= Ionospheric interference fringe spacing
{title2=$\Delta f\simeq c/\Delta L$}
Consider an original two-path model with a locally flat ground baseline $d$ and a symmetric equivalent return height $h$. Replace the <skywave> by two straight segments traveling at the <speed of light> $c$, and replace the <ground wave> by a straight segment of length $d$. Their lengths and positive excess distance are
$$
L_s=2\sqrt{h^2+(d/2)^2},\qquad L_g=d,\qquad \Delta L=L_s-L_g>0.
$$
This is an equivalent geometric construction, not a metallic layer overhead. It neglects the curvature of <Earth>, <refraction> within the return region, ground-induced <wave phase> shifts and <optical dispersion>.
For received <complex amplitudes> $A_g$ and $A_s$, absorb their constant relative <wave phase> into $\phi_0$ and choose their magnitudes nonnegative. An ideal detector reading proportional to squared received field has
$$
P(f)=K\left[A_g^2+A_s^2+2A_gA_s\cos\Phi(f)\right],\qquad
\Phi(f)=\frac{2\pi f\Delta L}{c}+\phi_0,\qquad K>0.
$$
Assume $\Delta L$ and $\phi_0$ are independent of <frequency> across the sweep, with slowly changing amplitudes. Consecutive maxima differ by $2\pi$ in <phase difference>, so
$$
\Phi(f+\Delta f)-\Phi(f)=\frac{2\pi\Delta f\Delta L}{c}=2\pi,
\qquad \Delta f=\frac{c}{\Delta L}.
$$
The often-used approximation $\Delta f\simeq c/\Delta L$ therefore carries a frequency-independent-path assumption. Counting $N$ complete fringe periods between $f_1$ and $f_2$ gives $\Delta L\simeq Nc/|f_2-f_1|$; equivalently, $N=\Delta L|1/\lambda_2-1/\lambda_1|$ when the vacuum <wavelength> is $\lambda=c/f$. Count whole periods, not maxima plus minima as though each were a whole period.
Choose $d=150\,\mathrm{km}$ and $h=100\,\mathrm{km}$, with $c\simeq3.00\times10^8\,\mathrm{m\,s^{-1}}$. Then $L_s=250\,\mathrm{km}$, $\Delta L=100\,\mathrm{km}$ and $\Delta f=3.00\,\mathrm{kHz}$. A $12.0\,\mathrm{kHz}$ sweep covers four full fringe periods. Conversely, the inferred equivalent height is
$$
h=\frac12\sqrt{(d+\Delta L)^2-d^2}=100\,\mathrm{km}.
$$
These chosen values illustrate the inversion; they are not a transcription of the 1924 measurements. With $A_g=1$ and $A_s=0.7$, the normalized reading ranges from $(1-0.7)^2=0.09$ to $(1+0.7)^2=2.89$: incomplete cancellation does not remove the distance information.
\Image[/ionosphere-ground-sky-interference.png]
{title=Two arrival paths and the <interference fringes> produced by a controlled <frequency> sweep}
{description=An original functional diagram distinguishes a curved skywave from a dashed equivalent straight-ray construction. The neighboring graph uses the stipulated nondispersive 150 km baseline and 100 km equivalent height, giving 3 kHz fringe spacing. The receiver detects the superposed arrivals.}
For a sweep with linearly changing <frequency> $f(t)=f_0+\dot f t$ and two constant delays $\tau_s>\tau_g$, the received instantaneous <frequency> difference has magnitude $|\dot f|(\tau_s-\tau_g)$. In this model $\tau_s-\tau_g=\Delta L/c$, giving a <beat> <frequency> $|\dot f|\Delta L/c$. Equivalently, traversing each $\Delta f$ of the sweep crosses one fringe. This calculation requires a sufficiently slow sweep that propagation, tuning and amplitude changes can be treated consistently; it is not a <beat> between two independent transmitters.
More generally, measure the total <phase difference> $\Phi(f)$, including propagation and any reflection or apparatus contribution. Define the differential <group delay>
$$
\Delta\tau_g=\frac{1}{2\pi}\frac{d\Phi}{df}.
$$
For a locally monotone <wave phase> with slowly varying <derivative>, the local spacing is $\Delta f\simeq1/|\Delta\tau_g|$. If $\Phi=2\pi fD(f)/c+\phi_r(f)$, this becomes
$$
\Delta\tau_g=\frac{D+fD'}{c}+\frac{1}{2\pi}\phi_r'(f).
$$
Here $D(f)$ is a phase-equivalent <optical path difference>, not necessarily the geometric excess distance $\Delta L$. A drifting <ionosphere>, several simultaneous <skywaves>, changing <wave polarization>, a frequency-dependent reflection <wave phase> or significant loss can invalidate the simple constant-spacing inversion.
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