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22
Physics Formulary by ir. J.C.A. Wevers
5.4
Green functions for the initialvalue problem
This method is preferable if the solutions deviate much from the stationary solutions, like pointlike excitations.
Starting with the wave equation in one dimension, with
∇
2
=
∂
2
/
∂
x
2
holds: if
Q
(
x,x
±
,t
)
is the solution with
initial values
Q
(
±
,
0) =
δ
(
x

x
±
)
and
∂
Q
(
±
,
0)
∂
t
=0
, and
P
(
±
)
the solution with initial values
P
(
±
,
0) = 0
and
∂
P
(
±
,
0)
∂
t
=
δ
(
x

x
±
)
, then the solution of the wave equation with arbitrary initial
conditions
f
(
x
)=
u
(
x,
0)
and
g
(
x
∂
u
(
x,
0)
∂
t
is given by:
u
(
x,t
∞
±
∞
f
(
x
±
)
Q
(
±
)
dx
±
+
∞
±
∞
g
(
x
±
)
P
(
±
)
dx
±
P
and
Q
are called the
propagators
. They are deFned by:
Q
(
±
1
2
[
δ
(
x

x
±

vt
)+
δ
(
x

x
±
+
)]
P
(
±
²
1
2
v
if

x

x
±

<vt
0
if

x

x
±

>vt
±urther holds the relation:
Q
(
±
∂
P
(
±
)
∂
t
5.5
Waveguides and resonating cavities
The boundary conditions for a perfect conductor can be derived from the Maxwell equations. If
±
n
is a unit
vector
⊥
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This note was uploaded on 11/16/2010 for the course ENGR 201 taught by Professor Elder during the Spring '10 term at Blinn College.
 Spring '10
 elder

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