Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/422

App. II. them, instead of being, according to hypothesis, composed of the fluid itself and therefore of the same density. It is not possible in the present work to go fully into the cause of this discrepancy which the author believes to be due to the mathematical theory regarding the vortex ring, as the result of an impulse distributed evenly over the disc area, instead of as the resultant of two equal and opposite impulses, the one applied over the disc area and the other to the confines of the fluid region. This is merely thrown out as a suggestion, but whatever the explanation may be, the case of a vortex ring travelling to and fro in a rigidly bounded region filled with incompressible fluid and carrying momentum is presented for consideration to the exponents of the Vortex Atom Theory as involving a flagrant violation of the third law of motion.

Example 2.—Momentum of Sound Waves.—This is a question that has been widely discussed of recent years, and one on which different authorities are not altogether in agreement.

If we take it as essential by definition that the passage of a complete wave or train of waves results in no permanent displacement of the particles of the fluid, that is to say, that each particle of the fluid occupies after the passage of the wave train the same position as before its passage, it immediately follows that the mean density of the wave train is equal to that of the undisturbed fluid.

It is therefore evident (as in § 5) that if such a wave train be supposed to travel to and fro in a box (Fig. 160), from end to end, being repeatedly reflected, no movement of the mass centre of