  We study 4 problems in the area of scattering of time harmonic acoustic or
electromagnetic waves by unbounded rough surfaces/inhomogeneous layers.
Specifically we study: i) a boundary value problem (BVP) for the Helmholtz
equation, in both 2 and 3 dimensions, modelling scattering of time harmonic
waves due to a source that lies within a finite distance of the boundary and
which decays along the boundary, by a layer of spatially varying refractive
index above an unbounded rough surface on which the field vanishes; ii) a BVP
for the Helmholtz equation with an impedance boundary condition, in 2 and 3
dimensions, modelling the scattering of time harmonic acoustic waves due to a
source that lies within a finite distance of the boundary and which decays
along the boundary, by an unbounded rough impedance surface; iii) a problem of
scattering of time harmonic waves by a layer of spatially varying refractive
index at the interface between semi-infinite half-spaces of fixed positive
refractive index (the waves arising due to a source that lies within a finite
distance of the layer and which decays along the layer); iv) a BVP for the
Helmholtz equation with a Dirichlet boundary condition, in 3 dimensions,
modelling the scattering of time harmonic acoustic waves due to a point source,
by an unbounded, rough, sound soft surface. We study i), ii) and iii) by
variational methods; via analysis of equivalent variational formulations we
prove these problems to be well-posed under various restrictions on the surface
(must be Lipschitz), the frequency, the rate of change of the refractive index
etc. We study iv) via a Brakhage-Werner type integral equation formulation,
based on an ansatz for the solution as a combined single- and double-layer
potential. We establish that when the rough surface is the graph of a Lipschitz
function, the problem is well-posed for arbitrary frequency.
