  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 two 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 due the
signal low velocity of propagation which prompt 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 velocity between transceivers, but also on
the absolute velocity 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 velocity between transceivers. We
illustrate our find using two couples of examples where, even though the
relative velocity between the mobiles is the same, their CIRs are not.
