In a fluid of given kinematic viscosity, the size of the rollers will vary inversely as the velocity, that is the velocity difference between the live stream and the dead water.
In a given fluid the frequency with which the vortices are generated will vary as the square of the velocity. It is probable that we have in this the origin of the “pitch note” that may be heard when a body is in rapid motion through the air, for example in the swish of a stick or the whistle of a projectile.
The foregoing conclusions are only strictly applicable so long as the vortex rollers are of small diameter compared to the body by which they are generated, for otherwise the motion will not be homomorphous, as required by hypothesis. It is probable that it is the relation between the size of the vortex rollers and that of the body that determines the point at which the discontinuous form of flow begins. Thus for velocities less than a certain minimum in any given fluid the value of will be so great that there is no room for the vortex to form; at a higher velocity it seems likely that a single vortex may be generated, which will follow in the wake of the body, as in § 104, and it will only be at velocities in excess of this that the vortices will detach themselves in accordance with the régime contemplated. The precise conditions must, however, be regarded as uncertain.
The ultimate fate of the vortices formed in the peripheral region of the wake is not altogether known; it would appear that they will break up into groups and sub-groups, after the manner described in § 93, till the whole wake of ““dead water” becomes a region of seething turbulence, the motion gradually becoming incoherent and dying out as the energy is absorbed in viscous strain.
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