We study acoustic waveguides with varying cross section and slowly bending axis. In particular, we consider waveguides with rough walls and cross sectional width that varies slowly. Roughness means fast and small fluctuations that occur on the scale of the wavelength. The roughness in the walls is unknown in applications and so we model it as a random process to study the propagation of uncertainty in the walls to uncertainty in the wavefield. The slow variations occur on a scale much larger than the wavelength and cause jumps in the number of propagating modes supported by the guide. Here we present a mathematical analysis from first principles of waves in waveguides with an arbitrary but finite number of turning points and use our analysis to quantify randomization of the wavefield and transport of power in the guide. Speaker(s): Derek Wood (University of Michigan)