In The Name of God
Computational and analytical analysis of wave scattering by
Elliptical and Spheroidal inhomogeneities.
y
x = x0
h = constant
h^
R2
x^
x=0
h=0
h=p
x = constant
a
a
R1
x
1. Different Curvature
2. Non-Symmetry
Numerous papers published by V.V. Varadan, J.E.
Burke, J.D. Alemar, F. Babick, R.H. Hackman, A.A.
Kleshchev,F. Leon, Y.S. Wang in this regard but There
is no paper to best of my knowledge which
1. Include Dissipation Effects
2. Used Eigen functions Expansion with
Explicit Galerkin Type Approach
4
5
,
and
,
P
Wave
U
φ
SV
Wave
×ψ
SH
Wave
 ×  ×(e ωχ)
6
,
.,
7
,
.,
Φ(r, t) = e
`
-iωt
Φ(r)
`
`
8
,
.,
Material
Solid
Elastic
Solid
Viscoelastic
Solid
Poroelastic
Solid
Fluid
Ideal
Fluid
Viscous
Fluid
9
,
,
U
2

t
2
=   U  ( λ   ) (.U )
2

( 
) = 0
 λs  2 s  /s
2


2
( 
) = 0
s /s


2
( 
) = 0
s /s
10
U = Φ   Ψ
1
 =  U U  
2
ij = λ δ ij kk  2 ij
11
,
,
,
E = E(ω) + iE(ω)
12
,
,
P  u   N    u   Q  U    2  11u  12U  = 0 ,
Q  u   R   U    2  12u   22U  = 0 ,
  k
2
s
2
solid
 =0
s
2
 2  f  k fluid
f = 0
  k
2
shear
 =0
  k
2
shear
 =0
2
2
s
s
s
s
13
u =  
s
s
12
s
U = 
 
 22
f
1
 =  U U  
2
 ij =  P  2 N  u  Q U   ij  2 N e
f
 ij = ( Q u  R  U )  ij
s
14
,
,
u

1 1
u
 c  (.u ) = 
 (    b ) (. )
2
t

t  3
t
2
2
2
u
 2
 2 1 4
= c  (    b )   φ
2
t
 3
t 

φ
2
ψ
t
=

t


=
ψ


2
 
2
15
u =  φ    ψ
ε = 1  u u  
2
 ij = [2 .u  iωρφ ]δ ij  2  ε ij
16
u
I
η
I
Σξξ
ξ=ξ0
ξ=ξ0
=u
II
η
=
u
ξ=ξ0
I
Σξξ
ξ=ξ0
I
η
ξ=ξ0
I
Σηη
=u
ξ=ξ0
II
η
=
ξ=ξ0
I
Σηη
ξ=ξ0
17
,
and
,
(  k )  = 0
2
=
2

 F (x )  G (h )   = F (x )  G(h )
n =0

n
n
G n  (x )  (b  2 q cosh2 x )G n (x ) = 0
Fn  (h )  (b  2 q cos2h ) Fn (h ) = 0

Fn (h ) = A n ce (h , q )  B n se (h , q )
G n (x ) =C n Mc n(1) (x , q )  D n Mc n(2) (x , q )
18
,
and
,
(  k ) X = 0
2
2

X =  Fn (x )  G n (h )  X = F (x )  G (h )
n =0





2



2
2

x 1 Gn (x )    q  c x  x 2  1   Gn (x ) = 0


2



2
2
2

1 h  Fn (h )    q c h  1 h 2   Fn (h ) = 0


2

Fn (h ) = An S0,n (q,h )  Bn S1,n (q,h )
Gn (x ) = Cn R0,n (q, x )  Dn R1,n (q, x )
19
Wave Type
Outgoing Wave
Incoming Wave
Boundary
Condition
Coefficient
Relation

Dn =iCn
Dn = 0

2π
0
B.C.(η)wi (η) = 0
i ≈ ACCURACY!
wi (η) = Sin(iη),Cos(iη)
Analytically
IsIntegrals
ThisCalculated
Method
Can be Used
With Any Boundary?
21
Convergence
 =?
Φ(η) =  Fn (η)
n=0
If Appropriate Function It Converge Soon But if
Not Convergence Would be very bad and we
prefer To Divide our domain into partiotions
22
23