 	imes \bold{u})

where:

$\lambda$ and $\mu$ are the so-called Lamé moduli describing the elastic properties of the medium,

$ho$ is density,

$\bold{f}$ is the source function (driving force),

and $\bold{u}$ is displacement.

Vector

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 Slinkie s) 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 exert on the mass

m

at the location

x+h

are:

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

:

(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

:

-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

:

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

A consequence is that
:

rac{T''}{c^2T} = rac{v''}{v} = -\lambda. \,

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

:

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.

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

:

rac{\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

:

abla \cdot 
abla v + \lambda v = 0, \,

in ''D'', and

:

rac{\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 .

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.

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.

	CATEGORIES ABOUT WAVE EQUATION

	fundamental physics concepts

	hyperbolic partial differential equations

	wave mechanics

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Wave Equation