1. In The Name of God
Computational and analytical analysis of wave scattering by Elliptical and Spheroidal inhomogeneities.

2. Elliptical , Spheroidal Coordinatesy x
h = 0
h = p R2 a
x = 0
x = constant
h = constant
x = x0 ^ h
Incident Plane Wave a R1

3. Why Ellipse?
1. Different Curvature
2. Non-Symmetry

4. Review Last Works
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

5. New Algorithm! 1
Convert Coupled Vector PDE to Uncoupled Helmholtz Equations Using Helmholtz Decomposition 2
Helmholtz Equation Solution is Wave Function Series 3
Determine Wave Function Coefficients By Satisfying Boundary Conditions with Galerkin Type Approach

6. Helmholtz Decompositon,
and , φ
×ψ
SV Wave
P Wave SH
Wave

7. P-Wave SH-Wave SV-Wave
Polarization , , .
P-Wave
SH-Wave
SV-Wave

8. Time Dependence , . , ` ` `

9. Material Fluid Solid Material??? Elastic Solid Viscoelastic Solid, . ,
Material
Solid
Elastic Solid
Viscoelastic Solid
Poroelastic Solid
Fluid
Ideal Fluid
Viscous Fluid

10. Elastic Solid , ,

11. Elastic Solid

12. Viscolastic Solid , , ,

13. Poroelastic Solid
(Biot Theory) , ,

14. Poroelastic Solid
(Biot Theory)

15. Viscous Fluid , ,

16. Viscous Fluid

17. Boundary Condition

18. Mathieu Functions ,
and ,

19. Spheroidal Functions ,
and ,

20. Wave Types Outgoing Wave Incoming Wave Wave Type Boundary Condition
Coefficient
Relation
Outgoing Wave
Incoming Wave 

21. Galerkin Type Approach Is This Method Can be Used With Any Boundary?
i ≈ ACCURACY!
Is This Method Can be Used With Any Boundary?
Integrals Calculated Analytically

22. Convergence
If Appropriate Function It Converge Soon But if Not Convergence Would be very bad and we prefer To Divide our domain into partiotions

23. Review Flowchart 1
Convert Coupled Vector PDE to Uncoupled Helmholtz Equations Using Helmholtz Decomposition 2
Helmholtz Equation Solution is Wave Function Series 3
Determine Wave Function Coefficients By Satisfying Boundary Conditions with Galerkin Type Approach