Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/429

Rh we may regard the variation from proportionality as an external force represented by $$X.$$ Thus in a wave of very considerable displacement and pressure excess, $$X / (P_M - P_N)$$ can be shown to be positive, and $$U$$ is greater than the value in (5). This agrees with certain experimental results given below.

The suggestion here appears to be that the straight line trace in the $$PV$$ diagram (which is the equivalent of the Poynting and Thomson hypothesis) is essential to the rigid application of theory for waves of sensible magnitude. This is contrary to the result here obtained, and surely must be incorrect. A gas obeying Boyle's law according to these authorities would share with the real gas the mutability of wave form consequent on the adiabatic law.

According to the present author the straight line diagram is to be found in the plotting of $$P$$ and $$\rho$$ for Boyle's law, Fig. 161 a, which corresponds to the hyperbola for the $$PV$$ diagram; and this straight line diagram is the looked-for analogue of the isochronous pendulum.

If we plot the analogous form of the adiabatic law. $$P / \rho^\gamma =$$ constant, Fig. 161 b, we no longer have a straight line diagram, but for small amplitude we may approximate by drawing a tangent $$EF$$ cutting the axis of $$y$$ at $$F .$$ We may regard the point $$F$$ as a new origin which will give the pressures proper to the limited portion of the curve approximated on the Boyle's law basis. From geometrical considerations we have $$AO$$ to $$AF$$ in the relation 1 is to $$\gamma$$, the relation of the real to the