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In biology, Denny's paradox refers to the apparent impossibility of surface-dwelling animals such as the water strider generating enough propulsive force to move. It is named after biologist
Mark Denny Mark W. Denny (born 1951) is a professor of biology at Stanford University. His research on the intertidal zone of wave-swept shores has led to increased understanding of this habitat. His most publicized research is his work on locomotion of wate ...
, and relates to
animal locomotion on the surface layer of water Animal locomotion on the surface layer of water is the study of animal locomotion in the case of small animals that live on the surface layer of water, relying on surface tension to stay afloat. There are two types of animal locomotion on water, ...
. If capillary waves are assumed to generate the momentum transfer to the water, the animal's legs must move faster than the
phase speed The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, ...
c_m of the waves, given by :c_m=\left(\frac\right)^, where g is the acceleration due to gravity, \sigma is the strength of surface tension, and \rho the
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
of
water Water (chemical formula ) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living organisms (in which it acts as ...
. For standard conditions, this works out to be about 0.23 m/s. In fact, water striders' legs move at speeds much less than this and, according to this physical picture, cannot move. Writing in the
Journal of Fluid Mechanics The ''Journal of Fluid Mechanics'' is a peer-reviewed scientific journal in the field of fluid mechanics. It publishes original work on theoretical, computational, and experimental aspects of the subject. The journal is published by Cambridge Uni ...
, David Hu and John Bush state that Denny's paradox "rested on two flawed assumptions. First, water striders' motion was assumed to rely on the generation of capillary waves, since the propulsive force was thought to be that associated with
wave drag In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
on the driving leg. Second, in order to generate capillary waves, it was assumed that the strider leg speed must exceed the minimum wave speed, c_m=\left(4g\sigma/\rho\right)^\simeq 0.23\mathrm. We note that this second assumption is strictly true only for steady motions".


References

{{DEFAULTSORT:Denny's Paradox Physical paradoxes Animal locomotion