  This paper is concerned with pulsating waves for multi-dimensional
reaction-diffusion equations in spatially periodic media. First, assuming the
existence of pulsating waves connecting two linearly stable steady states, we
study the continuity of wave speeds with respect to the direction of
propagation. The continuity was proved in [15] under the extra condition that
the speeds are nonzero in all directions. Here, we revisit this continuity
result without the extra condition. Secondly, we provide some sufficient
conditions ensuring the existence of pulsating waves in rapidly oscillating
media, which allow the equations to have multiple stable steady states.
