Some of the processes relevant to the propulsion of small organisms are investigated using the simple mathematical model of two-dimensional waves passing through a sheet immersed in a viscous fluid. This model was first used by Taylor, who considered an inextensible sheet moving in an unbounded fluid of negligible inertia. Here the effects of fluid inertia, of straining of the wave-bearing surface, and of nearby walls are included in the study. The applicability of the results is restricted both by the unrealistic geometry of the model and by the method of analysis which gives results valid for small Reynolds numbers and for small wave amplitudes only. However, the following general results may have counterparts in nature.

The effect of fluid inertia is to increase the propulsive speed for a particular wave amplitude. Straining of the waving surface will probably reduce the propulsive velocity for a given amplitude, although there exist modes of surface straining that give augmented propulsion. If the wave celerity and the energy output in swimming remain constant in the presence of a solid wall, the amplitude of the wave is reduced as the wall is approached while the propulsive speed first rises slightly and then drops. It appears further that an organism swimming near a wall may induce a shear pattern which directs it away from the wall.