The problem of two-dimensional acoustic scattering of an incident plane wave
by a semi-infinite lattice is solved. The problem is first considered for sound-soft
cylinders whose size is small compared to the wavelength of the incident field.
In this case the formulation leads to a scalar Wiener--Hopf equation, and this
in turn is solved via the discrete Wiener--Hopf technique. We then deal with a
more complex case which arises either by imposing Neumann boundary condition on
the cylinders' surface or by increasing their radii. This gives rise to a matrix
Wiener--Hopf equation, and we present a method of solution that does not require
the explicit factorisation of the kernel. In both situations, a complete description
of the far field is given and a conservation of energy condition is obtained.
For certain sets of parameters (`pass bands'), a portion of the incident energy
propagates through the lattice in the form of a Bloch wave. For other parameters
(`stop bands' or `band gaps'), no such transmission is possible, and all of the incident
field energy is reflected away from the lattice.
A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.