N. M. Khutoryansky, V. Genis
876
in 3-D case described in [15] reduces to the following expression for the dis-
placement component
( ) ( )
3
,,u xt u xt=
:
( ) ( ) ( )
( )
( ) ( )
( )
( ) ( )
( ) ( )
/
0
1
, 0, ,
2
1
0, 0, d
2
1
, ,d
2
1
, , d d in ,
2
t xc
t hxc
h t xc
u xt u t xc u ht h x c
e
D
c
e
h Dh
c
e
bq
c
ξ
στ τ τ
ρ
στ τ τ
ρ
ξτ ξτ τ ξ
ρ
−
−∞
−−
−∞
−−
−∞
= − + −−
−+
++
+ −Ω
∫
∫
∫∫
(8)
where
33
σ
is denoted by
σ
and it is taken into account that the outward
normals to the lower and upper boundaries of the layer
0 xh
≤≤
have opposite
directions.
In many practical applications, the electric volume charges are absent. There-
fore, we consider henceforth only the case when
0
q =
. Then the terms related
to
D
in the above expression can be simplified since, based on Equation (4) in
this case,
( )
,D xt
is spatially uniform:
( ) ( )
,.D xt Dt=
(9)
Due to the property (9) the representation formula (8) can be rewritten as
( ) ( ) ( )
(
)
( ) ( )
( )
( ) ( )
( )
0
1
, 0, ,
2
1
0, d
2
1
,d
2
1
, d d in .
2
t xc
t hxc
h t xc
u xt u t xc u ht h x c
e
D
c
e
hD
c
b
c
ξ
στ τ τ
ρ
στ τ τ
ρ
ξτ τ ξ
ρ
−
−∞
−−
−∞
−−
−∞
= − + −−
−+
++
+Ω
∫
∫
∫∫
(10)
To obtain a representation formula for
( )
,xt
φ
, let us consider an auxiliary
function
( ) (
) (
)
,, ,
e
xt xt u xt
ψφ
= −
(11)
that has the following connection to the electric displacement:
.D
ψ
′
= −
According to (3),
0D
ψ
′′ ′
=−=
. Then, using the corresponding Green’s
function
2x
and Equation (11), we get a representation formula for
( )
,xt
ψ
involving only boundary value of function
(
)
,
xt
ψ
and a spatially uniform
electric displacement:
( ) ( ) ( )
( )
12
, 0, , in .
22
hx
xt t ht D t
ψ ψψ
−
= ++ Ω
(12)
Formulas (12) and (11) lead to the following expression for
( )
,xt
φ
:
( ) ( ) (
) ( ) ( )
( )
( )
1
, 0, , 0, ,
22
2
, in .
2
e
xt t ht u t u ht
e hx
u xt Dt
φ φφ
= +− +
−
++ Ω
(13)
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