Sommerfeld radiation condition

E234762

The Sommerfeld radiation condition is a mathematical criterion in wave and scattering theory that selects physically meaningful, outward-radiating solutions to the Helmholtz equation at infinity.

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Sommerfeld radiation condition canonical 1

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Predicate Object
instanceOf boundary condition
concept in scattering theory
concept in wave theory
mathematical condition
radiation condition
appliesTo Helmholtz equation
acoustic waves
electromagnetic waves
scalar wave equation in frequency domain
scattering problems
time-harmonic wave fields
assumes time dependence e^{-iωt}
characterizes outgoing waves at infinity
contrastsWith Dirichlet boundary conditions
surface form: Dirichlet boundary condition at finite distance

Neumann boundary conditions in potential theory
surface form: Neumann boundary condition at finite distance

incoming radiation condition
periodic boundary conditions
domain mathematical physics
partial differential equations
scattering theory
wave propagation
ensures causal radiation behavior in frequency domain
finite energy flux at infinity
uniqueness of the scattered field
generalizationOf radiation conditions for other elliptic operators
hasConsequence existence of far-field pattern
representation of solutions via asymptotic expansions
well-posedness of exterior scattering problems
imposes asymptotic behavior on wave fields
mathematicalForm lim_{r→∞} r(∂u/∂r - iku) = 0 in three dimensions
lim_{r→∞} r^{(d-1)/2}(∂u/∂r - iku) = 0 for d-dimensional space
namedAfter Arnold Sommerfeld
purpose to enforce outward-radiating behavior at infinity
to ensure uniqueness of solutions to exterior boundary value problems
to exclude incoming waves from infinity
to select physically meaningful solutions of the Helmholtz equation
relatedTo Lippmann–Schwinger equation
limiting absorption principle
outgoing Green's function of the Helmholtz operator
relatesTo far-field behavior of solutions
radiation of energy to infinity
timePeriod early 20th century
usedBy applied mathematicians
engineers
physicists
usedIn acoustic scattering by obstacles
antenna theory
electromagnetic scattering by conductors
exterior Dirichlet problems
exterior Neumann problems
seismology wave propagation models

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Arnold Sommerfeld knownFor Sommerfeld radiation condition