Acoustic and Electromagnetic Equations

Integral Representations for Harmonic Problems

• Some Wave Equations
  • Introduction
  • Physical Background
    • The acoustic equation
    • The Maxwell equations**
    • Elastic waves
• The Helmholtz Equation
  • Introduction
  • Harmonic Solutions
  • Fundamental Solutions
  • The Case of the Sphere in R³ **
    • Spherical harmonics
    • Legendre polynomials
    • Associated Legendre functions
    • Vectorial spherical harmonics
  • The Laplace Equation in R³
    • The sphere
    • Surfaces and Sobolev spaces
    • Interior problems: Variational formulations
    • Exterior problems
    • Regularity properties of solutions in lRⁿ
    • Elementary differential geometry
    • Regularity properties
  • The Helmholtz Equation in 1R³
    • The spherical Bessel functions
    • Dirichlet and Neumann problems for a sphere
    • The capacity operator T
    • The case of a plane wave
    • The exterior problem for the Helmholtz equation
• Integral Representations and Integral Equations
  • Integral Representations
  • Integral Equations for Helmholtz Problems
    • Equations for the single layer potential .
    • Equations for the double layer potential
    • The spherical case
    • The far field
    • The physical optics approximation for the sphere
  • Integral Equations for the Laplace Problem
  • Variational Formulations for the Helmholtz Problems
    • The operator S
    • Fredholm operators
    • The operator N
    • Formulation with the far field
• Singular Integral Operators
• Maxwell Equations and Electromagnetic Waves
  • Introduction
  • Fundamental Solution and Radiation Conditions.
  • Multipole Solutions
    • Multipoles
    • The capacity operator
  • Exterior Problems
    • Trace and lifting associated with the space H(curl) .. .
    • Variational formulations for the perfect conductor problem
    • Coupled variational formulations for impedance conditions
  • Integral Representations
  • Integral Equations
    • The perfect conductor .
    • The zero frequency limit
    • The dielectric case
    • The infinite conductivity limit: The perfect conductor
  • The Far Field
    • Far field and scattering amplitude.
    • Integral equations and far field