Flexure waves in elastic rods

The order-1 theory of the dynamics of homogeneous elastic rods is treated with the emphasis on twist-free bending motions of inextensible rods. Various forms of the governing equations for such planar motions are presented, and their traveling wave solutions are shown to result in curves of the same form as those in Euler's theory of equilibrium configurations. Traveling waves are called subsonic or supersonic in accord with whether their speed is less than or greater than (E/p)1/2 with E the tensile modulus and p the density. The solitary traveling waves are loops and are given by elementary functions. At each prescribed level of tension below a critical value, for both subsonic and supersonic waves, the larger the loop the faster the wave. The periodic traveling waves, both with and without inflexion points, are given by elliptic functions and integrals. Small amplitude sinusoidal waves are a limiting case of inflexional waves. The solitary waves are obtainable as appropriate limits of both inflexional and noninflexional waves. Although, in general, traveling waves are motions of rods of infinite length, there are traveling waves that are possible planar modes of motion for rods of finite length with joined ends. A rod with such a traveling wave forms a figure eight in the inflexional case and a circular ring in the noninflexional case.