Type Ia supernovae (SNe) are presumed to arise from white dwarf progen
itors, which may not appreciably modify their ambient medium. We study
the interaction of the resulting supernova remnants with a constant d
ensity interstellar medium. Density profiles obtained from detailed ex
plosion models of Type Ia SN explosions can be complex, but an exponen
tial profile gives the best approximate representation of a set of pro
files, and we emphasize this case. We describe the time evolution of d
ynamical quantities (such as radius, velocity, and expansion parameter
) as a result of the interaction in terms of dimensionless variables a
nd present the profiles of physical quantities. We compare our results
to the power-law and constant ejecta density cases; a characteristic
feature of the exponential case is that the shocked ejecta have a rela
tively constant temperature. The effect of a possible circumstellar wi
nd region dose to the supernova is to create a dense, cool shell near
the contact discontinuity between the shocked ejecta and the surroundi
ng medium. The complex density structure found in some supernova model
s persists in the shocked layer, giving rise to density and temperatur
e variations. We apply our results to the two likely historical Type I
a SNe, SN 1006 and Tycho. The observed angular sizes and expansion rat
es are consistent with a distance of 1.95 +/- 0.4 kpc and an ambient H
density of 0.05-0.1 cm(-3) for SN 1006. For Tycho's SNR, the results
are not conclusive but indicate a distance around 2.3 kpc for an ambie
nt density of 0.6-1.1 cm(-3). In both cases, the low expansion rate li
mits the extent of a possible circumstellar wind region. The evidence
for temperature variations in the ejecta of Tycho's remnant suggests t
hat the supernova profile was more complex than an exponential profile
and contained density inhomogeneities, or that there was early intera
ction with a circumstellar wind.