  We have explored the effectiveness of a longitudinal gradient in Alfv\'en
speed in reducing the energy of propagating Alfv\'en waves under conditions
scaled to match solar coronal holes. The experiments were conducted in the
Large Plasma Device at the University of California, Los Angeles. Our results
show that the energy of the transmitted Alfv\'en wave decreases as the
inhomogeneity parameter, $\lambda/L_{\rm A}$, increases. Here, $\lambda$ is the
wavelength of the Alfv\'en wave and $L_{\rm A}$ is the scale length of Alfv\'en
speed gradient. For gradients similar to those in coronal holes, the waves are
observed to lose a factor of $\approx 5$ more energy than they do when
propagating through a uniform plasma without a gradient. We have carried out
further experiments and analyses to constrain the cause of wave energy
reduction in the gradient. The loss of Alfv\'en wave energy from mode coupling
is unlikely, as we have not detected any other modes. Contrary to theoretical
expectations, the reduction in the energy of the transmitted wave is not
accompanied by a detectable reflected wave. Nonlinear effects are ruled out as
the amplitude of the initial wave is too small and the wave frequency well
below the ion cyclotron frequency. Since the total energy must be conserved, it
is possible that the lost wave energy is being deposited in the plasma. Further
studies are needed to explore where the energy is going.
