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^πΌπ½.
If the potentials satisfy Lorentz condition
To find the solution we apply Green's function. A^π½ 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 cosπ = t
- sin π dπ = dt
π = 0, t=1
π = 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 π we get
Integrating over π
According to light cone condition
For πΌ = 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.