We consider the problem of seeking the source of a scalar signal using an autonomous vehicle modeled as the nonholonomic unicycle. The vehicle does not have the capability of sensing its position or the position of the source but is capable of sensing the scalar signal originating from the source. The signal field is assumed to decay away from the position of the source but the vehicle does not have the knowledge of the functional form of the field. We employ extremum seeking to steer the vehicle to the source. Our control strategy keeps the forward velocity constant and tunes the angular velocity, a setting suitable for most autonomous vehicles, including aerial ones. Because of the constant forward velocity constraint, after it has converged near the source, the vehicle exhibits extremely interesting and complex motions. Using averaging theory, we prove local exponential convergence to an "orbit-like" attractor around the source. We also present a thorough analysis of non-local behaviors and attractors that the vehicle can exhibit near the source. The richness and complexity of behaviors makes only some of them amenable to analysis, whereas others are illustrated through a carefully laid out simulation study.