Modal expansion analysis of nonlinear circumferential guided wave propagation in a circular tube

Within the second-order perturbation approximation, the nonlinear effect of primary circumferential guided wave propagation in a circular tube is investigated using modal expansion analysis for waveguide excitation. The nonlinearity of the wave equation governing the wave propagation ensures the second-harmonic generation accompanying primary circumferential guided wave propagation. This nonlinearity may be treated as a second-order perturbation of the linear elastic response. The fields of the second harmonics of primary circumferential guided wave propagation are considered as superpositions of the fields of a series of double frequency circumferential guided wave (DFCGW) modes. Based on the momentum theorem and mathematical formulae of nonlinear stress tensor and its divergence under the cylindrical coordinate system, the mathematical expressions of the corresponding double frequency traction stress tensors and bulk driving forces are deduced for a certain primary circumferential guided wave mode. Subsequently, the equation governing the DFCGW mode expansion coefficient is established. Finally, the mathematical expression of second-harmonic field of the primary circumferential guided wave mode in a tube is derived. The results of the theoretical analyses and numerical calculations indicate that the degree of cumulative growth of the DFCGW mode with circumferential angle is obviously influenced by that of phase velocity matching between the primary and double frequency wave modes. It is found that the amplitude of the DFCGW mode can grow with circumferential angle when its phase velocity matches with that of the primary circumferential guided wave, and that the amplitude of the DFCGW mode will show a beat effect with circumferential angle when its phase velocity is not equal to that of the primary wave mode. The DFCGW mode, whose phase velocity matches with that of the primary wave mode, plays a dominant role in the field of second harmonic generated by the primary wave mode propagation, and the contribution of the other DFCGW modes to the said second-harmonic field is negligible after the primary wave mode has propagated some circumferential angle.