Quasinormal ringing of thick braneworlds with a finite extra dimension

In this work, we investigate the quasinormal modes of the Poincar\'e thick brane with a finite extra dimension. Unlike the case with an infinite extra dimension, the gravitational effective potential exhibits three distinct shapes within different ranges of the parameter $n$ in the warp factor: harmonic oscillator potential, P"oschl-Teller potential, and volcanolike potential. We then study various types of perturbations in this system. Utilizing a combination of analytical, semianalytical, and numerical methods, we obtain the quasinormal modes of the perturbed fields. Our findings reveal a set of discrete quasinormal modes for the thick brane, similar to those of black holes. Interestingly, when $n=1$, the quasinormal modes exhibit purely imaginary behavior. This study may provide a new way to detect the existence of extra dimensions.