Higher order interaction of elastic waves in weakly nonlinear hollow circular cylinders. I. Analytical foundation

A mathematical framework to analyze the cumulative behavior of higher order harmonic generation due to the interaction of two collimated waves in a weakly nonlinear hollow circular cylinder is formulated in this article. A total number of (N+1)(N+2)/2-3 nonlinear boundary problems are formulated due to the Nth order mode interactions in a cylinder with Kth order nonlinearity (N≤K). The cumulative criteria for the second order harmonics (second harmonics, sum and difference harmonics) due to the quadratic interactions of two waves are examined based on the nonlinear forcing terms in curvilinear coordinates. These criteria are formulated by a synchronism condition, the circumferential orders of the primary modes, as well as the nature of the primary and the secondary wave fields, i.e., torsional or longitudinal. A generalized analysis that provides insight into the cumulative nature of the Nth order harmonics by Nth order interaction of two collimated waves is conducted by considering a cylinder with strain energy function written as Murnaghan's power series. The nature of the cumulative Nth order harmonics can be determined by the parity of the number of times the primary waves interact, and their circumferential orders.