Green Functions for the Wave Equation -...

Green Functions for the

Wave Equation

Golam Mustafa

University of South Dakota

The Wave Equation

Maxwell equations in terms of potentials in Lorenz gauge

Both are wave equations with known source distribution f(x,t) :

If there are no boundaries, solution by Fourier transform and the

Green function method is best.

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Green Functions for the Wave Equation

G. Mustafa

Fourier transforms

Fourier transformation in time

and inverse Fourier transformation

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Inhomogeneous Helmholtz wave equation

In the frequency domain, the wave equation

transforms to Inhomogeneous Helmholtz wave equation

where is the wave number associate with frequencyω

The Green function appropriate to Inhomogeneous Helmholtz

wave equation satisfies the equation:

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Free space Helmholtz Green function

In free space with no boundaries, the solution must be

spherically symmetric about x=x/. Let then

becomes

For

has the solution

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Free space Helmholtz Green function

In the limit of electrostatic,

then,

with the solution gives

and represent diverging and conversing spherical

waves respectively.

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Time dependent Green function

The time dependent Green functions corresponding to

and satisfy

Fourier transform of above equation leads to

The solutions of above equation are therefore,

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Time dependent Green function

Inverse Fourier transform of

gives rise to

where is the relative time.

Since ,

or, more explicitly,

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Retarded and advanced Green functions

The delta function requires to contribute

and R/c is always nonnegative. Therefore,

for G(+) only contributes, or sources only affect the

wave function after they act. Thus G(+) is called a retarded

Green function, as the affects are retarded (after) their

causes.

G(−) is the advanced Green function, giving effects which

precede their causes.

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In and Out Field

When a source distribution is localized in time and space, we

can envision two limiting situations:

At time , there exists a wave satisfying the

homogeneous equation.

G(+) ensures no contribution before the source is activated.

At time , exists.

G(-) makes sure no contribution after the source shuts off.

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In and Out Field

With the initial condition , we can write the

general solution

where [ ]ret means

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Green Functions for the Wave Equation -...

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