We expose some concepts concerning the channel impulse response (CIR) of linear time‐varying (LTV) channels to give a proper characterization of the mobile‐to‐mobile underwater channel. We find different connections between the linear time‐invariant (LTI) CIR of the static channel and 2 definitions of LTV CIRs of the dynamic mobile‐to‐mobile channel. These connections are useful to design a dynamic channel simulator from the static channel models available in the literature. Such feature is particularly interesting for overspread channels, which are hard to characterize by a measuring campaign. Specifically, the shallow water acoustic (SWA) channel is potentially overspread because of the signal low velocity of propagation, which prompts long delay spread responses and great Doppler effect. Furthermore, from these connections between the LTI static CIRs and the LTV dynamic CIRs, we find that the SWA mobile‐to‐mobile CIR does not only depend on the relative speed between transceivers, but also on the absolute speed of each of them referred to the velocity of propagation. Nevertheless, publications about this topic do not consider it and formulate their equations in terms of the relative speed between transceivers. We illustrate our find using 2 couples of examples where, even though the relative speed between the mobiles is the same, their CIRs are not.