We study acoustic waveguides with varying cross sections and slowly bending axes. 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.
We use our analysis to quantify randomization of the wavefield and transport of power in the guide. This is accomplished by obtaining a statistical description of coupled complex waveguide mode amplitudes in terms of the statistics of the fluctuations in the walls. Randomization is captured by decay of the means of the mode amplitudes with distance from the source. Transport of power is studied through differential equations for the second moments of the mode amplitudes. We show using these equations that the random fluctuations in the wall may increase or decrease net transmitted power depending upon the source excitation.PHDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/138640/1/derekmw_1.pd
