  The added mass effect is the contribution to a Brownian particle's effective
mass arising from the hydrodynamic flow its motion induces. For a spherical
particle in an incompressible fluid, the added mass is half the fluid's
displaced mass, but in a compressible fluid its value depends on a competition
between timescales. Here we illustrate this behavior with a solvable model of
two harmonically coupled Brownian particles of mass $m$, one representing the
sphere, the other the immediately surrounding fluid. The measured distribution
of the Brownian particle's velocity, $P(\bar{v})$, follows a Maxwell-Boltzmann
distribution with an effective mass $m^*$. Solving analytically for $m^*$, we
find that its value is determined by three relevant timescales: the momentum
relaxation time, $t_p$, the harmonic oscillation period, $\tau$, and the
velocity measurement time resolution, $\Delta t$. In limiting cases $\Delta t
\ll \tau,t_p$ and $\tau\ll\Delta t\ll t_p$, our expression for $m^*$ reduces to
$m$ and $2m$, respectively. We find similar behavior upon generalizing the
model to the case of unequal masses.
