The W.K.B. approximation

Consider a radio wave which is vertically incident, from below, on the horizontally stratified ionosphere. Since the wave normal is initially aligned along the $z$-axis, and since $n=n(z)$, we expect all field components to be functions of $z$ only, so that

$$\frac{\partial}{\partial x}\equiv\frac{\partial}{\partial y} \equiv 0.$$ (802)

In this situation, Eqs. (4.164) reduce to $E_z=cB_z=0$, with

$$-\frac{\partial E_y}{\partial z} = {\rm i}\,k\,c B_x,$$ (803)

$$\frac{\partial\, c B_x}{\partial z} = -{\rm i}\,k\,n^2\, E_y,$$ (804)

and

$$\frac{\partial E_x}{\partial z} = {\rm i}\,k\,c B_y,$$ (805)

$$-\frac{\partial\, c B_y}{\partial z} = -{\rm i}\,k\,n^2\, E_x.$$ (806)

Note that Eqs. (4.166) and (4.167) are isomorphic and completely independent of one another. It follows that, without loss of generality, we can assume that the wave is linearly polarized with its electric vector parallel to the $y$-axis. This means that we are only going to consider the solution of Eqs. (4.166). The solution of Eqs. (4.167) is of exactly the same form, except that it describes a linear polarized wave with its electric vector parallel to the $x$-axis.

Equations (4.166) can be combined to give

$$\frac{d^2 E_y}{d z^2} + k^2 n^2\,E_y = 0.$$ (807)

Since $E_y$ is a function of $z$ only, we now use the total derivative sign $d/dz$ instead of the partial derivative sign $\partial/\partial z$. The solution of the above equation for the case of a uniform medium, where $n$ is constant, is straightforward:

$$E_y = A\,{\rm e}^{\,{\rm i}\,\phi(z)},$$ (808)

where $A$ is a constant, and

$$\phi = \pm k\,n\,z.$$ (809)

Note that the ${\rm e}^{-{\rm i}\,\omega t}$ time dependence of all wave quantities is taken as read during this investigation. The solution (4.169) represents a wave of constant amplitude $A$ and phase $\phi(z)$. According to Eq. (4.170), there are, in fact, two independent waves which can propagate through the medium in question. The upper sign corresponds to a wave which propagates vertically upwards, and the lower sign corresponds to a wave which propagates vertically downwards. Both waves propagate with the constant phase velocity $c/n$.

In general, if $n=n(z)$ the solution of Eq. (4.168) does not remotely resemble the wave-like solution (4.169). However, in the limit in which $n(z)$ is a ``slowly varying'' function of $z$ (exactly how slowly varying is something which we shall establish later), we expect to recover wave-like solutions. Let us suppose that $n(z)$ is indeed a ``slowly varying'' function, and let us try substituting the wave solution (4.169) into Eq. (4.168). We obtain

$$\left(\frac{d\phi}{dz}\right)^2 = k^2 n^2 +{\rm i}\,\frac{d^2\phi} {dz^2}.$$ (810)

This is a non-linear differential equation which, in general, is very difficult to solve. However, we note that if $n$ is a constant then $d^2\phi/dz^2=0$. It is, therefore, reasonable to suppose that if $n(z)$ is a ``slowly varying'' function then the last term on the right-hand side of the above equation can be regarded as being small. Thus, to a first approximation Eq. (4.171) yields

$$\frac{d\phi}{dz} \simeq \pm k\,n,$$ (811)

and

$$\frac{d^2 \phi}{dz^2} \simeq \pm k\,\frac{dn}{dz}.$$ (812)

It is clear from a comparison of Eqs. (4.171) and (4.173) that $n(z)$ can be regarded as a ``slowly varying'' function of $z$ as long as its variation length-scale is far longer than the wavelength of the wave. In other words, provided that $(dn/dz)/(k\,n^2)\ll 1$.

The second approximation to the solution is obtained by substituting Eq. (4.173) into the right-hand side of Eq. (4.171):

$$\frac{d\phi}{dz} \simeq \pm \left(k^2 n^2 \pm {\rm i} \,k \,\frac{dn}{dz}\right)^{1/2}.$$ (813)

This gives

$$\frac{d\phi}{dz} \simeq \pm k\,n\left(1\pm \frac{{\rm i}}{k ... ... \right)^{1/2}\simeq \pm k\,n + \frac{\rm i}{2n}\frac{dn}{dz},$$ (814)

where use has been made of the binomial expansion. The above expression can be integrated to give

$$\phi \sim \pm k \int^z \!n\,dz +{\rm i}\,\log(n^{1/2}).$$ (815)

Substitution of Eq. (4.176) into Eq. (4.169) yields the final result

$$E_y \simeq A\,n^{-1/2}\,\exp\left(\pm {\rm i}\, k \int^z \!n\,dz\right).$$ (816)

It follows from Eq. (4.166a) that

$$cB_x\simeq \mp A\, n^{1/2}\,\exp\left(\pm {\rm i}\,k \int^z\... ...frac{dn} {dz}\,\exp\left(\pm {\rm i}\,k \int^z \!n\,dz\right).$$ (817)

Note that the second term is small compared to the first, and can usually be neglected.

Let us test to what extent the expression (4.177) is a good solution of Eq. (4.168) by substituting this expression into the left-hand side of the equation. The result is

$$\frac{A}{n^{1/2}}\left\{ \frac{3}{4}\left(\frac{1}{n} \frac{... ...2}\right\} {\rm exp}\left( \pm {\rm i}\,k\int^z\!n\,dz\right).$$ (818)

This must be small compared with either term on the left-hand side of Eq. (4.168). Hence, the condition for Eq. (4.177) to be a good solution of Eq. (4.168) becomes

$$\frac{1}{k^2}\left\vert \frac{3}{4}\left(\frac{1}{n^2} \frac... ...}\right)^2 -\frac{1}{2n^3}\frac{d^2 n}{dz^2}\right\vert \ll 1.$$ (819)

The solutions

$$E_y \simeq A\,n^{-1/2}\,\exp\left(\pm {\rm i}\, k \int^z \!n\,dz\right),$$ (820)

$$cB_x \simeq \mp A\, n^{1/2}\,\exp\left(\pm {\rm i}\,k \int^z\! n\,dz\right),$$ (821)

to the non-uniform wave equations (4.166) are most commonly called the W.K.B. solutions, in honor of G. Wentzel, H.A. Kramers, and L. Brillouin, who are credited with independently discovering these solutions (in a quantum mechanical context) in 1926. Actually, H. Jeffries wrote a paper on these solutions (in a wave propagation context) in 1923. Hence, some people call these the W.K.B.J. solutions (or even the J.W.K.B. solutions). In fact, these solutions were first discussed by Liouville and Green in 1837, and again by Rayleigh in 1912. We shall refer to Eqs. (4.181) as the W.K.B. solutions, since this is what they are most commonly called. However, it should be understand that, in doing so, we are not making any statement as to the credit due to various scientists in discovering these solutions. After all, this is not a history of science course!

Recall, that when a propagating wave is normally incident on an interface, where the refractive index suddenly changes (for instance, when a light wave propagating in the air is normally incident on a glass slab), there is generally significant reflection of the wave. However, according to the W.K.B. solutions (4.181), when a propagating wave is normally incident on a medium in which the refractive index changes slowly along the direction of propagation of the wave, then the wave is not reflected at all. This is true even if the refractive index varies very substantially along the path of propagation of the wave, as long as it varies slowly. The W.K.B. solutions imply that as the wave propagates through the medium its wavelength gradually changes. In fact, the wavelength at position $z$ is approximately $\lambda(z)= 2\pi/ k\,n(z)$. Equations (4.181) also imply that the amplitude of the wave gradually changes as it propagates. In fact, the amplitude of the electric field component is inversely proportional to $n^{1/2}$, whereas the amplitude of the magnetic field component is directly proportional to $n^{1/2}$. Note, however, that the energy flux in the $z$-direction, given by the the Poynting vector $-(E_y B_x^{~\ast} +E_y^{~\ast} B_x)/(4\mu_0)$, remains constant (assuming that $n$ is predominately real).

Of course, the W.K.B. solutions (4.181) are only approximations. In reality, a wave propagating into a medium in which the refractive index is a slowly varying function of position is subject to a small amount of reflection. However, it is easily demonstrated that the ratio of the reflected amplitude to the incident amplitude is of order $(dn/dz)/(k\,n^2)$. Thus, as long as the refractive index varies on a much longer length-scale than the wavelength of the radiation, the reflected wave is negligibly small. This conclusion remains valid as long as the inequality (4.180) is satisfied. There are two main reasons why this inequality might fail to be satisfied. First of all, if there is a localized region in the dielectric medium in which the refractive index suddenly changes (i.e., if there is an interface), then (4.180) is likely to break down in this region, allowing strong reflection of the incident wave. Secondly, the inequality obviously breaks down in the vicinity of a point where $n=0$. We would, therefore, expect strong reflection of the incident wave from such a point.