In this work, we implement two fourth-order compact finite differences (CFD) methods and use them to model wave propagation on a elastic string. The formulation of the first method employs the standard implicit CFD constructed on nodal grids and requires solving tridiagonal linear systems at each time iteration. A new second scheme is implemented using the staggered mimetic CFD operators that explicitly approximate derivatives at mid-cell points. The name mimetic stands for FD operators that satisfy in a discrete sense, some of conservative properties fulfilled by the continuous divergence and gradient operators. Both CFD methods are combined to high-order Runge-Kutta schemes for time integration. Numerical results show that the mimetic CFD scheme is slightly more accurate but yields similar fourth-order convergence than the nodal CFD method. In this paper, we also compare the CPU cost of both compact schemes to an explicit Leap-frog staggered solver to show the high computational efficiency of the mimetic CFD scheme.