In computational electromagnetics, the scatter boundary (also known as the radiation boundary or far-field boundary) is an artificial boundary that simulates the infinite extent of free space surrounding a finite computational domain. This boundary is used to terminate numerical simulations, ensuring that the outgoing electromagnetic waves do not reflect back into the computational domain.
For time harmonic propagation, the wave equation turns into the Helmholtz equation
`grad^2 u+k^2=0`
where k is the wave number, `k=(2 pi)/lambda`.
On the boundaries, there are inflow and outflows, so we assume the field at the boundaries to be the sum of an incident plane wave and a scattered wave. The incident wave propagates in an arbitrary direction and the scattered wave propagates in the normal direction.
`u=u_i+u_(sc)=u_0 e^(-i bbk*bbr )+u_s e^(-ik bbn* bbr)`
Where `bbn` is the outward boundary normal vector. At the boundary of the modeling domain, we have
`grad u =-k bbk u_0 e^(-i bbk*bbr)-ik bbn u_s e^(-ik bbn *bbr)=-ik bbn u|_Gamma-iu_0(bbk-k bbn)e^(-i bbk*bbr)`
There is no incident wave on the outflow boundary, the 2nd term vanishes. We make the further approximation for the inflow boundary that the incident wave propagates in the inward normal direction, so that `bbk=-kbbn`. Then we get the following boundary conditions:
Inflow: `bbn*gradu=-iku|_Gamma+2iku_0e^(ik bbn*bbr)`
Outflow: `bbn*grad u=-iku|_Gamma`
We usually call the `bbn * grad u ` as the Flux or Source condition.