This paper considers the joint identification of the wave speed and non-generic initial states of a one-dimensional wave equation from finite-time boundary measurements. An on-off boundary input partitions the observation horizon into zero-input and controlled phases, whose complementary information guarantees the unique identifiability of both the unknown wave speed and the initial states. By exploiting the exponential-series representation of the boundary response, the inverse problem is reformulated as the recovery of spectral data and modal coefficients from a perturbed exponential sequence. A multi-stage identification framework is then constructed by combining matrix-pencil spectral estimation, controlled-residual fitting, and regularized least-squares reconstruction. Perturbation bounds are established for the recovered discrete poles and characteristic exponents, quantifying the effect of modal truncation on the spectral estimates. Numerical experiments demonstrate accurate identification of the wave speed and reconstruction of the initial displacement and velocity. Monte Carlo simulations under multiplicative measurement noise further confirm the robustness of the proposed framework.
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