We present a non-linear navigation solution, referred to as the Reverse Bearing Only Target Motion Analysis (Reverse BO-TMA). Reverse BO-TMA is a passive method for the self-localization of an Autonomous Underwater Vehicle (AUV). Our method relies solely on bearing measurements based on the radiated noise of a passing vessel sailing along a known route. Compared to traditional range-based underwater localization methods, Reverse BO-TMA allows the AUV to remain farther from the reference vessel, and does not require collaboration or message exchange. We formalize the Reverse BO-TMA as an optimization problem, and offer both a least squares solution and an unscented Kalman filter solution. Numerical results show that Reverse BO-TMA provides accurate performance, in terms of both positioning and speed, which are close to the posterior Cramer-Rao lower bound. To demonstrate the effectiveness of our approach, we have implemented a prototype for Reverse BO-TMA and successfully tested it in three sea experiments. We show that the Reverse BO-TMA is suitable for the long-term deployment of an AUV and in cases where energy is scarce and cooperating anchors are not available.