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

Elliptical , Spheroidal Coordinates y x h = 0 h = p R2 a x = 0 x = constant h = constant x = x0 ^ h Incident Plane Wave a R1

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

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

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

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

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

Time Dependence , . , ` ` `

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

Elastic Solid , ,

Elastic Solid

Viscolastic Solid , , ,

Poroelastic Solid (Biot Theory) , ,

Poroelastic Solid (Biot Theory)

Viscous Fluid , ,

Viscous Fluid

Boundary Condition

Mathieu Functions , and ,

Spheroidal Functions , and ,

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

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

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

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