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A new analytical method is developed for transient solutions for bars in longitudinal vibration, shafts in torsional vibration and strings in transverse vibration. These vibrating continua have nonuniform distributions of geometric and material properties, and are governed by second-order partial differential equations (wave equations). Based on a transfer function formulation and a residue formula, the proposed method delivers exact solutions for several types of nonuniform systems subject to external, boundary and initial disturbances. The proposed method does not need system eigenfunctions, and is numerically efficient as it only involves simple operations of two-by-two matrices.