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We determine wave number $q$ and frequency $\omega$ dependent spin Hallconductivity $\sigma_{yx}^s(q, \omega)$ for a disordered two dimensionalelectron system with Rashba spin orbit interaction when $\q$ is {\ittransverse} to the electric field. Both the conventional definition of spincurrent and its new definition which takes care of the conservation of spins,have been considered. The spin Hall conductivitivities for both of thesedefinitions are qualitatively similar. $\sigma_{yx}^s(q, \omega)$ is zero at$q=0, \omega =0$ and is maximum at $q=0$ and at small but finite $\omega$ whosevalue depends on different parameters of the system. Interestingly for $\omega\to 0$, $\sigma_{yx}^s(q)$ resonates when $\Lambda \simeq L_{so}$ which are thewavelength $(\Lambda = 2\pi/q)$ of the electric field's spatial variation andthe length for one cycle of spin precession respectively. The sign of theout-of-plane component of the electrons' spin flips when the sign of electricfield changes due to its spatial variation along transverse direction. Itchanges the mode of spin precession from clockwise to anti-clockwise or {\itvice versa} and consequently a finite spin Hall current flows in the bulk ofthe system.Comment: 6 pages; 6 figures; major changes including the titl

Frequency- and transverse wave-vector-dependent spin Hall conductivity in two-dimensional electron gas with disorder