Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/418

in accordance with a regime

$\epsilon _{1}={\frac {U-V}{U+V}}.$

(1)

(Compare § 172 et seq.)

In the extreme case when $V$ becomes equal to Uno disturbance can precede the aerofoil in its flight, and the whole reaction will be due to the communication of downward momentum; the cyclic component in the peripteral system vanishes. In the above expression when $V=U,\epsilon _{1}=$ zero, which leads to the same conclusion.

Let us take the $\epsilon$ of Chapter VIII. to be the e$\epsilon$ proper to an incompressible fluid, and let the symbol employed above, $\epsilon _{1}$, be the corresponding value when $U$ is the velocity of sound. Then from the foregoing reasoning we have—

$\epsilon _{1}={\frac {U-V}{U+V}}\ \epsilon$

(2)

This expression is in harmony with equation (1), which relates to the special case where $\epsilon =$ unity.

Example.—Dealing with the highest result tabulated, i.e., 80 ft. sec., and taking $U=$ 1120,

$\epsilon _{1}={\frac {1120-80}{1120+80}}\ \epsilon ={\frac {13}{15}}\ \epsilon ;$

that is to say, for the velocity stated the value of $\epsilon$ employed in Chapter VIII. is too high in the relation 15/13.

But it is evident from the whole argument of Chapter VIII. that the constant $\epsilon$ is not the only one of the constants involved in the equations affected by the compressibility of air. In fact, from the reasoning employed (§§ 161, 172 et seq.) it would appear that the constants $c$ and $\kappa$ will also be affected, and we may fairly make the assumption^[1] that the constants will remain related in accordance with the equation of § 177, and that consequently we may regard the influence of compressibility as degrading the effective value of ${\mbox{n}}$ from its actual value to a less value in the proportion required by equation (2).

Thus in the case under discussion, if we have to deal with an

1. The method is evidently no more than an approximation.

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