Although the JASA paper appeared the following year, that paper was the one summarizing the work where the exact results were first discovered (although I did not know that they were exact at the time). The derivations and a thorough discussion of the exact results generalizing Gassmann's equations are summarized in the paper:

J. G. Berryman and G. W. Milton, ``Exact results for generalized Gassmann's equations in composite porous media with two constituents,'' Geophysics 56, 1950-1960 (1991).

And, yes, they really are exact results. We are not exaggerating!