In the paper, the definition and basic properties of the Fourier transform (FT) are discussed. It has been shown with specific examples that integral solutions of the model inhomogeneous equation, the nonstationary Cauchy problem on an inhomogeneous shear flow, and the boundary value problem on the transformation of internal waves in the vicinity of the focus in the inhomogeneous medium can be found by FT and using its properties. The constructed Fourier integrals refuted the widely held claim that the Fourier analysis is unusable for the study of inhomogeneous media.