  In this work, we study the localization of flexural waves in highly symmetric
clusters of scatterers. It is shown that when the scatterers are placed
regularly in the perimeter of a circumference the quality factor of the
resonances strongly increases with the number of scatterers in the cluster. It
is also found that in the continuous limit, that is to say, when the number of
scatterers tends to infinite, the quality factor is infinite so that the modes
belong to the class of the so called bound states in the continuum or BICs, and
an analytical expression for the resonant frequency is found. These modes have
different multipolar symmetries, and we show that for high multipolar orders
the modes tend to localize at the border of the circumference, forming
therefore a whishpering gallery mode with an extraordinarily high quality
factor. Numerical experiments are performed to check the robustness of these
modes under different types of disorder and also to study their excitation from
the far field. Although we have focused our study to flexural waves, the
methodology presented in this work can be applied to other classical waves,
like electromagnetic or acoustic waves, being therefore a promissing approach
for the design of high quality resonators based on finite clusters of
scatterers.

Classical Physics;Applied Physics;Optics