Asymptotics of the resonances for a continuously stratified layer

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.