The wave equation is a partial differential equation that describes how waves propagate over time and space. It is a fundamental equation in physics and is used to model a wide range of wave phenomena, including sound waves, electromagnetic waves, and water waves.

The inhomogeneous Maxwell's equation in the covarient form

The wave equation for 4 potential is obtained by taking 4 divergence of F^𝛼𝛽.

If the potentials satisfy Lorentz condition

To find the solution we apply Green's function. A^𝛽 is replaced by function D(x, x^1) called Green's function and right term is replaced by 4 dirac delta function. So,

In absence of boundary surfaces. Green's function depends only on

Let us introduce fourier integral

K.z is four scalar product

The integral over Ko is done by residue method considering the contour 'r' the integral

The Green's function is

Consider the integral

Let cos𝜃 = t

- sin 𝜃 d𝜃 = dt

𝜃 = 0, t=1

𝜃 = 𝜋, t =-1

Using this result the Green's function is

Substituting in Green's function

In the second and last integration. Let l K l = - l K l

Combining 1st and last and second and third part

For Zo> 0, R> 0

The second part is always zero

This is called retarded Green's function

If we take contour a we get

This is called advanced Green's function

Da gives unphysical result so we cut the value. Noting that

Since 2nd part is always zero, we can write Green's function

This is the required Green's function. Hence, the solution of the wave equation

Interchanging the integration over x' and 𝜏 we get

Integrating over 𝜏

According to light cone condition

For 𝛼 = 1

Solving the wave equation allows researchers and engineers to understand and predict how waves evolve in different physical systems, leading to advancements in fields such as acoustics, optics, electromagnetics, and fluid dynamics.

This note is a part of the Physics Repository.

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