Non-uniqueness in cylindrical shells optimization

In cylindrical shells, the lowest natural frequency is not necessarily associated with the lowest wave index. In fact, the natural frequencies do not fall in ascending order of the wave index either. Solution of the vibration problem of cylindrical shells also indicates repeated natural frequencies. These modes are referred to as double peak frequencies. Mode shapes associated with each one of the natural frequencies are usually a combination of (i) Radial (flexural), (ii) Longitudinal (axial), and (iii) Circumferential (torsional) modes.