Sonar signal propagation has been traditionally modeled as a linear stochastic process. This research is a major departure from that classical viewpoint and is aimed at modeling acoustic signal propagation dynamics deterministically using the analysis tools of Nonlinear Dynamics and Chaos Theory. Advantages of using this new paradigm over traditional approaches to sonar modeling are demonstrated by simulating a linear frequency modulated (LFM) active sonar signal and two nonlinear variations of it. When nonlinear performance metrics called the mutual information (MI) and the differential radius (DR) are applied to both linear and nonlinear variants of the LFM signal, both metrics are found to be useful detectors of nonlinearity in a simulated acoustic time series. Next we demonstrate the DR technique, on real ultrasonic time series data, that was found to be useful for the diagnosis and treatment of a heart disease called atrial fibrillation. After summarizing results, we conclude with recommended future directions in applied nonlinear sonar signal propagation research.