  This work deals with exact analytical modelling of transfer phenomena in
heterogeneous materials exhibiting one-dimensional continuous variations of
their properties. Regarding heat transfer, it has recently been shown that by
applying a Liouville transformation and multiple Darboux transformations,
infinite sequences of solvable profiles of thermal effusivity can be
constructed together with the associated temperature (exact) solutions, all in
closed-form expressions (vs. the diffusion-time variable and with a growing
number of parameters). In addition, a particular class of profiles, so-called
sech(xi)-type profiles, exhibit high agility and in the same time parsimony. In
this paper we go further into the description of these solvable profiles and
their properties. Most importantly, their quadrupole formulation is provided
which allows building smooth synthetic profiles of effusivity of arbitrary
complexity and thereafter getting very easily the corresponding temperature
dynamic response. Examples are given with increasing variability of effusivity
and increasing number of elementary profiles. These highly flexible profiles
are equally relevant for providing an exact analytical solution to wave
propagation problems in 1D graded media (i.e. Maxwell's equations, acoustic
equation, telegraph equation...). From now on, let it be for diffusion-like or
wave-like problems, when the leading properties present (possibly piecewise-)
continuously heterogeneous profiles, the classical staircase model can be
advantageously replaced by a "high-level" quadrupole model consisting of one or
more sech(xi)-type profiles, which makes the latter a true Swiss-Army knife for
analytical modelling.
