D i ¼ Àε 0 φ ,i :
ð2:69Þ
At the plate surfaces, there are six boundary conditions and four continuity
conditions:
T 23 Æh
ð Þ ¼ 0, J
p
2 Æh
ð Þ ¼ 0, J
n
2 Æh
ð Þ ¼ 0,
ð2:70Þ
φ h
þ
ð Þ ¼ φ h
À
ð Þ, D 2 h
þ
ð Þ ¼ D 2 h
À
ð Þ,
φ Àh
þ
ð
Þ¼φ Àh
À
ð
Þ, D 2 Àh
þ
ð
Þ¼D 2 Àh
À
ð
Þ:
ð2:71Þ
We look for waves propagating in the x 1 direction in the following form:
u
φ
Δp
Δn
8
> > > <
> > > :
9
> > > =
> > > ;
¼
A
B
C
D
8
> > > <
> > > :
9
> > > =
> > > ;
exp ηx 2
ð Þexp i ξx 1 À ωt
ð
Þ
½
,
ð2:72Þ
where A, B, C, and D are constants. ω is real and positive. ξ has a positive real part.
The substitution of Eq. (2.72) into Eqs. (2.66) and (2.67) results in four linear
homogeneous algebraic equations for A, B, C, and D. For nontrivial solutions the
determinant of the coefficient matrix of the equations has to vanish, which leads to a
polynomial equation of degree eight for η. Let the eight roots of this equation be
η
(m) (ξ, ω) and the corresponding nontrivial solution of A, B, C, and D be A
(m) , B
(m) ,
C
(m)
, and D
(m) . Only the ratios among A
(m)
, B
(m) , C
(m) , and D
(m) can be determined.
Then the general solution to Eqs. (2.66) and (2.67) can be written as
u
φ
Δp
Δn
8
> > > <
> > > :
9
> > > =
> > > ;
¼
X 8
m¼1
F
m
ð Þ
A
m
ð Þ
B
m
ð Þ
C
m
ð Þ
D
m
ð Þ
8
> > > <
> > > :
9
> > > =
> > > ;
exp η
m
ð Þ x 2
exp i ξx 1 À ωt
ð
Þ
½
,
ð2:73Þ
where F
(m) are undetermined constants. The solution in the free space is
φ ¼
G exp ξ h À x 2
ð
Þ
½
exp i ξx 1 À ωt
ð
Þ
½
, x 2 > h,
H exp ξ h þ x 2
ð
Þ
½
exp i ξx 1 À ωt
ð
Þ
½
, x 2 < Àh,
&
ð2:74Þ
where G and H are undetermined constants. Equation (2.74) satisfies Eq. (2.68).
Substituting Eqs. (2.73) and (2.74) into Eqs. (2.70) and (2.71), we obtain ten linear
algebraic equations for F
(m)
, G and H. For nontrivial solutions the coefficient matrix
of these equations has to vanish, which yields an equation that determines the
dispersion relations of ω versus ξ. The numerical procedure is carried out on a
computer using MATLAB.
28
2 Exact Solutions
ð2:69Þ
At the plate surfaces, there are six boundary conditions and four continuity
conditions:
T 23 Æh
ð Þ ¼ 0, J
p
2 Æh
ð Þ ¼ 0, J
n
2 Æh
ð Þ ¼ 0,
ð2:70Þ
φ h
þ
ð Þ ¼ φ h
À
ð Þ, D 2 h
þ
ð Þ ¼ D 2 h
À
ð Þ,
φ Àh
þ
ð
Þ¼φ Àh
À
ð
Þ, D 2 Àh
þ
ð
Þ¼D 2 Àh
À
ð
Þ:
ð2:71Þ
We look for waves propagating in the x 1 direction in the following form:
u
φ
Δp
Δn
8
> > > <
> > > :
9
> > > =
> > > ;
¼
A
B
C
D
8
> > > <
> > > :
9
> > > =
> > > ;
exp ηx 2
ð Þexp i ξx 1 À ωt
ð
Þ
½
,
ð2:72Þ
where A, B, C, and D are constants. ω is real and positive. ξ has a positive real part.
The substitution of Eq. (2.72) into Eqs. (2.66) and (2.67) results in four linear
homogeneous algebraic equations for A, B, C, and D. For nontrivial solutions the
determinant of the coefficient matrix of the equations has to vanish, which leads to a
polynomial equation of degree eight for η. Let the eight roots of this equation be
η
(m) (ξ, ω) and the corresponding nontrivial solution of A, B, C, and D be A
(m) , B
(m) ,
C
(m)
, and D
(m) . Only the ratios among A
(m)
, B
(m) , C
(m) , and D
(m) can be determined.
Then the general solution to Eqs. (2.66) and (2.67) can be written as
u
φ
Δp
Δn
8
> > > <
> > > :
9
> > > =
> > > ;
¼
X 8
m¼1
F
m
ð Þ
A
m
ð Þ
B
m
ð Þ
C
m
ð Þ
D
m
ð Þ
8
> > > <
> > > :
9
> > > =
> > > ;
exp η
m
ð Þ x 2
exp i ξx 1 À ωt
ð
Þ
½
,
ð2:73Þ
where F
(m) are undetermined constants. The solution in the free space is
φ ¼
G exp ξ h À x 2
ð
Þ
½
exp i ξx 1 À ωt
ð
Þ
½
, x 2 > h,
H exp ξ h þ x 2
ð
Þ
½
exp i ξx 1 À ωt
ð
Þ
½
, x 2 < Àh,
&
ð2:74Þ
where G and H are undetermined constants. Equation (2.74) satisfies Eq. (2.68).
Substituting Eqs. (2.73) and (2.74) into Eqs. (2.70) and (2.71), we obtain ten linear
algebraic equations for F
(m)
, G and H. For nontrivial solutions the coefficient matrix
of these equations has to vanish, which yields an equation that determines the
dispersion relations of ω versus ξ. The numerical procedure is carried out on a
computer using MATLAB.
28
2 Exact Solutions