We consider an approximation of second-order hyperbolic interface problems by partially penalized immersed finite element methods. In order to penalize the discontinuity of IFE functions, we add some stabilization terms at interface edges. Some semi-discrete and fully discrete schemes are presented and analyzed. We prove that the approximate solutions have optimal convergence rate in an energy norm. Numerical results not only validate the theoretical error estimates, but also indicate that our methods have smaller point-wise error over interface elements than classical IFE methods.