Page 471 - Electromagnetics

Solution to wave equation by specification of wave amplitudes. An alternative
to direct specification of boundary conditions is specification of the amplitude functions
˜
˜
A(x, y,ω) and B(x, y,ω) or their inverse transforms A(x, y, t) and B(x, y, t). If we specify
the time-domain functions we can write ψ(x, y, z, t) as the inverse transform of (A.35).
For example, a wave traveling in the +z-direction behaves as
+
ψ(x, y, z, t) = A(x, y, t) ∗ F (x, y, z, t) (A.42)
where
√
2
F (x, y, z, t) ↔ e −κz = e − z v p +2 p .
+
We can find F + using the following Fourier transform pair [26]):

2
√ I 1 σ t − (x/v) 2
x (p+ρ) −σ 2 − x σ x −ρt x
ρ
2
e − v ↔ e v δ(t − x/v) + e , < t. (A.43)
v t − (x/v) 2 v
2
Here x is real and positive and I 1 (x) is the modified Bessel function of the first kind and
order 1. Outside the range x/v < t the time-domain function is zero. Letting ρ = and
σ = we find
2

I
2
z − t 1 ( t − (z/v) )
2
+ − z
v δ(t − z/v).
F (x, y, z, t) = e U(t − z/v) + e (A.44)
v t − (z/v) 2
2
Note that F + is a real functions of time, as expected.
Substituting (A.44) into (A.42) and writing the convolution in integral form we have

I
∞ z − τ 1 ( τ − (z/v) )
2 2 2
ψ(x, y, z, t) = A(x, y, t − τ) e dτ +
2
z/v v τ − (z/v) 2
z

+ e − z , z > 0. (A.45)
v A x, y, t −
v
The 3-D Green’s function for waves in dissipative media
To understand the fields produced by bounded sources within a dissipative medium we
may wish to investigate solutions to the wave equation in three dimensions. The Green’s
function approach requires the solution to
2 ∂ 1 ∂
2
2

∇ − − G(r|r ; t) =−δ(t)δ(r − r )
2
2
v ∂t v ∂t 2
=−δ(t)δ(x − x )δ(y − y )δ(z − z ).

That is, we are interested in the impulse response of a point source located at r = r .

We begin by substituting the inverse temporal Fourier transform relations
1 ∞
˜ jωt
G(r|r ; t) = G(r|r ; ω)e dω,
2π
−∞
1 ∞ jωt
δ(t) = e dω,
2π −∞
obtaining
1 ∞ 2 2 1 2 jωt
˜

∇ − jω − ( jω) G(r|r ; ω) + δ(r − r ) e dω = 0.
2π v 2 v 2
−∞
© 2001 by CRC Press LLC