  A representation of solutions of the wave equation with two spatial
coordinates in terms of localized elementary ones is presented. Elementary
solutions are constructed from four solutions with the help of transformations
of the affine Poincar\'e group, i.e., with the help of translations, dilations
in space and time and Lorentz transformations. The representation can be
interpreted in terms of the initial-boundary value problem for the wave
equation in a half-plane. It gives the solution as an integral representation
of two types of solutions: propagating localized solutions running away from
the boundary under different angles and packet-like surface waves running along
the boundary and exponentially decreasing away from the boundary. Properties of
elementary solutions are discussed. A numerical investigation of coefficients
of the decomposition is carried out. An example of the field created by sources
moving along a line with different speeds is considered, and the dependence of
coefficients on speeds of sources is discussed.
