Aims. We aim to analytically and numerically explore small-amplitude Alfvén waves in the solar atmosphere. Methods. We transform the wave equations to obtain the cutoff-frequency and wave travel time for strictly linear Alfvén waves. The wave equations are solved numerically to find out spatial and temporal signatures of the waves. Results. The analytical predictions are verified by numerically solving the wave equations for linear Alfvén waves. The waves are impulsively generated and their characteristics and behavior in the solar atmosphere are investigated by the numerical simulations. The derived cutoff-frequency is used to determine regions in the solar atmosphere where strong reflection occurs for Alfvén waves of different frequencies. Conclusions. The numerical results reveal that impulsively generated small-amplitude waves exhibit characteristic spatial and temporal signatures which agree with the predictions of the analytical theory.
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