LAUSR

This paper investigates the quasinormal mode (QNM) vibrations of a rotating cylindrical black hole (or black string) spacetime that is surrounded by a thin shell rotating synchronously with the black string's axis. The existence of the thin shell leads to a piecewise metric of the black hole spacetime beyond the horizon, which is divided into two stationary spacetime parts by the radius of the thin shell. As a result, the potential function $V(r)$ of the QNM equation is also discontinuous. To solve the QNM equation with the discontinuous potential function, we propose two methods, matrix method and generalized Horowitz-Hubeny Method. We find that the influence of the thin shell can reduce the QNM frequency of the black string while alleviating their amplitude decay rate. Our suggested method can be easily applied to other QNM calculations of black hole spacetime with discontinuous potential function, thus facilitating investigations into more intricate and realistic black hole spacetimes, such as those with accretion disks. Additionally, the finite difference method is employed to investigate the spacetime too. This analysis discloses a substantial gap in the potential function when the thin shell's mass and charge achieve sufficiently high values, resulting in the outer spacetime nearing gravitational collapse and extreme black hole scenarios. Within this gap, the QNM wave displays oscillations, producing an echo effect. Moreover, it is established that the closeness of the spacetime to the collapse threshold and charge extremality have positive correlation with the beat interval of this echo.