  In this paper, we study the number of traveling wave families near a shear
flow under the influence of Coriolis force, where the traveling speeds lie
outside the range of the flow $u$. Under the $\beta$-plane approximation, if
the flow $u$ has a critical point at which $u$ attains its minimal (resp.
maximal) value, then a unique transitional $\beta$ value exists in the positive
(resp. negative) half-line such that the number of traveling wave families near
the shear flow changes suddenly from finite to infinite when $\beta$ passes
through it. On the other hand, if $u$ has no such critical points, then the
number is always finite for positive (resp. negative) $\beta$ values. This is
true for general shear flows under mildly technical assumptions, and for a
large class of shear flows including a cosine jet $u(y) = {1+\cos(\pi y)\over
2}$ (i.e. the sinus profile) and analytic monotone flows unconditionally. The
sudden change of the number of traveling wave families indicates that long time
dynamics around the shear flow is much richer than the non-rotating case, where
no such traveling wave families exist.
