  A body $\Theta$ containing two phases, which may form a periodic composite
with microstructure much smaller that the body, or which may have structure on
a length scale comparable to the body, is subjected to slowly time varying
boundary conditions that would produce an approximate uniform field in $\Theta$
were it filled with homogeneous material. Here slowly time varying means that
the wavelengths and attenuation lengths of waves at the frequencies associated
with the time variation are much larger than the size of $\Theta$, so that we
can make a quasistatic approximation. At least one of the two phase does not
have an instantaneous response but rather depends on fields at prior times. The
fields may be those associated with electricity, magnetism, fluid flow in
porous media, or antiplane elasticity. We find, subject to these
approximations, that the time variation of the boundary conditions can be
designed so boundary measurements at a specific time $t=t_0$ exactly yield the
volume fractions of the phases, independent of the detailed geometric
configuration of the phases. Moreover, for specially tailored time variations,
the volume fraction can be exactly determined frommeasurements at any time $t$,
not just at the specific time $t=t_0$. We also show how time varying boundary
conditions, not oscillating at the single frequency $\omega_0$, can be designed
to exactly retrieve the response at $\omega_0$.
