Self-buckling of undulating flagella: an elastohydrodynamic mechanism for double waves in spermatozoa

The relatively long flagella of spermatozoa from insects, birds, and octopuses display double waves, characterized by two superimposed helical waves. The prevalance of these highly organized waveforms across diverse taxa and distinct flagellar architectures hints at shared underlying physics, motivating a model of the flagellum as an elastic filament immersed in a viscous fluid, actively driven by a single set of internal bending moment waves. Simulations of a clamped filament show that it can buckle under its own activity into whirling and flapping states. A multiple-scales analysis of the elastohydrodynamic equations reveals how nonlinear interactions between fast undulations generate an effective compression driving buckling, and connects wave-driven buckling to classical follower-force instabilities. Extending the model to a swimming spermatozoon, the same instability produces double waves. Parameter estimates across species show that most observed double waves lie within the regime where buckling is permitted, supporting self-buckling as a generic physical mechanism for double waves.