In this paper we consider the extended Taylor-Goldstein problem of hydrodynamic stability dealing with the stability of stratified shear flows in sea straits of arbitrary cross section. For this problem we have obtained the upper and lower bounds on the phase velocity of neutral modes and estimates for the growth rate of unstable modes. Furthermore, we have found a semielliptical instability region depending on the minimum Richardson number and parabolic instability regions which intersect the semielliptical instability region for a class of basic flows.

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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 → 0 as the wave number k → ∞ is proved for two classes of basic flows.

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