Theory for the Plane, Spherical, and Cylindrical Radiation Boundary Conditions
Specify a Plane Wave Radiation, Spherical Wave Radiation, or Cylindrical Wave Radiation boundary condition to allow an outgoing wave to leave the modeling domain with minimal reflections. The condition can be adapted to the geometry of the modeling domain. The plane wave type is suitable for both far-field boundaries and ports (for plane waves only). For general radiation boundary conditions for waveguides (supporting multimodes) it is recommended to use the Port boundary condition.
Radiation boundary conditions are available for all types of studies. For the frequency-domain study, Givoli and Neta’s reformulation of the Higdon conditions (Ref. 1) for plane waves has been implemented to the second order. For cylindrical and spherical waves, COMSOL Multiphysics uses the corresponding 2nd-order expressions from Bayliss, Gunzburger, and Turkel (Ref. 2). The Transient, Mode analysis, and Eigenfrequency studies implement the same expansions to the first order.
The first-order radiation boundary conditions in the frequency domain read
where k is the wave number and κ ( r ) is a function whose form depends on the wave type:
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Plane wave: κ( r ) = 0
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Cylindrical wave: κ( r ) = 1 / (2 r)
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Spherical wave: κ( r ) = 1 / r
In the cylindrical and spherical wave cases, r is the shortest distance from the point r = (x, y, z) on the boundary to the source. The right-hand side of the equation represents an optional incoming pressure field pi (see Incident Pressure Field).
The second-order radiation boundary conditions in the frequency domain are defined below. In these equations, Δ|| at a given point on the boundary denotes the Laplace operator in the tangent plane at that particular point.
Plane Wave
In the notation of Givoli and Neta (Ref. 1), the above expressions correspond to the parameter choices C0 = C1 = C2 = ω/k. For normally incident waves, this gives a vanishing reflection coefficient.
Cylindrical Wave
The cylindrical wave boundary condition is based on a series expansion of the outgoing wave in cylindrical coordinates (Ref. 2), and it assumes that the field is independent of the axial coordinate. Specify the axis of this coordinate system by giving an orientation (nx,  ny, nz) and a point (x0, y0, z0) on the axis. In axisymmetric geometries, the symmetry axis is the natural and only choice.
Spherical Wave
Use a spherical wave to allow a radiated or scattered wave — emanating from an object centered at the point (x0, y0, z0) that is specified — to leave the modeling domain without reflections. The boundary condition is based on an expansion in spherical coordinates from Bayliss, Gunzburger, and Turkel (Ref. 2), implemented to the second order.
Transient Analysis
The transient radiation boundary condition is the first-order expression
where κ ( r ) is the same wave-type dependent function as for the eigenfrequency case and pi the optional Incident Pressure Field.
An estimate of the reflection coefficient Rs for spurious waves off the plane wave radiation boundary, for incident plane waves at angle θ is given by the expression:
where N is the order of the boundary condition (here 1 or 2). So at normal incidence (θ = 0) there are no spurious reflections, while, for example, at an incidence angle of 30° for N = 2 (plane wave radiation in the frequency domain) the amplitude of the spurious reflected wave is 0.5% of the incident.