Similarity laws for supersonic flows

The non-linear differential equation for the velocity potential of three-dimensional steady irrotational supersonic flow past wings of finite span has been investigated. It is found that the whole Mach number range from 1 to $\infty$ may be divided into two regions (not strictly divided), in each of which similarity laws are obtained, with two parameters ${K_1} = {\left ( {{M^2} - 1} \right )^{1/2}}/{\tau ^n}$ and ${K_2} = A{\left ( {{M^2} - 1} \right )^{1/2}}$; $\tau$ is the non-dimensional thickness ratio, $A$ the aspect ratio of the wing, $M$ the Mach number of the uniform stream in which the wing is placed. The factor $n$ is given explicitly as a function of $M$ and $\tau$; in the lower region of Mach numbers it tends to $1/3$ as $M \to 1$, for all $\tau$, giving the ordinary transonic rule, and in the upper region it tends to $- 1$ as $M \to \infty$, for all $\tau$, as in the ordinary hypersonic rule.