We consider the equation
u
t
=
u
x
x
+
f
(
t
,
u
)
u_t=u_{xx} +f(t,u)
,
x
∈
R
x\in \mathbb {R}
,
t
>
0
t>0
, where
f
(
t
,
x
)
f(t,x)
periodically depends on
t
t
and is of bistable type. Classical results showed that for a large class of initial functions, the solutions converge to a periodic traveling wave if it connects two linearly stable time-periodic states. Under some conditions on the initial functions, we prove this convergence result by a new approach which allows the time-periodic states to be degenerate.