Many treatments of wave scattering by irregular media take the incident field to be that of a monochromatic plane wave. There are, however, important cases encountered in practice, in particular in Radio Astronomy, when the incident field is not a plane wave, but is produced by a source of a finite angular size. Moreover, the radiation is not monochromatic, but is received over a continuous band of frequencies. The present paper is concerned with fluctuations produced when such a field is scattered by a medium containing weak random irregularities of refractive index. The important statistical properties of these fluctuations are shown to depend on three quantities which are double Fourier integrals with respect to the spatial frequencies present in the scattered field. These functions are assembly averages of products of scattered field components, and are derived by studying the scattering of a wave obliquely incident on a weakly irregular medium. Physical considerations are used to show how a spread of frequencies and angles of arrival, in the incident wave, affect the fluctuations imposed by the medium. The integrands of the Fourier integrals can be used later for the general problem of multiple scatter, but the results in this paper are only for weak single scattering, when the scatterer is either a thin phase changing screen or an extended irregular medium. Formulae are given for the scintillation index, the scintillation visibility, and the spatial autocorrelation function of the fluctuations of received intensity, for some particular cases of source brightness function, receiver bandpass function and autocorrelation function of irregularities in the medium. They are discussed briefly, to illustrate some properties of fluctuations of the scattered field when the incident field is only partially coherent.