  Space-time modulation adds another powerful degree of freedom to the
manipulation of classical wave systems. It opens the door for complex control
of wave behavior beyond the reach of stationary systems, such as nonreciprocal
wave transport and realization of gain media. Here we generalize the transfer
matrix method and use it to create a general framework to solve wave
propagation problems in time-varying acoustic, electromagnetic, and electric
circuit systems. The proposed method provides a versatile approach for the
study of general space-time varying systems, which allows any number of
time-modulated elements with arbitrary modulation profile, facilities the
investigation of high order modes, and provides an interface between space-time
modulated systems and other systems.
