"Wandering" natural frequencies of an elastic cuspidal plate with the clamped peak


Cuspidal irregularities of solids have been recognized as Vibrating Black Holes for elastic and acoustic waves. The corresponding absorption phenomenon is caused, in particular, by the appearance of the continuous spectrum [k, +∞) of the Lame system in a two-dimensional plate with the sharp cusp that provokes for wave processes in a finite volume. However, if the plate is clamped in the small h-neighborhood of the cusp top, the spectrum becomes discrete and consists of isolated natural frequencies kjh of finite multiplicity. The asymptotics of kjh as h → +0 is constructed that describes the effect of the "wandering" of the natural frequencies above the threshold k > 0, namely the asymptotic formula
kjh = Kj (ln h) + O (hδ)   with δ > 0 is valid where Kj is a periodic function. In other words, some of frequencies flounce in the semi-axis (k, +∞) at a quite high rate O (h-1). At the same time, natural frequencies below the threshold get the sustainable behaviour kph = kp0 + O (hδ), δ > 0, as h → +0.