An alternative to using standard perturbation theory to compute the mode attenuation coefficients is to use a reflection coefficient argument. For an iso velocity waveguide, assume the magnitude of the bottom reflection coefficient to be close to unity, i.e., approximately 
a. Derive an expression for the cycle distance associated with a mode. Using this cycle distance, express the change in the acoustic field as a function of the acoustic field itself, the cycle distance and the loss per bounce. This simple differential equation gives the modal attenuation coefficient.
b. What happens for the non-isovelocity case? Compute a skip distance by taking advantage of the fact that the horizontal wavenumber of a mode is constant whereas the vertical wavenumber varies with depth.
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