  We formulate an effective medium (mean field) theory of a material consisting
of randomly distributed nodes connected by straight slender rods, hinged at the
nodes. Defining novel wavelength-dependent effective elastic moduli, we
calculate both the static moduli and the dispersion relations of ultrasonic
longitudinal and transverse elastic waves. At finite wave vector $k$ the waves
are dispersive, with phase and group velocities decreasing with increasing wave
vector. These results are directly applicable to networks with empty pore
space. They also describe the solid matrix in two-component (Biot) theories of
fluid-filled porous media. We suggest the possibility of low density materials
with higher ratios of stiffness and strength to density than those of foams,
aerogels or trabecular bone.
