Scattering theory is concerned with the effect obstacles or inhomogeneities have on an incident waves. There are two types of problems in this area:

**The Direct Problem:**This problem is concerned with determining the scattered field from the knowledge of the incident field and the scattering obstacle.**The Inverse Problem:**Here one tries to determine the shape and/or physical properties of the scatterer from the measurement of the scattered field for a number of incident fields.

The propagation of waves in a homogeneous, isotropic medium is mathematically described by the wave equation,

The total field is usually represented as the sum of the incident and the scattered field. The scattering problem can then be formulated as a boundary value problem for the scattered field in the region outside the scattering obstacle, consisting of

- the Helmholtz equation,
- a boundary condition on the boundary of the scattering obstacle,
- Sommerfeld's radiation condition.

The image above is a visualisation of the scattered field for such a bounded obstacle problem. A plane wave is incident from the left-hand side and scattered by the kite-shaped object. A Dirichlet boundary condition (i.e. the total field vanishes) was used on the surface. For more detail on this problem, there is a separate page including an animation (Java applet) of the wave field.

A diffraction grating is a very simple type of unbounded scatterer: It is a surface
that is invariant in the, say *x _{3}* coordinate direction while showing
some periodicity in the

If the incident field also is periodic with a certain, possibly different, period
(which is the case for an incident plane wave!), both the incident and the scattered
field will be **quasi-periodic**. This property allows substantial simplifications
as the problem can thus be dealt with in spaces of periodic functions.

A further difficulty arises, in that the scattered field in a diffraction problem
will, in general, not show any decay in directions parallel to the surface. Thus,
Sommerfeld's radiation condition will not apply. It is replaced by the **Rayleigh
Expansion Radiation Condition** (RERC), that the scattered field has, at some height
above the surface, an expansion in plane waves propagating upwards and evanescent waves,
decaying exponentially with distance from the surface.

Numerically, the problem can be treated, for example, by a boundary integral ansatz with a Nyström method for the numerical treatment of the resulting boundary integral equation. This type of method is extremely efficient, and a code developed by the Brunel Rough Surface Scattering group clearly demonstrates the theoretically expected exponential convergence rates. There is also a more detailed description of the problem available, that includes an animation of the total field based on a Java applet.

Of special interest to the Brunel Rough Surface Scattering group
are problems involving effectively unbounded obstacles. Such an
obstacle is termed a rough surface, if it can be represented as
the graph of a bounded, continuous function *f*. Often,
*f* will be required to satisfy some additional smoothness
properties, usually Lipschitz continuity of its first derivative.

A number of difficulties arise when dealing with such problems.
Firstly, to formulate the problem as a boundary value problem,
a suitable radiation condition has to be imposed. Such a condition
is the **upward propagating radiation
condition** introduced by
Chandler-Wilde.
It was shown that this
radiation condition ensures uniqueness of solution to the sound soft
and impedance surface scattering problems for quite general classes
of incident fields, including cylindrical and plane waves.

Secondly, when deriving boundary integral equation formulations, the standard fundamental solution, i.e. the Green's function in free field conditions, turns out to be poor choice as the kernel in single- and double-layer potentials. This is due to the fact that we cannot expect the surface density to be decaying along the infinite surface for general classes of incident waves. Thus, the decay rate of the free field Green's function, i.e.

However, and this is a further difficulty, the arising integral operators have an unbounded range of integration. Even for the simplest case, a convolution operator, one observes that such an operator is not compact. Thus, an entirely new solvability theory is required to prove that the boundary integral equation admits a unique solution.

The extension of these results to the full, 3-dimensional case, is the object of much ongoing research in the Brunel Rough Surface Scattering group.

- iterative methods relying on a Newton type iteration,
- methods splitting up the problem in the ill-posed linear problem of reconstructing the scattered field from the far-field pattern and the well-posed non-linear problem of finding the boundary from the scattered field,
- methods using a direct characterisation of the obstacle.

- D. Colton and R. Kress,
*Integral Equation Methods in Scattering Theory*, Wiley, 1983. - D. Colton and R. Kress,
*Inverse Acoustic and Electromagnetic Scattering Theory*, 2nd edition, Springer-Verlag, 1998. - A. Kirsch,
*An Introduction to the Mathematical Theory of Inverse Problems*, Springer-Verlag, 1996.

- There are some nice examples of scattering problems and related numerical methods on the home page of Klaus Giebermann at Bonn University.

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