  We study a 2D lattice model of forward-directed waves in which the integrated
intensity for classical waves (or probability for quantum mechanical particles)
is conserved. The model describes the time evolution of 1D quantum particle in
a time-varying potential and also applies to propagation of electromagnetic
waves in two dimensions within the parabolic approximation. We present a closed
form solution for propagation in a uniform system. Motivated by recent studies
of non-unitary directed models for localized 2D electrons tunneling in a
magnetic field, we then address related theoretical questions of how the
interference pattern between constrained-forward paths in this unitary model is
affected by the addition of phases corresponding to such a magnetic field. The
behavior is found to depend sensitively on the value of $\Phi/\Phi_{0}$, where
$\Phi$ is flux per plaquette and $\Phi_{0}$ is the unit of flux quantum. For
$\Phi/\Phi_{0} = p/q$ we find the amplitude to be more collimated the larger is
the value of $q$. We next consider propagation in a random forward scattering
media. In particular the scaling properties associated with the transverse
width $x$ of the wave, as function of its distance $t$ from point source, are
addressed. We find the moments of $x$ to scale with $t$ in a very different way
from what is known for either off-lattice unitary or on-lattice non-unitary
systems. The scaling of the moments of the probability $[P^{n}(x,t)]$ (or
intensity) at a point $(x=0,t)$ is found to be consistent with a simple
behavior $[P^{n}(0,t)] \sim t^{-\frac{n}{2}}$. Implications to the behavior of
one-dimensional lattice quantum particles in a dynamically fluctuating
