Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/417

APPENDIX I.

INFLUENCE OF COMPRESSIBILITY OF AIR ON THE ENERGY EXPENDED IN FLIGHT.^[1]

The influence of compressibility as affecting the expenditure of energy in flight is best computed from the velocity of wave motion—sound.

The whole theory of Chapter VIII., based on the hypothesis of constant sweep, relates, strictly speaking, as set forth, to the incompressible fluid; it will be shown that the effects of compressibility can be dealt with as a correction, or rather by a preliminary correction, to the figures involved.

Let us write $U$ for the velocity of sound, and, as before, let $V$ be the velocity of flight. Then it is evident that any disturbance will travel forward relatively to the body in flight less rapidly than it will travel backward in the opposite direction in the relation ${\frac {U-V}{U+V}},$ as in the case of "Döppler's principle." Now, regarding the fluid motion as due to a field of force (Chapter IV., § 113), we have the communication of upward momentum diminished, and the communication of downward momentum increased, in like proportion.

Thus in the ideal case of Chapter IV., if we have to deal with a compressible fluid, an expenditure of power becomes necessary

1. The method here given is founded on a suggestion made by the author in his paper to the Birmingham Natural History and Philosophical Society in 1894. Owing to repeated rearrangements and revisions, it was accidentally omitted from the MS, of the present work.

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