  We study the Casimir force on a single surface immersed in an inhomogeneous
medium. Specifically we study the vacuum fluctuations of a scalar field with a
spatially varying squared mass, $m^{2}+\lambda\Delta(x-a)+V(x)$, where $V$ is a
smooth potential and $\Delta(x)$ is a unit-area function sharply peaked around
$x=0$. $\Delta(x-a)$ represents a semi-penetrable thin plate placed at $x=a$.
In the limits $\{\Delta(x-a)\to\delta(x-a), \lambda\to\infty \}$ the scalar
field obeys a Dirichlet boundary condition, $\phi=0$, at $x=a$. We formulate
the problem in general and solve it in several approximations and specific
cases. In all the cases we have studied we find that the Casimir force on the
plate points in the direction opposite to the force on the quanta of $\phi$: it
pushes the plate toward higher potential, hence our use of the term buoyancy.
We investigate Casimir buoyancy for weak, reflectionless, or smooth $V(x)$, and
for several explicitly solvable examples. In the semiclassical approximation,
which seems to be quite useful and accurate, the Casimir buoyancy is a local
function of $V(a)$. We extend our analysis to the analogous problem in
$n$-dimensions with $n-1$ translational symmetries, where Casimir divergences
become more severe. We also extend the analysis to non-zero temperatures.
