Scattering wave systems that are periodically modulated in time offer many new degrees of freedom to control waves in both the spatial and frequency domains. Such systems, albeit linear, do not conserve frequency and require the adaptation of the usual theories and methods. In this paper, we provide a general extension of transmission-line or telegraph equations to periodically time-modulated systems. As a byproduct of the theory, we obtain a general approach to compute and measure the complete scattering matrix of such systems. Finally, the proposed theory and methods are applied and validated on a concrete practical example in the realm of airborne acoustics: a time-modulated actively controlled loudspeaker membrane terminating a monomode waveguide. Different modulation functions and parameters are tested. The experimental results are compared to both numerical simulation and an analytical model based on a two-time-scale method.