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# Green and Delta Functions, Eigenmode Theory of Waveguides

DOI link for Green and Delta Functions, Eigenmode Theory of Waveguides

Green and Delta Functions, Eigenmode Theory of Waveguides book

# Green and Delta Functions, Eigenmode Theory of Waveguides

DOI link for Green and Delta Functions, Eigenmode Theory of Waveguides

Green and Delta Functions, Eigenmode Theory of Waveguides book

## ABSTRACT

As we have seen before, the general driven wave equation for the electromagnetic four-potential is

(5.1)

One of the most fundamental properties of the wave equation is the fact that it is linear: if we know the solutions for two different four-current sources,

and

(5.2)

then, the solution for a linear superposition of sources is also known;

(5.3)

Now, we can express any four-current as a linear superposition of deltafunctions:

(5.4)

Therefore, the solution to the wave equation driven by a delta-function is of particular importance, as we can write, in the Lorentz gauge,

(5.5)

Aµ µ0 jµ+ ∂ µ ∂νAν( ).=

µ,

A1 µ µ0 j1µ+ ∂ µ ∂νA1ν( ),=

A2 µ µ0 j2µ+ ∂ µ ∂νA2ν( ),=

λ1A1µ λ2A2µ+( ) µ0 λ1 j1µ λ2 j2µ+( )+ ∂ µ ∂ν λ1A1µ λ2A2µ+( )[ ].=

jµ xλ( ) jµ xλ( )δ4 xλ xλ-( ) d4x.∫∫∫∫=

G xλ xλ-( ) µ0δ4 xλ xλ-( )+ 0,=

and the general solution to the wave equation, with an arbitrary four-current source is

(5.6)

The solution is known as the Green function or propagator for the linear wave equation under consideration.