  We propose simple, explicit, analytical approximations for the kinematics of
an adiabatic blast wave propagating in an exponentially stratified ambient
medium, and for the onset of radiative cooling, which ends the adiabatic era.
Our method, based on the Kompaneets implicit solution and the Kahn
approximation for the radiative cooling coefficient, gives straightforward
estimates for the size, expansion velocity, and progression of cooling times
over the surface, when applied to supernova remnants (SNRs). The remnant shape
is remarkably close to spherical for moderate density gradients, but even a
small gradient in ambient density causes the cooling time to vary substantially
over the remnant's surface, so that for a considerable period there will be a
cold dense expanding shell covering only a part of the remnant. Our
approximation provides an effective tool for identifying the approximate
parameters when planning 2-dimensional numerical models of SNRs, the example of
W44 being given in a subsequent paper.
