The Jeffries connection formula

In the preceding section there is a tacit assumption that the square of the refractive index, $q(x)\equiv n^2(x)$, is a real function. As is apparent from Eq. (4.162), this is only the case in the ionosphere as long as electron collisions are negligible. Let us generalize our analysis to take electron collisions into account. In fact, the main effect of electron collisions is to move the zero of $q(\hat{z})$ a short distance off the real axis (the distance is relatively short provided that we adopt the physical ordering $\nu\ll \omega$). The arrangement of Stokes and anti-Stokes lines around the new zero point, located at $\hat{z}=\hat{z}_0$, is sketched in Fig. 20. Note that electron collisions only significantly modify the form of $q(\hat{z})$ in the immediate vicinity of the zero point. Thus, sufficiently far away from $\hat{z}=\hat{z}_0$ in the complex $\hat{z}$-plane, the W.K.B. solutions and the locations of the Stokes and anti-Stokes lines are exactly the same as in the preceding section.

Figure: The arrangement of Stokes lines (dashed) and anti-Stokes lines (solid) in the complex $\hat{z}$ plane. Also shown is the branch cut (wavy line).

The W.K.B. solutions (4.284) and (4.285) are valid all the way along the real axis, except for a small region close to the origin where electron collisions significantly modify the form of $q(\hat{z})$. Thus, we can still adopt the physically reasonable decaying solution (4.286) on the positive real axis. Let us trace this solution in the complex $\hat{z}$-plane until we reach the negative real axis. We can achieve this by moving in a semi-circle in the upper half-plane. Since we never move out of the region in which the W.K.B. solutions (4.284) and (4.285) are valid, we conclude, by analogy with the preceding section, that the solution on the negative real axis is given by Eq. (4.292). Of course, in all of the W.K.B. solutions the point $\hat{z}=0$ must be replaced by the new zero point $\hat{z}=\hat{z}_0$. The new formula for the reflection coefficient, which is just a straightforward generalization of Eq. (4.296), is

$$R = -{\rm i}\, \exp\left(2\,{\rm i}\, h\int_a^{\hat{z}_0} q^{1/2} \,d\hat{z}\right).$$ (953)

This is called the Jeffries connection formula, after H. Jeffries, who discovered it in 1923. The general expression for the reflection coefficient is incredibly simple. We just integrate the W.K.B. solution in the complex $\hat{z}$-plane from the phase reference level $\hat{z}=a$ to the zero point, square the result, and multiply by $-{\rm i}$. Note that the path of integration between $\hat{z}=a$ and $\hat{z}=\hat{z}_0$ does not matter, because of Cauchy's theorem. Note, also, that since $q^{1/2}$ is, in general, complex along the path of integration, we no longer have $\vert R\vert=1$. In fact, it is easily demonstrated that $\vert R\vert\leq 1$. Thus, when electron collisions are included in the analysis we no longer obtain perfect reflection of radio waves from the ionosphere. Instead, some (small) fraction of the radio energy is absorbed at each reflection event. This energy is ultimately transfered to the particles in the ionosphere with which the electrons collide.