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## Exterior Boundary Value Problem: Scattering

Let us apply the properties of the Bessel function to solve the following exterior boundary value (scattering'') problem:

Find that solution to the Helmholtz equation in the Euclidean plane which satisfies

1. the Dirichlet boundary condition on the circular boundary and
2. the condition that its asymptotic form, as , is that of a plane wave propagating into the -direction,

plus only outgoing waves, if any; i.e. no incoming waves.

Mathematically the second condition is a type of boundary condition at infinity. It is evident that this boundary condition states that the solution consists of plane wave + outgoing wave''. The physical meaning of this condition is that it represents a scattering process.

If the circular boundary were absent, then there would have been no scattering. The Dirichlet boundary condition 1. would have been replaced by the regularity requirement that finite at while the second boundary condition 2. at would have remained the same. In that case the resulting no scattering'' solution is immediate, namely,

By contrast, if the circular boundary is present as stipulated by the problem, then this solution must be augmented so that the Dirichlet boundary conditions are satisfied,

This augmentation can be implemented with Hankel functions of the first kind, or of the second kind, or with a combination of the two. The boundary condition that the solution represent a plane wave plus a scattered wave, outgoing only, demands that the augmentation have the form

It expresses the requisite outgoing wave condition for time dependence. This is because as ,

By contrast, at , the Dirichlet boundary condition demands that
 0

The fact that this holds for all angles implies that each term (= partial wave amplitude'') in the sum must vanish. Consequently,

It follows that the hard cylinder scattering process yields the modified plane wave

This solution to the Helmholtz equation represents the incident plane wave plus the scattered outgoing cylinder waves.

Next: Finite Interior Boundary Value Up: Applications of Hankel and Previous: Applications of Hankel and   Contents   Index
Ulrich Gerlach 2010-12-09