This paper addresses the leader-follower consensus problem for a network of dynamical agents modeled by a class of boundary-controlled wave PDEs affected by boundary perturbations. An agent acts as the leader, while the remaining ones, referred to as followers, are required to asymptotically track the leader state. When linear local interaction protocols are employed, the possibly time-varying leader input signal must be available to all followers, which represents a restrictive assumption. By introducing a decentralized discontinuous local interaction protocol inspired by the Twisting second-order sliding mode control algorithm, suitably adapted to the distributed-parameter multi-agent systems setting, the leader input signal no longer needs to be communicated to the followers. In addition, uniformly bounded boundary disturbances are fully rejected by the proposed discontinuous local interaction rule. Consensus is proven to be achieved pointwise-in-space and with an exponential convergence rate through a Lyapunov-based analysis by means of which constructive tuning rules for the control parameters are derived. Simulation results are presented and discussed to corroborate the theoretical findings.