Magnetohydrodynamic wave interactions in a linear shear flow are investigated using the Lagrangian fluid displacement ξ and entropy perturbation Δ S, in which a spatial Fourier solution is obtained in the frame of the background shear flow (Kelvin's method). The equations reduce to three coupled oscillator equations, with time-dependent coefficients and with source terms proportional to the entropy perturbation. In the absence of entropy perturbations, the system admits a wave action conservation integral consisting of positive and negative energy waves. Variational and Hamiltonian forms of the equations are obtained. Examples of wave amplification phenomena and sharp resonant-type wave interactions are obtained. Implications for the interaction of magnetohydrodynamic waves in the shear flow between fast, polar coronal-hole solar wind and slow, streamer belt solar wind are discussed.