  Many swimming microorganisms, such as bacteria and sperm, use flexible
flagella to move through viscoelastic media in their natural environments. In
this paper we address the effects a viscoelastic fluid has on the motion and
beating patterns of elastic filaments. We treat both a passive filament which
is actuated at one end, and an active filament with bending forces arising from
internal motors distributed along its length. We describe how viscoelasticity
modifies the hydrodynamic forces exerted on the filaments, and how these
modified forces affect the beating patterns. We show how high viscosity of
purely viscous or viscoelastic solutions can lead to the experimentally
observed beating patterns of sperm flagella, in which motion is concentrated at
the distal end of the flagella.
