  In this paper, we studied the long-wave instability of the shear flows. When
the wavenumber of perturbation is larger than the critical value, the flow is
always neutrally stable. First, we obtain a new upper bound for the neutral
wavenumber $k_1\leq (p^2-1)\mu_1$, where $p>1$ and $\mu_1$ is the smallest
eigenvalue of Poincar\'{e}'s problem. Second, we find a new upper bound for the
imaginary part of the complex phase velocity $c_i \leq k_1 \Delta
U/\sqrt{\mu_1}$, where $\Delta U$ is the variance of the velocity. The new
bound is finite for all $k>0$ similar to the Howard's semicircle theorem, while
the previous ones by Craik and Banerjee et al would be infinity as
$k\rightarrow 0$. Third, we find a new upper bound of growth rate $\omega_i
\leq (p-1) \sqrt{\mu_1} \Delta U$. All the new bounds are much more strict than
the previous ones by H{\o}iland, Howard, Craik and Banerjee et al. Our results
also extend the inverse energy cascade theory by Kraichnan. As shear
instability is due to long-wave instability, it implies that the truncation of
long-waves may change the instability of shear flows.
