  Long wavelength oscillations (Tkachenko waves) of the triangular lattice of
quantized vortices in superfluid neutron stars have been suggested as one of
the possible explanations for the timing noise observed in many radio pulsars,
in particular for the 100-1000 day variations in the spin of PSR B1828-11. Most
studies to date have, however, been based on the hydrodynamics developed for
superfluid Helium. In this paper we extend the formulation to a two fluid
neutron and proton system, relevant for neutron star interiors and include the
effect of chemical coupling, compressibility and mutual friction between the
components. In particular we find that chemical coupling and compressibility
can have a drastic effect on the mode structure. However, for the slower
pulsars rotating at 1-10 Hz (such as PSR 1828-11), most choices of parameters
in the equation of state lead to Tkachenko oscillations with frequencies in the
correct range to explain the timing noise. We also investigate the case of more
rapidly rotating pulsars (above 100 Hz) for which we find that there is a vast
portion of parameter space in which there are no Tkachenko modes, but only
modified sound waves at much higher frequencies.
