  We study a class of oscillating bounce solutions to the Euclidean field
equations for gravity coupled to a scalar field theory with two, possibly
degenerate, vacua. In these solutions the scalar field crosses the top of the
potential barrier $k>1$ times. Using analytic and numerical methods, we examine
how the maximum allowed value of $k$ depends on the parameters of the theory.
For a wide class of potentials $k_{\rm max}$ is determined by the value of the
second derivative of the scalar field potential at the top of the barrier.
However, in other cases, such as potentials with relatively flat barriers, the
determining parameter appears instead to be the value of this second derivative
averaged over the width of the barrier. As a byproduct, we gain additional
insight into the conditions under which a Coleman-De Luccia bounce exists. We
discuss the physical interpretation of these solutions and their implications
for vacuum tunneling transitions in de Sitter spacetime.
