‘Blinking’ and ‘gliding’ eigenfrequencies of oscillations of elastic bodies with blunted cuspidal sharpenings

The spectrum of a two-dimensional problem in elasticity theory is investigated for a body $\Omega^h$ with a cuspidal sharpening with a short tip of length $h>0$ that is broken off. It is known that when the tip is in place, the spectrum of the problem for $\Omega^0$ has a continuous component $[\Lambda_\dagger,+\infty)$ with positive cut-off point $\Lambda_\dagger>0$. We show that each point $\Lambda>\Lambda_\dagger$ is a ‘blinking’ eigenvalue, that is, it is an actual eigenvalue of the problem in $\Omega^h$ ‘almost periodically’ in the scale of $|\ln h|$. Among families of eigenvalues $\Lambda^h_{m(h)}$, which continuously depend on $h$, we discover ‘gliding’ eigenvalues, which fall down along the real axis at a great rate, $O((\Lambda^h_{m(h)}-\Lambda_\dagger)h^{-1}|\ln h|^{-1})$, but then land softly on the threshold $\Lambda_\dagger$. This reveals a new way of forming the continuous spectrum of the problem for a cuspidal body $\Omega^0$ from the system of discrete spectra of the problems in the $\Omega^h$, $h>0$. In addition, there may be ‘hardly movable’ eigenvalues, which remain in a small neighbourhood of a fixed point for all small $h$, in contrast to ‘gliding’ eigenvalues. Bibliography: 30 titles.

blunted cuspidal sharpening, two-dimensional elastic isotropic body, discrete and continuous spectrum, asymptotic behaviour, ‘blinking’ and ‘gliding’ eigenfrequencies