The problem of acoustic wave diffraction on a finite body in half-space or in a homogeneous waveguide with ideally plane boundaries is treated. The decision method is based on the scattering diagram (amplitude of scattering, in particular). A formula to calculate the scattering amplitude for an arbitrary incident wave by the scattering amplitude for plane incident waves is given. An algorithm to solve the problem of acoustic wave radiation by an acoustically opaque source in half-space with an arbitrary reflection factor is proposed. As an example, the problem of low-frequency acoustic wave radiation by a sphere with acoustically soft boundaries near the acoustically hard or soft boundary of half-space is solved. The problem of the field of a spherical source with the account of diffraction on it in an ideal acoustic waveguide is next considered. The results of scattering on a sphere in half-space with ideal boundaries are used. It is shown that the resulting field is formed by an infinite sum of sources. The errors caused by restriction of their quantity are defined.  The results of numerical calculations of the field of a sphere in the ideal waveguide with and without  account of scattering are given.