  The phase transitions in the transverse field Ising model in a competing
spatially modulated (periodic and oscillatory) longitudinal field are studied
numerically. There is a multiphase point in absence of the transverse field
where the degeneracy for a longitudinal field of wavelength $\lambda$ is
$(\frac {1 + \sqrt{5}}{2})^{2N/\lambda}$ for a system with $N$ spins, an exact
result obtained from the known result for $\lambda =2$. The phase transitions
in the $\Gamma $ (transverse field) versus $h_0$ (amplitude of the longitudinal
field) phase diagram are obtained from the vanishing of the mass gap $\Delta$.
We find that for all the phase transition points obtained in this way, $\Delta
$ shows finite size scaling behaviour signifying a continuous phase transition
everywhere. The values of the critical exponents show that the model belongs to
the universality class of the two dimensional Ising model. The longitudinal
field is found to have the same scaling behaviour as that of the transverse
field, which seems to be a unique feature for the competing field. The phase
boundaries for two different wavelengths of the modulated field are obtained.
Close to the multiphase point at $h_c$, the phase boundary behaves as $(h_c -
h_0)^b$, where $b$ is also $\lambda$ dependent.
