Vibrational cavity modes in a free cylindrical disk

Abstract

We analyze theoretically the vibrational properties of a free cylindrical elastic disk of thickness $2h$ and radius $a$. In particular, the acoustic cavity modes, i.e., the low-lying resonant vibrations with a large angular wave number $n$ ($=2\ensuremath{\pi}a/\ensuremath{\lambda}\ensuremath{\sim}20$, with $\ensuremath{\lambda}$ as the wavelength) along the circumference are considered. These modes are either confined at the corners between the sidewall and the top and bottom planes of the disk (the edge modes), or trapped near the circumference of the cylinder (the surface modes). They are understood as the modes modified from the whispering-gallery and Rayleigh waves on the curved surface of an infinitely long cylinder by the presence of the stress-free end planes. Small mode volumes occupied by these vibrations and their expected large quality factors in the disk with smooth boundary surfaces make the cavity modes attractive for the fundamental studies as well as for the applications to micro- and nanophononics. The mathematical formulation follows the works originally developed by Rasband [J. Acoust. Soc. Am. 57, 899 (1975)] and Hutchinson [J. Appl. Mech. 47, 901 (1980)]. Numerical examples are presented for polycrystalline aluminum disks with aspect ratios $h/a=0.5$ and 0.05.