  Existing theoretical results for attenuation of surface waves propagating on
water of random fluctuating depth are shown to over predict the rate of decay
due to the way in which ensemble averaging is performed. A revised approach is
presented which corrects this and is shown to conserve energy. New theoretical
predictions are supported by numerical results which use averaging of
simulations of wave scattering over finite sections of random bathymetry for
which transfer matrix eigenvalues are used to accurately measure decay. The
model of wave propagation used in this paper is derived from a linearised long
wavelength assumption whereby depth averaging leads to time harmonic waves
being represented as solutions to a simple ordinary differential equation. In
this paper it is shown how this can be adapted to incorporate a model of a
continuous covering of the surface by fragmented floating ice. Attenuation of
waves through broken ice of random thickness is then analysed in a similar
manner as bed variations previously and some comparisons are made with
published field data for attenuation of waves in the marginal ice zones. Key
features of the data are reproduced by theory including the attenuation being
proportional to a power of frequency between 2 and 4 as well as capturing the
"roll-over effect" at high frequencies.
