  A weakly interacting, spin-orbit coupled, ultracold, dilute Bose gas on a
two-dimensional square lattice with an external Zeeman field is studied. We
explore the plane and stripe wave phases of the system involving nonzero
condensate momenta, which occur when the Zeeman field is below a critical
value. Their excitation spectra are found using Bogoliubov theory and by two
different routes. The validity of each method to obtain the excitation spectrum
is discussed, and it is found that projection on the lowest single-particle
band is an excellent approximation in the plane wave phase, while it is a poor
approximation in the stripe wave phase. While the plane wave phase has a phonon
minimum at its single condensate momentum, revealing a nonzero sound velocity
of the excitations, the stripe wave phase has quadratic minima at its two
condensate momenta showing zero sound velocity of the excitations. We discuss
how the presence of more than one condensate momentum is essential for these
differences between the two phases. Additionally, it is emphasized that the
zero sound velocity in the stripe wave phase is a lattice effect, since
continuum studies of the same phase have shown nonzero sound velocity.
