  We explore bosonic string sigma models on warped $ BTZ\times S^3 $ both in
the plane wave as well as beyond plane wave limit. Using the light cone gauge,
we obtain the corresponding Hamiltonian and therefore the spectrum associated
with the pp wave strings. Our analysis reveals a constant shift in the spectrum
arising as a result of TsT transformations along the isometries of the target
space manifold. Imposing the energy positivity constraint on the CFT$_2 $
spectrum, we estimate an upper bound on the shift. We also estimate corrections
as we go beyond the plane wave limit. Finally, we perform calculations using
conformal gauge which reveals identical spectrum for the pp wave strings and
thereby shows the equivalence between the two approaches.
