  Using Leaver's continue fraction and time domain method, we study the wave
dynamics of phantom scalar perturbation in a Schwarzschild black string
spacetime. We find that the quasinormal modes contain the imprint from the
wavenumber $k$ of the fifth dimension. The late-time behaviors are dominated by
the difference between the wavenumber $k$ and the mass $\mu$ of the phantom
scalar perturbation. For $k<\mu$, the phantom scalar perturbation in the
late-time evolution grows with an exponential rate as in the four-dimensional
Schwarzschild black hole spacetime. While, for $k=\mu$, the late-time behavior
has the same form as that of the massless scalar field perturbation in the
background of a black hole. Furthermore, for $k>\mu$, the late-time evolution
of phantom scalar perturbation is dominated by a decaying tail with an
oscillation which is consistent with that of the usual massive scalar field.
Thus, the Schwarzschild black string is unstable only against the phantom
scalar perturbations which satisfy the wavelength $\lambda>2\pi/\mu$. These
information can help us know more about the wave dynamics of phantom scalar
perturbation and the properties of black string.
