Wave dynamics reflect a broad spectrum of natural phenomena and are often characterized by wave equation such as in the development of meta-devices used to steer wave propagation. Modeling synchronization of wave dynamics is critical in various applications such as in communications and neuroscience. In this paper, we study the synchronization problem for oscillations governed by wave equation with nonlinear (van der Pol type) boundary conditions through a single boundary coupling. The dynamics of the master system is self-excited and presents sensitive and rapid oscillations. With the only signal received at one end of the boundary, by constructing a mathematical model, we show the existence of a slave system that can be synchronized with the master system via the study of wave reflections on the boundary to recover the actual wave dynamics. The coupling gain, which represents the strength of the connection between the master system and the slave system, has been identified. The obtained result can be also viewed as an observer construction when the measurable output is only on the boundary. Numerical simulations are provided to demonstrate the effectiveness of the theoretical outcomes.