Whip waves

The sound created by a whip as it cracks is produced by a mini-sonic boom created by a supersonic motion of the end of the whip. To create such a motion, one sends an impulse to the handle of the whip that travels to the end and accelerates the tip to supersonic speed. The impulse which creates a whip crack is studied as a wave travelling along an elastic rod. The whip is modeled as an inextensible, unshearable, inhomogeneous planar elastic rod. A crack is produced when a section of the whip breaks the sound barrier. We show by asymptotic analysis that a wave travelling along the whip increases its speed as the radius decreases—as the whip tapers. A numerical scheme adapted to account for the varying cross-section and realistic boundary conditions is presented, and results of several numerical experiments are reported and compared to theoretical predictions. Finally, we describe the shape of the shock waves emitted by a material point on the whip travelling faster than the speed of sound.