  This paper constructs an approximate sinusoidal wave packet solution to the
equations of loop quantum gravity (LQG). There is an SU(2) holonomy on each
edge of the LQG simplex, and the goal is to study the behavior of these
holonomies under the influence of a passing gravitational wave. The equations
are solved in a small sine approximation: holonomies are expanded in powers of
sines, and terms beyond $\sin^2$ are dropped; also, fields vary slowly from
vertex to vertex. The wave is unidirectional and linearly polarized. The
Hilbert space is spanned by a set of coherent states tailored to the symmetry
of the plane wave case. Fixing the spatial diffeomorphisms is equivalent to
fixing the spatial interval between vertices of the loop quantum gravity
lattice. This spacing can be chosen such that the eigenvalues of the triad
operators are large, as required in the small sine limit, even though the
holonomies are not large. Appendices compute the energy of the wave, estimate
the lifetime of the coherent state packet, discuss coarse-graining, and
determine the behavior of the spinors used in the U(N) SHO realization of LQG.
