Asymptotics of the resonances for a continuously stratified layer

TL;DR

This paper investigates how the distribution of acoustic resonances in a stratified elastic layer depends on boundary coefficient smoothness, with implications for ultrasound diagnostics of cartilage degeneration.

Contribution

It provides a detailed analysis of the asymptotic distribution of resonances based on coefficient smoothness, linking mathematical results to ultrasound cartilage testing.

Findings

Resonances are asymptotically aligned along a line if coefficients have boundary jumps.

Resonances follow a logarithmic curve if coefficients are continuous.

Resonance spacing is sensitive to cartilage degeneration.

Abstract

Ultrasound wave propagation in a nonhomogeneous linearly elastic layer of constant thickness is considered. The resonances for the corresponding acoustic propagator are studied. It is shown that the distribution of the resonances depends on the smoothness of the coefficients. Namely, if the coefficients have jump discontinuities at the boundaries, then the resonances are asymptotically distributed along a straight line parallel to the real axis on the unphysical sheet of the complex frequency plane. In the contrary, if the coefficients are continuous, then it is shown that the resonances are asymptotically distributed along a logarithmic curve. The spacing between two successive resonances turns out to be sensitive to articular cartilage degeneration. The application of the obtained results to ultrasound testing of articular cartilage is discussed.