For the extended Taylor–Goldstein problem of hydrodynamic stability governing the stability of shear flows of an inviscid, incompressible but density stratified fluid in sea straits of arbitrary cross-section a new estimate for the growth rate of an arbitrary unstable normal mode is given for a class of basic flows. Furthermore the Howard’s conjecture, namely, the growth rate $kc_i\to 0$ as the wave number $k\to\infty$ is proved for two classes of basic flows.

We consider the extended Rayleigh problem of hydrodynamic stability dealing with the stability of inviscid homogeneous shear flows in sea straits of arbitrary cross section. We prove a short wave stability result, namely, if $k>0$ is the wave number of a normal mode then $k>k_c$ (for some critical wave number $k_c$) implies the stability of the mode for a class of basic flows. Furthermore, if $K(z)=\frac{-({U''}_0-T_0{U'}_0)}{U_0-U_{0s}}$, where $U_0$ is the basic velocity, $T_0$ (a constant) the topography and prime denotes differentiation with respect to vertical coordinate 𝑧 then we prove that a sufficient condition for the stability of basic flow is $0 < K(z)\leq\left(\frac{\pi^2}{D^2}+\frac{T^2_0}{4}\right)$, where the flow domain is $0\leq z\leq D$.