Non-stationary Electromagnetics

July 6, 2012 10:33 PSP Book - 9in x 6in 09-Alexander-App

566 Appendix B: Taking into Account Solutions of a Homogeneous Equation

If v

2

> v, then in the sub-region vt < x < v

2

t of the region

T

1

solutions to the homogeneous equation must be added to the

solution determined by the resolvent R

(1)

. These solutions have the

form of finite waves

θ

(

v

2

t − x

)

· A



1

exp



iω

2



t −

x

v

2



θ

(

v

2

t − x

)

· A



3

exp



−iω

2



t −

x

v

2



θ

(

v

2

t − x

)

· A



4

exp



iω

2



t −

x

v

2



When A



1

=−A

1

then the first solution compensates the first wave

in this sub-region and the second and the third ones give missing

waves.

It is worth noting that the functions satisfying the homogeneous

equation are integrated ov