Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/131

Rh § 84. Energy of Superposed Systems.—The superposition of systems of flow containing energy may in certain cases result in the addition of their separate energies of motion, but it is evident that this is the exception rather than the rule. The energy of two combined systems is given by the number of $$\psi$$, $$\phi$$ elements in the combined field.

In the special case, for example, of the superposition of two motions of translation at right angles, as along the axes respectively of $$x$$ and $$y$$, it is found that the energy of the combined field is the sum of the separate energies, a fact which is otherwise obvious (Eucl. 47, L). In general, it can be shown that, if on a general motion of translation he superposed any system of flow whose mean velocity in the direction of the translation is zero, the energy of the resultant is the sum of the energies of the component fields.

Let us suppose the translation to take place along the axis of $$x$$, and let the velocity of translation be $$U$$; let the $$x$$ component of the velocity of the superposed system be $$u$$ a variable in respect of $$x$$, $$y$$ and $$z$$. Then the mass of each small element of the fluid is $$\rho\ \delta x\ \delta y\ \delta z$$, and the energy of the combined field is $$\frac{1}{2}\ \Sigma\ \rho\ \delta x\ \delta y\ \delta z\ (U + u)^2$$, but $$\Sigma\ \rho\ \delta x\ \delta y\ \delta z\ u$$ is zero; we therefore have energy $$= \frac{1}{2}\ m\ U^2 + \Sigma\ \rho\ \delta x\ \delta y\ \delta z\ u^2$$, where $$m$$ is the total mass; which proves the proposition in respect of motion along the axis of $$x$$. But the energy of any components of the superposed motion in the direction of the axes of $$y$$ and $$z$$, which may be regarded as translations at right angles to the main motion, we have already seen also comply. Therefore the total energy is the sum of the components.

§ 85. Example: Cyclic Superposition.—An example may be given in two-dimensional motion in the case of the cyclic superposition (Fig. 48). We know that the energy contained in a case of cyclic motion around a cylinder or cylindrical filament in space is infinite, for the linear size of the $$\psi$$, $$\phi$$ squares forms a geometrical progression, and any finite number of such squares, however