Pulse propagation in the ionosphere

Suppose that we possess a generator of radio waves which sends radio pulses vertically upwards into the ionosphere. For the sake of argument, we shall assume that these pulses are linearly polarized such that the electric field vector lies parallel to the $y$-axis. The pulse structure can be represented as

$$E_y(t) = \int_{-\infty}^{\infty} F(\omega)\,{\rm e}^{-{\rm i} \, \omega \,t}\,d\omega,$$ (859)

where $E_y(t)$ is the electric field produced by the generator (i.e., the field at $z=0$). Suppose that the pulse is a signal of roughly constant (angular) frequency $\omega_0$, which lasts a time $T$, where $T$ is long compared to $1/\omega_0$. It follows that $F(\omega)$ possesses narrow maxima around $\omega=\pm \omega_0$. In other words, only those frequencies which lie very close to the central frequency $\omega_0$ play a significant role in the propagation of the pulse.

Each component frequency of the pulse yields a wave which travels independently up into the ionosphere, in a manner specified by the appropriate W.K.B. solution [see Eqs. (4.181)]. Thus, if Eq. (4.209) specifies the signal at ground level ($z=0$), then the signal at height $z$ is given by

$$E_y(z,t) = \int_{-\infty}^{\infty} \frac{F(\omega)}{n^{1/2}(\omega, z)}\,\, {\rm e}^{\,{\rm i}\, \phi(\omega, z,t)}\,d\omega,$$ (860)

where

$$\phi(\omega, z,t) = \frac{\omega}{c} \int_0^{z} \!n(\omega, z)\,dz - \omega \,t.$$ (861)

Here, we have used $k=\omega/c$.

Equation (4.210) can be regarded as a contour integral in $\omega$-space. The quantity $F/n^{1/2}$ is a relatively slowly varying function of $\omega$, whereas the phase $\phi$ is a large and rapidly varying function of $\omega$. As described in Section 4.11, the rapid oscillations of $\exp(\,{\rm i}\,\phi)$ over most of the path of integration ensure that the integrand averages almost to zero. However, this cancellation argument does not apply to those points on the path of integration where the phase is stationary; i.e., those points where $\partial\phi/\partial\omega =0$. It follows that the left-hand side of Eq. (4.210) averages to a very small value, expect for those special values of $z$ and $t$ at which one of the points of stationary phase in $\omega$-space coincides with one of the peaks of $F(\omega)$. The locus of these special values of $z$ and $t$ can obviously be regarded as the equation of motion of the pulse as it propagates through the ionosphere. Thus, the equation of motion is specified by

$$\left(\frac{\partial\phi}{\partial\omega}\right)_{\omega=\omega_0} = 0,$$ (862)

which yields

$$t = \frac{1}{c} \int_0^z \left[\frac{\partial(\omega \,n)}{\partial\omega} \right]_{\omega=\omega_0}\, dz.$$ (863)

Suppose that the $z$-velocity of a pulse of central frequency $\omega_0$ at height $z$ is given by $u_z(\omega_0,z)$. The differential equation of motion of the pulse is then $dt = dz/u_z$. This can be integrated, using the boundary condition $z=0$ at $t=0$, to give the full equation of motion:

$$t =\int_0^z \frac{dz}{u_z}.$$ (864)

A comparison of Eqs. (4.213) and (4.214) yields

$$u_z(\omega_0,z) = c\left/ \left\{\frac{\partial[\omega \,n(\omega,z)]}{\partial\omega} \right\}_{\omega=\omega_0}\right..$$ (865)

The velocity $u_z$ is usually called the group velocity. It is easily demonstrated that the above expression for the group velocity is entirely consistent with that given previously [see Eq. (4.135)].

The dispersion relation (4.164) yields

$$n(\omega,z) = \left(1-\frac{\omega_p^{~2}(z)}{\omega^2}\right)^{1/2},$$ (866)

in the limit where electron collisions are negligible. The phase velocity of radio waves of frequency $\omega$ propagating vertically through the ionosphere is given by

$$v_z(\omega,z) = \frac{c}{n(\omega,z)} = c\,\left(1-\frac{\omega_p^{~2}(z)}{\omega^2}\right)^{-1/2}.$$ (867)

According to Eqs. (4.215) and (4.216), the corresponding group velocity is

$$u_z(\omega,z) = c \,\left(1-\frac{\omega_p^{~2}(z)}{\omega^2}\right)^{1/2}.$$ (868)

It follows that

$$v_z\,u_z = c^2.$$ (869)

Note that as the reflection point $z=z_0$ [defined as the solution of $\omega =\omega_p(z_0)$] is approached from below, the phase velocity tends to infinity, whereas the group velocity tends to zero.

Let $\tau$ be the time taken for the pulse to travel from the ground to the reflection level, and back to the ground again. The product $c\,\tau/2$ is termed the equivalent height of reflection, and is denoted $h(\omega)$, since it is a function of the pulse frequency, $\omega$. The equivalent height is the height to which the pulse would have to go if it always traveled with the velocity $c$. Since we know that a pulse of dominant frequency $\omega$ propagates at height $z$ with the $z$-velocity $u_z(\omega, z)$ (this is true for both upgoing and downgoing pulses), and also that the pulse is reflected at the height $z_0(\omega)$, where $\omega =\omega_p(z_0)$, it follows that

$$\tau = 2\int_0^{z_0(\omega)} \frac{dz}{u_z(\omega,z)}.$$ (870)

Hence,

$$h(\omega) = \int_0^{z_0(\omega)}\frac{c}{u_z(\omega,z)} \,dz.$$ (871)

Note that the equivalent height of reflection, $h(\omega)$, is always greater than the actual height of reflection, $z_0(\omega)$, since the group velocity $u_z$ is always less than the velocity of light. The above equation can be combined with Eq. (4.218) to give

$$h(\omega) = \int_0^{z_0(\omega)} \!\left(1-\frac{\omega_p^{~2}(z)} {\omega^2}\right)^{-1/2} dz.$$ (872)

Note that the integrand diverges as the reflection point is approached, but the integral remains finite.