Ultrasound wave propagation in a nonhomogeneous linearly elastic layer of constant thickness immersed between homogeneous fluid and solid media is considered. The resonances (scattering poles) for the corresponding acoustic propagator are studied. It is shown that the distribution of the resonances depends on the smoothness of the coefficients that characterize physical properties of the layer and the ambient media. 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. On the contrary, if the coefficients are continuous, then it is shown that the resonances are asymptotically distributed along a logarithmic curve. The developed mathematical model is applied to the ultrasonic testing of the articular cartilage (AC) layer attached to the subchondral bone from one side and being in contact with a solution on the other side. It is conjectured that the spacing between two successive resonances may be sensitive to AC degeneration. The application of the obtained results to the development of ultrasonic testing for quantitative evaluation of AC is discussed.