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Wave propagation in a nonperiodically time-varying medium provides a means for realizing in simple physical structures a variety of signal-processing operations, such as frequency translation and coding, variable delay recall, gating, time-scale stretching or shrinking, and time reversal. The use of low-velocity modes, such as acoustic, spin, or magnetoelastic waves in solids, reduces the length of the propagation structure required to less than an inch. A general review is given of the principles of wave propagation in a spatially uniform medium with nonperiodic time variation. Both abrupt and gradual time variations are discussed. Illustrations are given for the cases of spin-wave and magnetoelastic-wave propagation and signal processing operations in these media are explained. Consideration is given to the problem of spin-wave propagation in a time-and space-varying magnetic field of the form encountered in experiments; and it is shown that a simple separated variable solution exists. For the more difficult problem of magnetoelastic propagation with both space and time variations, an approximate space-time ray theory is described. Experimental results for pulsed-field processing of spin and magnetoelastic waves are given and related to the theory.