  We present a mathematical theory of time-harmonic wave propagation and
reflection in a two-dimensional random acoustic waveguide with sound soft
boundary and turning points. The boundary has small fluctuations on the scale
of the wavelength, modeled as random. The waveguide supports multiple
propagating modes. The number of these modes changes due to slow variations of
the waveguide cross-section. The changes occur at turning points, where waves
transition from propagating to evanescent or the other way around. We consider
a regime where scattering at the random boundary has significant effect on the
wave traveling from one turning point to another. This effect is described by
the coupling of its components, the modes. We derive the mode coupling theory
from first principles, and quantify the randomization of the wave and the
transport and reflection of power in the waveguide. We show in particular that
scattering at the random boundary may increase or decrease the net power
transmitted through the waveguide depending on the source.
