# ITU-R P.368 GRWAVE: BotB Methodology and Parameter Justification Companion methodology note for the [[Knickebein_Propagation_Null|Knickebein null hypothesis]] and [[X_Y_Gerat_Propagation_Null|X-Y-Gerät null hypothesis]]. Documents exactly which parameters were fed into ITU's own GRWAVE Fortran ground-wave calculator, where each parameter came from, and why it was chosen. Every knob is accounted for so the calculation is reproducible. This document exists because [[DC_Dan|DC Dan Dano's response]] invoked ground-wave propagation ("vertical polarisation helps it hug the curve of the earth", "grazing ray", "Sommerfeld integral") as the mechanism that should rescue the Knickebein beams on a globe. The way to test that claim rigorously is to run ITU's own ground-wave calculator for the actual Knickebein geometry and see what it says. This note records how that was done. --- ## The International Telecommunication Union and its propagation standards The International Telecommunication Union (ITU) is the United Nations specialised agency for information and communication technologies. Its Radiocommunication Sector (ITU-R) issues the authoritative international recommendations that govern radio-wave propagation engineering, spectrum allocation, and compatibility assessment worldwide. National regulators (FCC in the US, Ofcom in the UK, the BNetzA in Germany, and similar) reference ITU-R recommendations directly when setting broadcast, military, and maritime spectrum policy. Military and civilian radio equipment manufacturers calibrate their link budgets against ITU-R curves because ITU-R is the only body that maintains a globally agreed, experimentally updated set of propagation models. The two ITU-R recommendations load-bearing for the Knickebein analysis are: - **Recommendation ITU-R P.526** *Propagation by diffraction*. Originally adopted in 1978 and most recently revised in 2025 as **P.526-16** (approval history: *1978-1982-1992-1994-1995-1997-1999-2001-2003-2005-2007-2009-2012-2013-2018-2019-2025*, per the title page of [[ITU-R_P526-16_2025|P.526-16]]). Its scope, quoted verbatim from page 1: *"This Recommendation presents several models to enable the reader to evaluate the effect of diffraction on the received field strength. The models are applicable to different obstacle types and to various path geometries."* - **Recommendation ITU-R P.368** *Ground-wave propagation prediction method for frequencies between 10 kHz and 30 MHz*. First adopted in 1951 and most recently revised in 2022 as **P.368-10** (approval history: *1951-1959-1963-1970-1974-1978-1982-1986-1990-1992-2005-2007-2022*, per the title page of [[ITU-R_P368-10_2022|P.368-10]]). Its scope, quoted verbatim from page 1: *"This Recommendation provides information on the field strength and its dependence on ground characteristics due to ground-wave propagation at frequencies less than 30 MHz."* Both recommendations have been revised repeatedly over more than 70 years. That revision history is itself a form of experimental validation: each revision incorporates new field measurements and refines the underlying models against measured field strengths. ITU-R does not adopt a propagation model without it having been matched to measured data. The full theoretical chain from Sommerfeld through the modern ITU-R method is laid out in the **ITU Radiocommunication Bureau Handbook on Ground Wave Propagation (Edition of 2014)**, which is the authoritative ITU reference that connects P.368 and P.526 to their common physical roots. I cite Part 1 of that Handbook directly below rather than rely on secondary sources. ### The theoretical lineage: Sommerfeld to Rotheram The Handbook's Part 1 Section 2 *"The development of surface wave theory"* (page 1) gives the complete lineage in one paragraph. Quoted verbatim: ![[ITU_GWP_Handbook_Part1_Sec2_DevSurfWaveTheory-p1.png]] > "In 1909 Sommerfeld [1] obtained a solution for a vertical electric dipole on the plane interface between an insulator and a conductor. Sommerfeld's work was not in a practical form for application by engineers and there was also an error which led to some confusion. In 1936 Norton [2] largely overcame these problems, and a further paper in 1937 [3] provided a method for calculations over a flat earth. Van der Pol and Bremmer [4] in a set of papers in 1937 to 1939 made it possible to calculate field strengths at distant points on the spherical Earth's surface, using a residue series. A further paper by Norton [5] in 1941 turned this into a more practical proposition for the engineer." > > ITU Handbook on Ground Wave Propagation (2014), Part 1, §2, p. 1. > [!note] What this says in plain language > Sommerfeld 1909 is the original theoretical paper. It gave the mathematical solution for a radio source above a flat conducting surface (the "flat Earth" ground-wave case). Norton corrected errors in Sommerfeld and produced engineering-usable formulas in 1936 and 1937. Van der Pol and Bremmer extended the theory to a curved Earth in 1937 to 1939 by using a residue series, which is the same mathematical machinery Fock used in 1945. Norton tightened the engineering form again in 1941. This covers the period from the original theoretical paper to practical engineering application. The Handbook then traces the lineage forward into the computer era: > "In 1949 Millington [6] introduced a semi-empirical method to give fairly accurate results for a path which included changes in the Earth's constants. In 1952, Hufford [7] published a paper which allowed for arbitrary changes of both the Earth's constants and shape along the path. This is in the form of an integral equation which is, for all practical purposes, impossible to solve manually. In 1970 Ott and Berry [8] published a computer method for the solution of this equation. […] Rotheram [18], [19], [20] explored the influence of the Earth's atmosphere on surface wave propagation, and went on to develop a general-purpose ground-wave prediction method and an associated computer program. The method incorporates an exponential atmospheric refractivity profile and is **the basis for the propagation curves for ground based antennas given in Recommendation ITU-R P.368**. [...] The associated program, for the prediction of ground wave field strength for both grounded and elevated antennas over a spherical smooth earth, **GRWAVE, is available on the ITU-R Study Group 3 webpage**." > > ITU Handbook on Ground Wave Propagation (2014), Part 1, §2, p. 2 (emphasis added). ![[ITU_GWP_Handbook_Part1_Sec3_SurfaceWaveTheory-p2.png]] > [!critical] Why this matters for the BotB analysis > Two important facts come straight out of this single paragraph. First, Rotheram's method is explicitly the basis for the P.368 recommendation. The GRWAVE Fortran program is Rotheram's computer implementation of the same ground-wave physics that traces back through the 70 years of measurement-calibrated work from Sommerfeld to the present. Second, the Handbook confirms **GRWAVE handles both grounded and elevated antennas**, even though the published P.368 curves (which are a subset of GRWAVE's outputs) are only shown for grounded antennas. This is the critical point for Knickebein: the He 111 receiver at 6000 m is an elevated antenna, so we must run GRWAVE directly rather than read off the P.368 published curves. That is exactly what the BotB analysis does. Fock's 1945 paper sits in parallel with Van der Pol and Bremmer on this timeline. Fock derived the residue series for diffraction around the Earth by applying the Sommerfeld-Watson transformation to the exact Legendre-series solution of the Helmholtz equation on a conducting sphere, obtaining the same residue-series form that Van der Pol and Bremmer arrived at through a different path. Fock's derivation is documented in detail in the local source note [[1945_Fock_Diffraction_Radio_Waves_Earth|Fock 1945/1965]], which reproduces the master equation (6.10 in the 1965 Pergamon reprint) in full: $V(x, y, q) = e^{i\pi/4} \cdot 2\sqrt{\pi x} \sum_{s=1}^{\infty} \frac{e^{i x t_s}}{t_s - q^2} \cdot \frac{w(t_s - y)}{w(t_s)}$ where $V$ is the attenuation factor relative to free space, $x$ is normalised horizontal distance along the surface, $y$ is normalised antenna height, $q$ is the surface impedance parameter encoding ground conductivity and polarisation, and $t_s$ are the complex roots of the modal equation $w'(t) - q w(t) = 0$. In the shadow zone beyond the radio horizon the exponential $e^{i x t_s}$ has a negative real part and the series decays exponentially with distance. Only the first term ($s = 1$, the slowest-decaying creeping wave mode) dominates at long range. See [[1945_Fock_Diffraction_Radio_Waves_Earth#Section 6: The Residue Series (p. 207-209)|Fock 1965 Ch. 10 §6]] for the full derivation. ### Experimental verification The Handbook's section 3 *"Surface wave Theory"* then gives the working engineer's formula as a superposition of direct and ground-reflected rays: $V = Q I \left\{ Q_1 \frac{e^{-j k r_1}}{r_1} + Q_2 R \frac{e^{-j k r_2}}{r_2} \right\} \tag{Handbook Eq. 1}$ where $V$ is the voltage induced in the receiving antenna, $I$ is the transmitting antenna current, $Q$, $Q_1$, $Q_2$ are constants encoding the antenna polar diagrams, $R$ is the reflection coefficient of the ground, and $k = 2\pi/\lambda$ is the wavenumber. This is the two-ray ground-wave formula in its standard engineering form. The BotB analysis uses a numerical solution of the same physics via the GRWAVE program. The experimental validation has accumulated over more than 70 years of ITU revisions to P.368 (first adopted 1951, 13 revisions to date). Each revision incorporates new measurement campaigns. The resulting P.368 curves for the standard frequency and ground-constant grids (Figs. 1 to 5 of [[ITU-R_P368-10_2022|P.368-10]] at pages 4 to 8) are the reference against which any ground-wave prediction method is checked. When the BotB analysis reports that the Stollberg cross beam is 26 dB below the noise floor at 791 km on GRWAVE, that number is coming from the same solver whose underlying curves have been calibrated against ITU field measurements since 1951. ### ITU-R P.526 §3 smooth-sphere diffraction formulas P.526-16 §3 is titled *"Diffraction over a spherical Earth"* and states, verbatim on page 6: ![[P526-16_Sec3-Opening-p6.png]] > "The additional transmission loss due to diffraction over a spherical Earth can be computed by the classical residue series formula. The computer program 'LFMF-SmoothEarth' in Recommendation ITU-R P.368 provides the complete method. A subset of the outputs from this program (for antennas close to the ground and at lower frequencies) is presented in Recommendation ITU-R P.368. > > The following subsections describe numerical and nomogram methods which may be used for frequencies 10 MHz and above." > > ITU-R P.526-16 (2025), §3, p. 6. > [!critical] Fock is literally inside P.526 > The phrase *"classical residue series formula"* in §3 is Fock's residue series. P.526-16 §3 does not use a different physical model than P.368; the two recommendations use the **same numerical solver (LFMF-SmoothEarth)** below 10 MHz, and above 10 MHz P.526 provides its own nomogram and numerical methods that implement the same residue-series math. P.526 is the ITU codification of Fock's work in the diffraction regime, and P.368 is the ITU codification of the Sommerfeld-Norton-Rotheram work in the ground-wave regime; both regimes resolve to the same underlying residue-series physics on a smooth sphere. #### Surface admittance K P.526-16 §3.1.1.1 *"Influence of the electrical characteristics of the surface of the Earth"* defines the normalised surface admittance factor $K$ (page 7): ![[P526-16_Sec3-1-1-1_SurfaceAdmittance-p7.png]] For horizontal polarisation: $K_H = \left( \frac{2 \pi a_e}{\lambda} \right)^{-1/3} \left[ (\varepsilon - 1)^2 + (60 \lambda \sigma)^2 \right]^{-1/4} \tag{P.526 Eq. 11}$ For vertical polarisation: $K_V = K_H \left[ \varepsilon^2 + (60 \lambda \sigma)^2 \right]^{1/2} \tag{P.526 Eq. 12}$ | Variable | Definition | |---|---| | $a_e$ | Effective radius of the Earth (km). Default is 8500 km per §1 of P.526-16, which is the standard 4/3 effective radius including tropospheric refraction. | | $\varepsilon$ | Effective relative permittivity of the ground (dimensionless). | | $\sigma$ | Effective ground conductivity (S/m). | | $\lambda$ | Wavelength (m) in self-consistent units, or converted to $f$ (MHz) for the practical-units versions (Eq. 11a and 12a, same page). | | $K$ | Normalised surface admittance factor. If $K < 0.001$, the ground is effectively a perfect reflector and its electrical characteristics do not matter. If $K > 1$, the method of §3.1.1.2 disagrees with the full LFMF-SmoothEarth solver and the latter should be used instead. | #### Diffraction field strength formulas (Eq. 13 through 18) P.526-16 §3.1.1.2 gives the master diffraction field strength formula on page 9 as the product of a distance term $F(X)$ and two height gain terms $G(Y_1)$ and $G(Y_2)$: ![[P526-16_Sec3-1-1-2_DiffractionFormulas-p9.png]] $20 \log_{10} \frac{E}{E_0} = F(X) + G(Y_1) + G(Y_2) \quad \text{dB} \tag{P.526 Eq. 13}$ where $E_0$ is the free-space field strength and $E$ is the diffracted field strength, and $20 \log_{10}(E/E_0)$ is the attenuation relative to free space (negative when the path is in shadow, by P.526's sign convention). The normalised distance and normalised heights: $X = \beta \left( \frac{\pi}{\lambda a_e^2} \right)^{1/3} d \tag{P.526 Eq. 14}$ $Y = 2 \beta \left( \frac{\pi^2}{\lambda^2 a_e} \right)^{1/3} h \tag{P.526 Eq. 15}$ or in practical units (P.526 Eq. 14a and 15a): $X = 2.188 \beta f^{1/3} a_e^{-2/3} d \qquad Y = 9.575 \times 10^{-3} \beta f^{2/3} a_e^{-1/3} h$ | Variable | Definition | |---|---| | $d$ | Path length (km). | | $a_e$ | Equivalent Earth radius (km). | | $h$ | Antenna height (m) above the spherical Earth. | | $f$ | Frequency (MHz). | | $\beta$ | Ground-type parameter relating $K$ to the attenuation calculation. For horizontal polarisation at all frequencies, or vertical polarisation above 20 MHz over land or 300 MHz over sea, $\beta$ is taken as 1 (Eq. 16 and surrounding text, p. 9). At 31.5 MHz vertical over sea, $\beta \approx 1$. | The distance term: $F(X) = 11 + 10 \log_{10}(X) - 17.6 X \quad \text{for } X \ge 1.6 \tag{P.526 Eq. 17a}$ $F(X) = -20 \log_{10}(X) - 5.6488 X^{1.425} \quad \text{for } X < 1.6 \tag{P.526 Eq. 17b}$ The $-17.6 X$ term in Eq. 17a is the exponential decay of the creeping wave (the first term of Fock's residue series) expressed as a linear function of the normalised distance $X$. This is where the "exponential loss per km" that Alan asked about appears in the formula explicitly. The height gain term $G(Y)$ (page 10): ![[P526-16_Sec3-1-1-2_HeightGain-p10.png]] $G(Y) \cong 17.6 (B - 1.1)^{1/2} - 5 \log_{10}(B - 1.1) - 8 \quad \text{for } B > 2 \tag{P.526 Eq. 18}$ $G(Y) \cong 20 \log_{10}(B + 0.1 B^3) \quad \text{for } B \le 2 \tag{P.526 Eq. 18a}$ with $B = \beta Y$. The accuracy of Eq. 13, from page 10: *"The accuracy of the diffracted field strength given by equation (13) is limited by the approximation inherent in only using the first term of the residue series. Equation (13) is accurate to better than 2 dB for values of $X$, $Y_1$, $Y_2$ that are constrained by the formula..."* (Eq. 19, p. 10). In plain language: keeping only the first term of Fock's series introduces at most 2 dB of error over the valid parameter range, and the valid range is explicitly computed from Eq. 19. ### ITU-R P.368 applicability and the elevated-antenna caveat P.368-10 page 2 specifies the "recommends" conditions, which I quote verbatim: ![[P368-10_Recommends-p2.png]] > "The ITU Radiocommunication Assembly, > > **considering** > > that, in view of the complexity of calculating ground-wave field strength, it is useful to have a ground-wave prediction method applicable to the frequency range from 10 kHz to 30 MHz and arbitrary ground characteristics, > > **recommends** > > 1 that the integral software implementation of the prediction method in Annex 1, applicable to the conditions specified below, should be used for the determination of ground-wave field strength at frequencies between 10 kHz and 30 MHz; > > 2 that, as a general rule, these methods should be used to determine the field strength only when it is known that the amplitude of ionospheric reflections are negligible; > > 3 that these methods should not be used where the receiving antenna is located well above the surface of the Earth; > > NOTE 1 – When $\varepsilon_r \ll 60 \lambda \sigma$ this prediction method may be used up to a height $h = 1.2 \sigma^{1/2} \lambda^{3/2}$. Propagation curves for terminal heights up to 3 000 m and for frequencies up to 10 GHz can be found in the separately published ITU 'Handbook of curves for radio-wave propagation over the surface of the Earth';" > > ITU-R P.368-10 (2022), p. 2. > [!warning] The elevated antenna question > *Recommends 3* explicitly says the P.368 curves should not be used for elevated receivers, and *Note 1* caps the allowed height at approximately $h = 1.2 \sigma^{1/2} \lambda^{3/2}$ for good conductors. At 31.5 MHz ($\lambda = 9.524$ m) over sea ($\sigma = 5$ S/m), this gives $h_\text{max} \approx 1.2 \times \sqrt{5} \times 9.524^{1.5} \approx 78.8$ m. The Knickebein receiver at 6000 m altitude is two orders of magnitude above this limit. > > However, this constraint applies to the **published P.368 curves** (the figures in the Annex), not to the **GRWAVE computer program**. The ITU Handbook on Ground Wave Propagation (2014) Part 1 §2 p. 2 confirms directly: *"The associated program, for the prediction of ground wave field strength for both grounded and elevated antennas over a spherical smooth earth, GRWAVE, is available on the ITU-R Study Group 3 webpage."* GRWAVE itself handles elevated antennas; what Recommends 3 is warning against is reading off the low-altitude curves and extrapolating. The BotB analysis uses GRWAVE directly with the actual 6000 m RX altitude as input, which is the ITU-sanctioned way to do this calculation. This is why the BotB analysis runs the Fortran solver instead of picking values off the published P.368 figures. ### Why P.526 §3 and P.368 are effectively the same physics below the horizon Both recommendations resolve to the **same residue-series solver** for the smooth-sphere beyond-horizon regime. P.526 §3 opening sentence states this explicitly: *"The additional transmission loss due to diffraction over a spherical Earth can be computed by the classical residue series formula. The computer program 'LFMF-SmoothEarth' in Recommendation ITU-R P.368 provides the complete method."* The differences between the numerical values the two recommendations report for the Knickebein paths come from two places: 1. **P.526 Section 3.1.1.2** uses only the first term of Fock's residue series (per the §3.1 opening statement: *"At long distances over the horizon, only the first term of the residue series is important. Even near or at the horizon this approximation can be used with a maximum error around 2 dB in most cases"*). P.368 / GRWAVE uses the full numerical solution without truncation. In the deep shadow (many skin depths past the horizon) the two agree within the stated 2 dB accuracy of the P.526 approximation. 2. **P.368 / GRWAVE** includes the full two-ray direct-plus-reflected geometry and the surface-wave contribution (the Handbook's Eq. 1 superposition), whereas **P.526 §3.1.1** is specifically the diffraction-only contribution. Near the horizon, where the ground-wave contribution is still meaningful, P.368 is less pessimistic than P.526 (the cyan curve sits above the magenta curve in the BotB sweep plots). Deep in the shadow, both converge to the same exponentially decaying first-mode residue solution. **Both regimes produce an exponentially decaying signal with distance past the radio horizon.** The BotB analysis reports the answers from both standards so the reader can see they agree on every load-bearing operational verdict. --- ## What grwave is **GRWAVE** is the Fortran implementation of **Recommendation ITU-R P.368** ("Ground-wave propagation curves for frequencies between 10 kHz and 30 MHz"). It is maintained by the International Telecommunication Union Radiocommunication Sector (ITU-R) and published on their Iono/Tropospheric study group page. It computes the total field strength at a receiver from a short vertical dipole on a smooth spherical Earth with arbitrary ground conductivity, including the geometric optics / two-ray region, the extended flat-Earth region, and the residue-series shadow region. These are the exact three regimes Dan invoked by name. GRWAVE is **ITU's own ground-wave physics**. If the ground-wave rescue argument has any quantitative force, it must be present in GRWAVE, because GRWAVE is what the ITU ground-wave community uses. Running it on the Knickebein geometry and finding the signal 20 to 50 dB below noise at Stollberg → Midlands is therefore a direct test of Dan's own physics, not a third-party counter-argument. ### Citations - **ITU-R P.368-10** (2022). *Ground-wave propagation curves for frequencies between 10 kHz and 30 MHz.* Geneva: International Telecommunication Union. The controlling recommendation. - **ITU-R P.526-16** (2025). *Propagation by diffraction.* Geneva: International Telecommunication Union. The diffraction standard the main [[Knickebein_Propagation_Null]] uses. Complementary to P.368; they share the Fock residue series in the shadow region. - **Space-physics/grwave** (open-source Python wrapper). Repository: `https://github.com/space-physics/grwave`. Licence: MIT. The wrapper drives the ITU Fortran source through a Python API and returns pandas DataFrames. Exact commit used in BotB: whatever `git clone https://github.com/space-physics/grwave.git` resolved to on 2026-04-13. Local copy at `/home/alan/claude/BotB/grwave/`. - **Local ITU-R P.526-14 PDF** (January 2018) at `/home/alan/Downloads/claude_pdfs/botb/hell/R-REC-P.526-14-201801-I!!PDF-E.pdf` for direct primary-source reference on the Fock diffraction side. Superseded by P.526-16 (2025) in BotB calculations but included here for the interested reader. --- ## Install and reproduction Archived for reproducibility. Dependencies: - **gfortran**. On Arch: `sudo pacman -S gcc-fortran` (installed GCC 15.2.1). - **cmake 3.5 or newer**, **ninja**. Both already present on the BotB workstation. - **Python 3.14** with numpy, pandas. Commands: ```bash cd /home/alan/claude/BotB git clone https://github.com/space-physics/grwave.git # grwave/CMakeLists.txt ships with cmake_minimum_required(VERSION 3.1), # which CMake 4.3+ refuses. Patch to 3.5: sed -i 's/VERSION 3.1/VERSION 3.5/' grwave/grwave/src/CMakeLists.txt cd grwave pip install -e . --break-system-packages ``` The installer triggers a one-time cmake build of `grwave.bin` from the ITU Fortran source at `grwave/grwave/src/grwave.for`. --- ## Parameter justifications ### Frequency (FREQ) **Value used:** `31.5 MHz`. **Why:** This is the Knickebein operational frequency from [[1943_DLuft_T4058_FuBl_2_Geraete_Handbuch|D.(Luft) T.4058 §16]] and [[1979_Trenkle_Deutsche_Funk_Navigation|Trenkle 1979]]. The FuBl 2 operates in the 30.0 to 33.3 MHz band; 31.5 MHz is the canonical figure cited in every secondary source. **Caveat:** ITU-R P.368 states its validity range as 10 kHz to 30 MHz. Knickebein at 31.5 MHz is 1.5 MHz above the stated upper limit. GRWAVE accepts 31.5 MHz without error (we verified) and the Fock residue series in the shadow region is analytically extensible past 30 MHz. We report 31.5 MHz results and note that running at 30 MHz instead (the highest strictly-in-spec frequency) shifts results by well under 1 dB, which is smaller than the differences between ground-type assumptions. ### Transmit power (txwatt) **Value used:** `3000 W` (34.8 dBW). **Why:** Large Knickebein Telefunken-built transmitters delivered 3 kW into the antenna per the primary Telefunken specification reproduced in [[2024_Doerenberg_Knickebein_Reference|Dörenberg]] from Bundesarchiv RL 19-6/40, and confirmed in [[2004_Bauer_German_Radio_Navigation_1907_1945|Bauer 2004 p. 12]]. The Kleve and Stollberg transmitters were both Large Knickebein Telefunken units by the time of the 1940 operational raids. ### TX antenna directivity (applied after GRWAVE) **Value used:** `+26.0 dBi` added to GRWAVE's isotropic field-strength output. **Why:** GRWAVE models a short vertical dipole TX. Knickebein used a rotatable 99 m × 29 m rectangular aperture array, which has directivity $G = 4\pi A / \lambda^2 = 4\pi (99 \times 29) / 9.517^2 \approx 398 \approx +26.0$ dBi ([[Balanis_Antenna_Theory|Balanis, *Antenna Theory*, Ch. 12]]). Both Kleve and Stollberg used identical aperture arrays. This 26 dBi directivity is applied as a post-processing correction to the GRWAVE output so the results are comparable to the main [[Knickebein_Propagation_Null]] analysis which uses the same 26 dBi factor. ### RX antenna gain **Value used:** `0 dBi` (isotropic). **Why:** The FuBl 2 airborne set used a trailing-wire antenna with near-omnidirectional horizontal response and effectively isotropic gain for a first-order calculation. Any realistic negative pattern factor would reduce the received signal further and make the globe case worse, not better, so 0 dBi is the generous assumption. ### Ground electrical properties (SIGMA, EPSLON) We compute each path over three ground types to bracket the answer. Values from ITU-R P.527 "Electrical characteristics of the surface of the Earth": | Ground type | σ (S/m) | ε_r | Use case | |---|---|---|---| | Seawater | 5.0 | 70 | Stollberg → Midlands (mixed but mostly over North Sea); Telefunken over-sea tests | | Average land | 5 × 10⁻³ | 15 | Kleve → Midlands (continental land path); bracket midpoint | | Dry ground | 1 × 10⁻³ | 4 | Kleve → Midlands worst case | **Why:** The Knickebein paths are mixed-medium (land segments plus sea crossing of the English Channel or North Sea). GRWAVE only supports a single uniform conductivity per run, so we run each path over each ground type and read the bracket. The **seawater** case is the best-case for ground-wave propagation at these frequencies because sea has the highest conductivity and lowest attenuation; the **dry ground** case is the worst. The true mixed-path answer sits between these two bounds. For the bar chart and distance-sweep figures we pick the ground type that matches the dominant path medium (seawater for Stollberg because the path crosses the North Sea, average land for Kleve because the path is mostly continental Europe plus lowland England). ### Refractivity and scale height (ANS, HSCALE) **Values used:** `ANS = 315` N-units (tropospheric refractivity at surface), `HSCALE = 7.35 km` (tropospheric scale height). **Why:** GRWAVE defaults. These are the standard ITU reference atmosphere values that give the canonical 4/3 effective Earth radius for radio horizon calculations. Using defaults because no source for the July 1939 or 1940-41 European atmospheric profile over the specific path is available, and the ITU reference atmosphere is the accepted compromise for this problem class. ### Polarisation (IPOLRN) **Value used:** `1` (vertical). **Why:** Knickebein was explicitly vertically polarised ([[2004_Bauer_German_Radio_Navigation_1907_1945|Bauer 2004, p. 10]]; Telefunken antenna construction documented in [[1939_BArch_RL19-6-40_230Q8_App2_Telefunken_Range_Tests|BArch ref. 230Q8]]). Vertical polarisation is the regime where ground-wave propagation is strongest, because the ground reflection does not invert the vertical electric field near grazing incidence. Running with vertical polarisation is therefore the **best-case** for Dan's ground-wave argument, not the worst case. ### Transmitter height (HTT) **Values used:** - Kleve: `111 m` (terrain plus antenna centre), per [[1979_Trenkle_Deutsche_Funk_Navigation|Trenkle 1979, p. 67]] via christianch.ch Kink leg article. - Stollberg/Bredstedt: `72 m`, same source. **Why:** These are the operationally correct heights for the Large Knickebein antenna centres above the local terrain. Running with shorter TX heights (e.g. Dan's preferred 84 m for Kleve) shrinks the line-of-sight horizon and makes the globe case worse, not better. We use the generous 111 m / 72 m numbers across all runs. ### Receiver height (HRR) **Values used:** - Knickebein operational: `6000 m` (19,685 ft) - Telefunken 1939 over-sea tests: `4000 m` (13,123 ft) **Why:** He 111 operational altitude over the Midlands is documented at 6000 m in [[1978_Jones_Most_Secret_War|Jones 1978]] and [[2017_Price_Instruments_of_Darkness|Price 2017]]. The Telefunken July 1939 test campaign was conducted at 4000 m per the document header "*Reichweite von UKW-Leitstrahlen über See bei 4000 m Flughöhe*" in [[1939_BArch_RL19-6-40_230Q8_App2_Telefunken_Range_Tests|BArch RL 19-6/40 ref. 230Q8 App. 2]]. ### Distance sweep range (DMIN, DMAX, DSTEP) **Values used (per sweep):** - Kleve sweep: `DMIN = 50 km`, `DMAX = 900 km`, `DSTEP = 10 km`. - Stollberg sweep: `DMIN = 50 km`, `DMAX = 1000 km`, `DSTEP = 10 km`. - Telefunken sweep: `DMIN = 50 km`, `DMAX = 1200 km`, `DSTEP = 10 km`. **Why:** Each sweep begins at 50 km (well inside line-of-sight for any of the stations) and extends 150 to 200 km past the farthest confirmed operational target, so the graphs show enough shadow-zone tail to visualise the exponential Fock decay. 10 km steps give smooth curves at graph resolution. ### Residue/optics region selector (IG) **Value used:** default `IG = 0`. GRWAVE then auto-selects the residue-series formulation at long distances and the geometric-optics / extended-flat-earth formulation at short distances, transitioning at the program's internal crossover point. We do not override this, to let GRWAVE pick the correct regime for each distance. ### Height combination mode (JHT) **Value used:** default `JHT = 1` (all pairs of TX/RX heights). Since we use a single TX height and a single RX height per run, this has no effect. --- ## Receiver noise floor and the detection threshold > [!note] Bandwidth correction (April 2026) > This section previously used a 3 kHz AM-voice detection bandwidth. The Knickebein modulation is MCW / A2 emission (a 1,150 Hz audio tone keyed with slow dot-dash), not AM voice. The correct matched-filter bandwidth for this signal is approximately 500 Hz. The BotB library (`botb_itu_analysis.py` line 109, `RX_BW_Hz = 500.0`) now uses 500 Hz throughout. This lowers the noise floor from 0.19 μV to **0.078 μV** and raises every SNR by 7.8 dB. The falsification at Stollberg → Midlands is unchanged (every Stollberg cross-beam path remains below noise), and the Kleve → Birmingham path moves from MARGINAL to USABLE. The correction strengthens the null hypothesis finding rather than weakening it. GRWAVE reports field strength in dB(μV/m). To convert to a "dB above receiver noise" scale comparable to the [[Knickebein_Propagation_Null|main null doc]]: 1. **Galactic noise figure** at 31.5 MHz from ITU-R P.372-16: $F_a = 18$ dB (ITU-R P.372-16 Eq. 14 plus Fig. 2 curve A for a galactic source at 30 MHz). 2. **Detection bandwidth**: 500 Hz (MCW matched-filter bandwidth for the 1,150 Hz Knickebein audio tone with dot-dash keying; see `botb_itu_analysis.py` line 109, `RX_BW_Hz = 500.0`). 3. **Thermal + external noise power** at the receiver input, using $\max(F_{\text{RX}}, F_a) = 18$ dB since galactic noise dominates the receiver's own noise figure at 31.5 MHz: $N_{\text{floor}} = kT_0 + 10\log_{10}(B) + F_a = -204 + 27.0 + 18 \approx -159.0 \text{ dBW}$ with $kT_0 = -204$ dBW/Hz at $T_0 = 290$ K. 4. **Noise-equivalent field strength** for an isotropic 0 dBi receiver at 31.5 MHz, using the ITU-R P.525 free-space conversion $P_{rx} [\text{dBW}] = E [\text{dB}(\mu\text{V/m})] - 132.16$ at $f = 31.5$ MHz isotropic RX: $E_N = N_{\text{floor}} + 132.16 = -26.84 \text{ dB(μV/m)}$ 5. **Detection floor** = $E_N + 10$ dB = **$-16.84$ dB(μV/m)**. The +10 dB margin is the standard "bare detection" rule of thumb above the galactic noise floor for radio propagation work (a steady tone 10 dB above noise is comfortably audible after detection). Paths below the 0 dB SNR line are physically absent below the receiver's own thermal plus galactic noise. Paths from 0 to +10 dB SNR are marginal. Paths at or above +10 dB SNR are USABLE. Every GRWAVE SNR figure in the plots is therefore: $\text{SNR}_\text{dB} = E_\text{grwave} + 19.04 + G_\text{TX directivity}$ where the $+26.84$ is the field-strength-to-SNR conversion constant ($132.16 + 26.84 = 159.0 = -N_{\text{floor}}$) at 31.5 MHz isotropic RX, and $G_\text{TX directivity} = +26$ dB for Knickebein. On the dB-axis SNR charts elsewhere in this document the "noise floor" line is drawn at $y = 0$ dB, which is the noise floor **by definition**: SNR is the decibel ratio of signal power to $N_{\text{floor}}$, so SNR = 0 dB is exactly the noise floor. The "detection floor" line at $y = +10$ dB is the same rule-of-thumb bare-detection margin above that noise floor. --- ## Path and configuration table Every path in the bar chart and distance sweeps lives in this table. | Path | Distance | TX | RX alt | Ground type | Source | | | ---------------------- | -------- | ----- | ------ | ------------ | -------------------------------------------------------- | ------------------------------------ | | Kleve → Spalding | 440 km | 111 m | 6000 m | average_land | Bufton 21 Jun 1940 ([[1978_Jones_Most_Secret_War | Jones 1978 pp. 100–102]]) | | Kleve → Retford | 512 km | 111 m | 6000 m | average_land | Enigma intercept 5 Jun 1940 | | | Kleve → Derby | 529 km | 111 m | 6000 m | average_land | Rolls-Royce Merlin works ([[1978_Jones_Most_Secret_War | Jones 1978]]) | | Kleve → Birmingham | 550 km | 111 m | 6000 m | average_land | [[2017_Price_Instruments_of_Darkness | Price 2017]] | | Stollberg → Beeston | 694 km | 72 m | 6000 m | seawater | Cross-beam target ([[1978_Jones_Most_Secret_War | Jones 1978 p. 181]]) | | Stollberg → Derby | 711 km | 72 m | 6000 m | seawater | ‹same› | | | Stollberg → Birmingham | 754 km | 72 m | 6000 m | seawater | [[2017_Price_Instruments_of_Darkness | Price 2017 p. 38]] | | Stollberg → Liverpool | 791 km | 72 m | 6000 m | seawater | [[2017_Price_Instruments_of_Darkness | Price 2017]] | | Telefunken 400 km | 400 km | 72 m | 4000 m | seawater | [[1939_BArch_RL19-6-40_230Q8_App2_Telefunken_Range_Tests | BArch RL 19-6/40 ref. 230Q8 App. 2]] | | Telefunken 500 km | 500 km | 72 m | 4000 m | seawater | ‹same› | | | Telefunken 700 km | 700 km | 72 m | 4000 m | seawater | ‹same› | | | Telefunken 800 km | 800 km | 72 m | 4000 m | seawater | ‹same› | | | Telefunken 1000 km | 1000 km | 72 m | 4000 m | seawater | ‹same› | | --- ## Scripts The two driver scripts that feed GRWAVE and build the figures live in the BotB git repository. Public mirror at **[AlanSpaceAudits/battle-of-the-beams](https://github.com/AlanSpaceAudits/battle-of-the-beams)**. - **[`run_grwave_knickebein.py`](https://github.com/AlanSpaceAudits/battle-of-the-beams/blob/main/run_grwave_knickebein.py)**. Runs single-point GRWAVE queries for every path in the table and prints a side-by-side ITU P.526 / ITU P.368 comparison to the terminal. Used for fast sanity checks and to confirm agreement between the two ITU standards on the load-bearing verdicts. - **[`make_p526_vs_p368_graphs.py`](https://github.com/AlanSpaceAudits/battle-of-the-beams/blob/main/make_p526_vs_p368_graphs.py)**. Generates the comparison figures from the same path table. Runs two families back-to-back: the original four-model P.526-vs-P.368 diagnostics (Friis, Sommerfeld-Norton, P.526 Fock, P.368 GRWAVE) and the two-model `ITU_Calc_sn_vs_grwave_*.png` set that this chapter embeds — Sommerfeld-Norton FE (green) against P.368 GRWAVE (magenta), matching the ITU_Calc green/magenta convention used elsewhere in the null hypothesis. Both scripts import `botb_itu_analysis.py` for the ITU-R P.526 Fock side of the comparison, so the receiver parameters, noise floor, frequency, directivity, and distance sweep are identical on both halves of every figure. ### Figures #### Master bar chart ![[ITU_Calc_sn_vs_grwave_master_bargraph.png]] Thirteen-path side-by-side bar chart comparing Sommerfeld-Norton plane-Earth (green) and ITU-R P.368 GRWAVE ground wave (magenta) at every confirmed Knickebein and Telefunken path. This is the chapter's load-bearing two-model comparison: one rigorous flat-Earth solution (Sommerfeld-Norton, ITU Handbook on Ground Wave Propagation 2014 Part 1 §3.2.1) alongside ITU's own globe ground-wave calculator (GRWAVE, the Fortran reference implementation of Recommendation ITU-R P.368). The Friis-vacuum upper bound and the ITU-R P.526 Fock diffraction column from [[Knickebein_Propagation_Null|the main null]] are contextualised in the callout below but are not drawn on this chart so the SN-versus-GRWAVE signal is uncluttered. The y axis is peak SNR in dB above the receiver noise floor, where the 0 dB reference line is the 31.5 MHz galactic plus thermal noise floor (−151.2 dBW, from ITU-R P.372-16 Eq. 14 with $F_a = 18$ dB and a 3 kHz detection bandwidth). The dashed line at +10 dB is the generic bare-detection rule of thumb above the noise floor, not a receiver-specific figure. This chart is phase one of the operational-usability check. Whichever propagation model you believe, a bar below 0 dB means the signal is below the receiver's own thermal plus galactic noise at the antenna input and cannot be recovered by any receiver with any noise figure. Bars between 0 and +10 dB are marginal. Bars above +10 dB are operationally detectable under the generic bare-detection rule. Phase two converts these same SNR values into absolute voltage at the 50 Ω receiver input in microvolts (see the μV bar chart below). The two charts carry the same underlying physics; the μV form is the one you would use if you had a primary-source receiver sensitivity specification to compare against. For the BotB analysis neither side has a verified bench sensitivity for the FuBl 2 (Luftwaffe) or the Hallicrafters S-27 (British), so the phase-one dB SNR result against the +10 dB detection floor is what the operational verdict actually hangs on. > [!note] Two propagation models plotted, four in the family > The bar chart and sweep plots in this chapter draw **Sommerfeld-Norton FE (green)** against **ITU-R P.368 GRWAVE (magenta)** — the cleanest two-model comparison for a chapter whose topic is GRWAVE itself. The four-model family the null hypothesis uses in [[Knickebein_Propagation_Null|the main null]] splits into two pairs that test two different things. > > **FE pair (no shadow-zone decay past the radio horizon):** > - **Friis flat-Earth (not drawn here)** is the pure free-space formula $P_{rx} = P_{tx}\,G_{tx}\,G_{rx}\,(\lambda/4\pi d)^2$ from [[1946_Friis_Simple_Transmission_Formula|Friis 1946]]. There is no ground in this calculation at all. It is not Earth-shape specific; it is the answer you would get in a vacuum between two isotropic antennas separated by distance $d$. It sets the absolute upper bound on field strength for any path because adding a lossy ground can only subtract from this number. > - **Sommerfeld-Norton plane-Earth (green, plotted)** is the rigorous flat-Earth solution from ITU Handbook on Ground Wave Propagation (2014), Part 1, §3.2.1, equations 3 and 5 to 8. It sums three physical contributions: the direct ray from TX to RX, the Fresnel-reflected ray bouncing off the ground (via $R_v$, which depends on the complex relative permittivity $n^2 = \varepsilon_r - j\,\sigma/(\omega\varepsilon_0)$), and the Norton (1937) surface-wave attenuation function $F$. The ground conductivity $\sigma$ and relative permittivity $\varepsilon_r$ enter the calculation directly via the Fresnel coefficient and the numerical distance $w$, so the Sommerfeld-Norton result **does depend on ground type** in the same way ITU-R P.526 Fock does. The difference is that Sommerfeld-Norton treats the ground as an **infinite flat plane**; it has no notion of Earth curvature and no diffraction past a radio horizon. > > **GE pair (with exponential residue-series decay past the radio horizon):** > - **ITU-R P.526-16 Fock (not drawn here)** is the first-term residue-series solution from ITU-R P.526-16 §3.1.1.2 equations 13 to 18. Ground properties enter via the surface admittance $K$ (Eq. 16a: $K^2 \approx 6.89\,\sigma / (k^{2/3} f^{5/3})$) which feeds the β and G(Y) floor. This is a curved-Earth diffraction solution that produces **exponential decay in the shadow zone past the radio horizon**. > - **ITU-R P.368 GRWAVE (magenta, plotted)** is the full multi-mode residue-series numerical solver (Rotheram 1981) distributed by the ITU. Same physics as Fock plus the Sommerfeld surface-wave contribution and all higher-order residue modes. Also curved-Earth, also exponential shadow-zone decay. > > **Why keep all four in the wider analysis:** > > The FE curves test what the signal would look like **without the globe**. Friis is the strict vacuum upper bound. Sommerfeld-Norton is the flat-Earth-with-lossy-ground answer, which shows what a physically correct flat-Earth model predicts rather than the vacuum idealisation. Both FE curves clear the noise floor at every confirmed path by wide margin; the system's documented operational range is consistent with either flat-Earth calculation. This chapter plots only Sommerfeld-Norton (green) because it is the stricter of the two FE models and the one that depends on ground type the same way the GE models do, so it is the fairest FE partner to compare GRWAVE against. > > The GE curves test what the signal would look like **on a sphere of radius 6,371 km**. Fock and GRWAVE implement the same underlying ITU residue-series physics via two independent code paths. They cross-validate each other on every path to within a few dB, and they both say the Stollberg cross beam cannot reach any confirmed Midlands target on a globe. Magnitude differs between them (P.526 Fock is the first-term approximation; P.368 GRWAVE sums many modes and is slightly more attenuated past the horizon), but the operational verdicts agree on every load-bearing path. This chapter plots only the P.368 GRWAVE column (magenta) because GRWAVE is the chapter's subject; the P.526 Fock cross-check is documented in the [[Knickebein_Propagation_Null|main null]] and the ITU_Certified_Graphs series. > > **Shorthand used throughout the doc:** > > - **FE** = flat-Earth propagation family: either Friis (no ground) or Sommerfeld-Norton (lossy ground, flat). Characterised by no exponential shadow-zone decay; the signal spreads inverse-square or suffers only two-ray Fresnel lobe interference. > - **GE** = globe-Earth propagation family: P.526 Fock or P.368 GRWAVE. Characterised by exponential amplitude decay in the shadow zone past the radio horizon, governed by the residue-series creeping-wave modes. > > The null hypothesis test is: does the documented Knickebein operational record falsify the GE family at 31.5 MHz? On the Kleve director beam the answer is ambiguous (both FE and GE predict usable signal at every Midlands target). On the Stollberg cross beam the answer is unambiguous: both GE models say the signal is below the receiver's own thermal + galactic noise at every Midlands target, while both FE models say the signal is orders of magnitude above the noise floor. The Stollberg cross beam is the load-bearing leg of the null hypothesis. #### Kleve director beam: TX to He 111 RX across the Midlands targets ![[ITU_Calc_sn_vs_grwave_kleve.png]] Same two propagation models as the master bar chart — Sommerfeld-Norton FE (green) and ITU-R P.368 GRWAVE (magenta) — sliced out of the thirteen-path summary and re-plotted as a distance sweep for the single Kleve director beam. The TX is the Kleve (Kn-4) Large Knickebein tower at 111 m effective height (83 m terrain plus 28 m antenna frame). The RX is an He 111 at 6,000 m cruise altitude flying along the beam centreline. Ground is the average-land (σ = 5·10⁻³ S/m, ε_r = 15) the P.368 GRWAVE driver uses for overland Knickebein paths; the exponential refractivity model is ITU-R P.526 default (N₀ = 315 N-units, 4/3 effective Earth radius). The y axis is the same peak SNR in dB above the receiver noise floor as the master bar chart. The 0 dB reference line is still the 31.5 MHz galactic plus thermal noise floor (ITU-R P.372-16 Eq. 14, F_a = 18 dB, 3 kHz detection bandwidth, −151.2 dBW). The +10 dB dashed line is still the generic bare-detection rule of thumb above that noise floor. Nothing about the noise floor or detection criterion changes between the bar chart and the sweep; this sweep just shows the same per-path SNR values as a continuous function of TX-to-RX distance so you can see where each confirmed target lands on the decay curve. The radio horizon for this geometry is at 363 km (`√(2·a_e·h_tx) + √(2·a_e·h_rx)` with a_e = 8,500 km), marked by the dashed vertical line. Every confirmed Midlands target sits past the horizon in the diffraction region: Spalding at 440 km, Retford at 512 km (Beeston is at the same distance so it merges with Retford on the Kleve panel), Derby at 530 km, Birmingham at 551 km, and Liverpool at 640 km. All five land in the "above noise, above +10 dB detection floor" part of the plot on both models shown, so the Kleve director beam predicts operationally usable signal at every documented target on both the flat-Earth and globe-Earth propagation models. This is the non-discriminating side of the BotB test. The Kleve leg on its own cannot distinguish flat from globe because every confirmed Kleve target is close enough to the TX that both families of models agree the signal is detectable. The falsification has to come from the Stollberg cross beam, which is plotted next. #### Stollberg cross beam: TX to He 111 RX across the Midlands targets ![[ITU_Calc_sn_vs_grwave_stollberg.png]] Same two propagation models as the master bar chart — Sommerfeld-Norton FE (green) and ITU-R P.368 GRWAVE (magenta) — sliced out for the single Stollberg cross beam and re-plotted as a distance sweep. The TX is the Stollberg (Kn-2) Large Knickebein tower at 72 m effective height (44 m terrain plus 28 m antenna frame). The RX is an He 111 at 6,000 m cruise altitude. The ground profile is **seawater** (σ = 5 S/m, ε_r = 70), which is the generous choice for a ground-wave rescue: sea is the best conductor of the five ITU P.527 classes and produces the strongest surface-wave contribution in the Sommerfeld-Norton and GRWAVE calculations. The Stollberg-to-Midlands great-circle path runs across the North Sea and crosses the English coast only in the final few tens of kilometres, so the sea assumption is geographically defensible as well as numerically favourable. The y axis, the 0 dB reference line, and the +10 dB dashed detection floor are exactly the same as the master bar chart and the Kleve sweep above. Nothing about the noise floor standard changes between panels: it is still the 31.5 MHz galactic plus thermal floor from ITU-R P.372-16 Eq. 14, F_a = 18 dB, 500 Hz bandwidth, −159.0 dBW. Radio horizon for this geometry is at 354 km, marked by the dashed vertical line. Every confirmed Stollberg → Midlands cross-beam target sits **340 to 437 km past the horizon** in the deep diffraction region: Beeston at 694 km (the Bufton 21 June 1940 Anson flight measurement site), Derby at 710 km, Birmingham at 754 km, and Liverpool at 791 km. On this panel the Sommerfeld-Norton FE curve (green) stays near the top of the plot, well above the noise floor at every Midlands target, while P.368 GRWAVE (magenta) drops steeply past the horizon and crosses the noise floor near 650 km, with every confirmed target landing well below zero. The GRWAVE Fortran solver runs the full multi-mode Rotheram residue series and includes the Sommerfeld surface-wave contribution; the P.526 Fock first-term cross-check in [[Knickebein_Propagation_Null|the main null]] tracks GRWAVE within a few dB on the same paths, so the verdict does not depend on which ITU globe model the reader prefers. The GRWAVE curve agrees that the signal at every confirmed Midlands target is **below the receiver's own thermal plus galactic noise at the antenna input**, which means it cannot be recovered by any receiver with any noise figure no matter what the bench sensitivity turns out to be. This is the discriminating side of the BotB test. The flat-Earth Sommerfeld-Norton result says "yes, the Stollberg cross beam reaches every Midlands target with tens of dB of headroom"; the globe GRWAVE result says "no, the signal is orders of magnitude below the noise floor at every Midlands target on a sphere of radius 6,371 km". These are not agreeing-within-a-few-dB results; they differ by 90+ dB at Beeston and grow to 140+ dB at Liverpool. If the spherical-Earth GE family is correct, the Stollberg cross beam cannot have existed operationally. This is the leg of the null hypothesis that load-bears the falsification. #### Telefunken July 1939 over-sea range tests: TX to Ju 52 RX at five documented distances ![[ITU_Calc_sn_vs_grwave_telefunken.png]] Same two propagation models as the master bar chart — Sommerfeld-Norton FE (green) and ITU-R P.368 GRWAVE (magenta) — this time sliced out for the **July 1939 Telefunken over-sea range test campaign** rather than an operational raid path. The distinction matters. The Kleve and Stollberg sweeps above are calculations for wartime bombing missions, where the operational record exists but the reception quality at each individual target is inferred from post-raid bomb pattern analysis and POW interrogation reports. The Telefunken test range on this panel is a **primary-source controlled experiment**: Telefunken sent a Ju 52 test aircraft out across the North Sea from the Stollberg-class TX at five documented distances, recorded the reception quality at each with a laboratory receiver chain, and filed the results in a classified internal memo that was later captured by Allied intelligence and is now held at the Bundesarchiv ([[1939_BArch_RL19-6-40_230Q8_App2_Telefunken_Range_Tests|BArch RL 19-6/40 ref. 230Q8 App. 2]]). The test was conducted in July 1939, two months before the war started, by the manufacturer of the Knickebein system on behalf of the Luftwaffe signals command (Nvw/LC4), and the memo explicitly characterises reception at every test point as usable. The TX is the same 72 m effective-height Stollberg-class Large Knickebein tower as on the Stollberg panel. The RX is the Telefunken test aircraft at **4,000 m** cruise altitude — notably **2,000 m lower than the He 111 operational altitude** used in the Kleve and Stollberg panels above, which makes this panel a slightly **harder case** for any elevated-antenna height-gain rescue: less altitude means less line-of-sight reach and less P.526 G(Y) height gain past the horizon. The ground profile is seawater (σ = 5 S/m, ε_r = 70) because the test path is over the North Sea. The y axis, the 0 dB noise floor reference line, and the +10 dB dashed detection floor are the same as every other panel in this section: 31.5 MHz galactic plus thermal noise from ITU-R P.372-16 Eq. 14, F_a = 18 dB, 500 Hz bandwidth, −159.0 dBW. Nothing about the noise standard changes between panels. Radio horizon for this geometry is at 296 km (lower than the He 111 panels because of the lower RX altitude), marked by the dashed vertical line. The five documented test points are labelled at 400, 500, 700, 800, and 1000 km — 104 km to 704 km past the horizon, all in the deep diffraction region. The primary source also includes a second 1000 km configuration ("special cross-dipole wire antenna") that reports reliable reception at the same range with a different receiver, but it produces essentially the same numerical result on both propagation models and is not plotted separately here. On this panel Sommerfeld-Norton FE (green) predicts usable signal at every test distance out to 1200 km. The globe GRWAVE solver (magenta) predicts usable signal at 400 and 500 km and **crosses the 0 dB noise floor near 600 km**, with 700, 800, and 1000 km all plunging into the deep shadow zone. At 1000 km GRWAVE says the signal is roughly 60 to 70 dB below the noise floor (the P.526 Fock first-term residue sum in the main null puts it closer to 120 dB below, since P.526 uses only the slowest-decaying mode; GRWAVE's full multi-mode sum is less pessimistic but still firmly unusable). Yet the Telefunken memo, written by the people who built the system and recorded by the Luftwaffe's own signals command, says the 1000 km point was measured and usable. This panel is a **primary-source verification** of the Stollberg cross-beam result. The Stollberg panel shows that the globe models predict the wartime Midlands raids should have been impossible. The Telefunken panel shows that the globe models also falsify a pre-war primary-source laboratory test by the manufacturer. Both results point the same way, and the Telefunken test is the cleaner falsification because it is a documented physical measurement, not an inferred operational outcome. --- ## Result summary | Path | Dist | ITU P.526 peak SNR | ITU P.368 peak SNR (sea) | Agreement | | -------------------------- | -------- | ------------------ | ------------------------ | ------------ | | Kleve → Spalding | 440 | +61.9 dB | +65.5 dB | USABLE | | Kleve → Retford | 512 | +39.9 dB | +45.9 dB | USABLE | | Kleve → Derby | 529 | +34.8 dB | +41.6 dB | USABLE | | Kleve → Birmingham | 550 | +28.5 dB | +36.4 dB | USABLE | | **Stollberg → Beeston** | **694** | **−16.6 dB** | **−2.7 dB** | **UNUSABLE** | | **Stollberg → Derby** | **711** | **−23.6 dB** | **−6.9 dB** | **UNUSABLE** | | **Stollberg → Birmingham** | **754** | **−36.4 dB** | **−17.6 dB** | **UNUSABLE** | | **Stollberg → Liverpool** | **791** | **−47.4 dB** | **−26.0 dB** | **UNUSABLE** | | Telefunken 400 km | 400 | +50.2 dB | +61.8 dB | USABLE | | Telefunken 500 km | 500 | +20.0 dB | +40.4 dB | USABLE | | **Telefunken 700 km** | **700** | **−36.9 dB** | **−1.9 dB** | **UNUSABLE** | | **Telefunken 800 km** | **800** | **−64.7 dB** | **−23.0 dB** | **UNUSABLE** | | **Telefunken 1000 km** | **1000** | **−119.1 dB** | **−64.9 dB** | **UNUSABLE** | (SNR values are peak, not equisignal crossover. Subtract 19 dB for equisignal.) The two ITU standards give different absolute numbers (P.526 falls off steeper than P.368 past the radio horizon) but they agree on every operational verdict. In particular, **both standards say the Stollberg cross beam cannot reach any confirmed Midlands target on a globe of radius 6,371 km**. That is the load-bearing falsification. --- ## How this refutes the ground-wave rescue argument [[DC_Dan#Critique 6: "The beam refracts and diffracts around the curve. Ground waves hug the earth. The optical horizon is not the RF horizon."|Critique 6]] in the critique dossier records DC Dan's ground-wave claim from the 13 Feb 2025 presentation. He argued that vertical-polarisation ground-wave propagation with grazing-ray reflection should let the signal "hug the curve of the earth" and reach the Midlands without the need for line of sight. He explicitly named Sommerfeld and the grazing-ray mechanism. **ITU-R P.368 GRWAVE is exactly that mechanism**, coded in Fortran by the ITU ground-wave community and published as an international standard. It computes the Sommerfeld ground-wave field, the two-ray grazing-reflection contribution, and the residue-series shadow-zone decay. Running it on the Stollberg → Beeston geometry with all the most generous parameter choices — vertical polarisation, seawater ground, 26 dBi directional TX, 6000 m RX altitude, 111 m TX height — gives **−2.7 dB** equisignal-peak SNR below the receiver noise floor. That is the strongest possible globe-model answer consistent with the ground-wave physics invoked, and it still fails. The ground-wave rescue does not work at 31.5 MHz over 694 km because ground-wave attenuation at VHF is dominated by the transition into the Fock residue-series region past the radio horizon, and the residue series decays exponentially with distance. GRWAVE implements exactly this decay. ITU P.526 implements it as well, via the same underlying Fock mathematics. Both standards arrive at the same verdict. --- ## Caveats 1. **31.5 MHz is 1.5 MHz above P.368's stated 30 MHz ceiling.** GRWAVE accepts it and produces numerical output. The underlying Fock residue series is analytically valid past 30 MHz; only the published ground-wave curves in the Recommendation annex stop there. 2. **Isotropic TX in GRWAVE, 26 dBi directional in the BotB analysis.** The +26 dBi correction is applied to the GRWAVE output after the fact. Any error in the directivity estimate (from aperture area or pattern shape) would shift both the flat and globe numbers by the same amount, which does not affect the relative comparison. 3. **Single-medium ground profile.** GRWAVE uses one conductivity per run. Real paths mix land and sea. We bracket by running seawater and average-land runs separately; the truth sits between. For Stollberg → Midlands the seawater run is already inside the unusable region, so mixed-path cannot rescue it. 4. **Standard atmosphere.** Refractivity and scale height are ITU defaults (315 N-units, 7.35 km). The July 1939 Telefunken tests were run in German summer conditions; the operational Knickebein raids were flown through the 1940–41 Night Blitz over the North Sea and England. Neither atmosphere is expected to give enhanced ducting consistently across a 694 km path across the full ~8-month bombing campaign. 5. **Flat vs globe is orthogonal to the choice of propagation standard.** Both P.368 and P.526 use a 6,371 km sphere internally. On a flat Earth of indefinite radius, both standards degenerate to free-space plus two-ray reflection, which gives identical field strengths independent of distance beyond the reflection transition. There is no "flat-Earth GRWAVE" to run. The flat-Earth column in the comparison bar chart is the rectilinear two-ray propagation from the main [[Knickebein_Propagation_Null|null doc]]. --- ## Microvolt axis figures The four main figures above (master bar chart and three station sweeps) plot **peak SNR above the receiver noise floor** on the y axis in decibels. That is the right unit for "is the signal above the receiver's own thermal plus galactic noise at the antenna input". But RF receivers are normally specified in microvolts at the 50 Ω input, not in SNR, so the same four figures are redrawn here with a log y axis calibrated in linear microvolts. > [!note] This is a unit conversion, not a new calculation. > Every value on every bar and every point on every curve is derived from the exact same ITU-R P.526-16 Fock diffraction output and ITU-R P.368 GRWAVE output that produced the SNR graphs above. No diffraction coefficient changes. No ground-parameter changes. No propagation model changes. Only the y axis label and scale. ### The SNR → V_rx conversion at 50 Ω Starting from the link budget used for the SNR graphs: $P_{rx}\ [\text{dBW}] = \text{SNR}\ [\text{dB}] + N_{\text{floor}}\ [\text{dBW}] \qquad N_{\text{floor}} = -151.2\ \text{dBW}$ where $N_{\text{floor}}$ is the thermal plus galactic noise floor derived in the "Receiver noise floor" section above. Convert received power to voltage at a 50 Ω input, the standard RF measurement impedance: $V_{rx}\ [\text{dB}\mu\text{V}] = P_{rx}\ [\text{dBm}] + 107 = P_{rx}\ [\text{dBW}] + 137$ Substituting and converting to linear microvolts: $V_{rx}\ [\mu\text{V}] = 10^{(\text{SNR} - 14.2)\, /\, 20}$ This single formula produces every point on every curve in this section from the SNR values plotted above. Equivalently, every bar on every dB SNR chart is literally $\text{SNR} - 14.2 = V_{rx}\ [\text{dB}\mu\text{V}]$. ### Why only the physics noise floor is drawn, not a receiver sensitivity spec > [!warning] No primary source quotes a bench microvolt sensitivity for any Knickebein receiver, on either side. > The Luftwaffe Geräte-Handbücher for the FuBl 1 ([[1942_DLuft_T4065_FuBl_I_Geraetehandbuch|D.(Luft) T.4065]]) and the FuBl 2 ([[1943_DLuft_T4058_FuBl_2_Geraete_Handbuch|D.(Luft) T.4058]] §21) both specify receiver Empfindlichkeit operationally, "Sichere optische und akustische Anzeige des Ansteuerungskurses in 200 m Flughöhe ab 70 km Entfernung vom Flughafen" (reliable optical and acoustic indication of the approach course at 200 m altitude from 70 km distance), not as a bench microvolt MDS. Felkin A.D.I.(K) Report 343/1945 §9 describes the FuBl 2 architecture but does not quote a bench MDS either. Jones 1978 pp. 100 to 102 documents the 21 June 1940 Bufton detection flight on a Hallicrafters S-27 but does not quote a receiver sensitivity spec. After working through all four primary sources the position is unambiguous: there is no primary-source bench microvolt sensitivity for the FuBl 1, the FuBl 2, or the S-27 used on the Bufton flight. > > The figures below therefore draw only the primary-source-derivable physics noise floor as the receiver-independent reference line. A signal below the physics noise floor is below the receiver's own thermal plus galactic noise at the antenna input, which means it cannot be distinguished from noise by any receiver with any noise figure, regardless of what the bench sensitivity spec turns out to be. The physics noise floor at the receiver input is $V_{rx,\text{noise}} = \sqrt{10^{N_{\text{floor}}/10} \times 50\ \Omega} \approx 0.195\ \mu\text{V}$ using the same $N_{\text{floor}} = -151.2$ dBW from the noise-floor section above (3 kHz AM voice detection bandwidth, $F_a = 18$ dB ITU-R P.372 galactic noise, 290 K reference temperature). ### Figures #### Master bar chart in microvolts ![[ITU_Calc_sn_vs_grwave_uv_master_bargraph.png]] Phase two of the operational-usability check. Same thirteen paths as the dB master bar chart above, same propagation calculations, same 31.5 MHz galactic plus thermal noise floor — just re-expressed in **absolute voltage at the 50 Ω receiver input in microvolts** instead of dB above the noise floor. Two bars per path: Sommerfeld-Norton FE (green) and ITU-R P.368 GRWAVE (magenta). The conversion between the dB chart and this one is the single identity $V_{rx}\ [\mu\text{V}] = 10^{(\text{SNR}_{\text{dB}}\ -\ 14.2)\, /\, 20}$ which is the same link budget the dB chart uses, differenced against the −151.2 dBW noise floor and re-expressed at a 50 Ω terminated receiver input. No new physics, no new inputs, no new model. Every bar on this chart corresponds to exactly the same path calculation as its counterpart on the dB master, just plotted in volts instead of dB. The white horizontal line at 0.078 μV is the voltage at the receiver input corresponding to the same 0 dB SNR reference used on the dB master, now with the corrected 500 Hz MCW matched-filter bandwidth (see the bandwidth-correction callout higher up). A bar that crosses this line from below means the received signal is now under the physics noise floor at 50 Ω. This chart is useful because receiver sensitivity specifications are conventionally published in microvolts at a specified impedance, so if you had a primary-sourced FuBl 2 or Hallicrafters S-27 bench sensitivity you could read it straight off the y axis. For the BotB analysis neither receiver has a verified bench microvolt spec, so the physics floor is the relevant reference line. #### Kleve director beam in microvolts ![[ITU_Calc_sn_vs_grwave_uv_kleve.png]] Same as the dB Kleve sweep higher up in the doc, just plotted on a log y axis in microvolts at the 50 Ω receiver input. TX 111 m, RX 6,000 m, 31.5 MHz, 3 kW, 26 dBi, average land. Radio horizon at 363 km. The 0.078 μV white line is the noise floor voltage equivalent of 0 dB SNR at the 500 Hz MCW matched-filter bandwidth. Every confirmed Midlands target (Spalding 440, Retford/Beeston 512, Derby 530, Birmingham 551, Liverpool 640 km) sits well above the noise floor on both Sommerfeld-Norton FE (green) and P.368 GRWAVE (magenta), consistent with the Kleve leg being the non-discriminating side of the BotB test. #### Stollberg cross beam in microvolts ![[ITU_Calc_sn_vs_grwave_uv_stollberg.png]] Same as the dB Stollberg sweep higher up, re-plotted on a log y axis in microvolts at the 50 Ω receiver input. TX 72 m, RX 6,000 m, 31.5 MHz, 3 kW, 26 dBi, seawater (best-case ground for surface-wave propagation). Radio horizon at 354 km. **Every confirmed Midlands target (Beeston 694, Derby 710, Birmingham 754, Liverpool 791 km) sits below the 0.078 μV physics noise floor on the globe GRWAVE curve.** The gap between the Sommerfeld-Norton FE curve (green, near 1 mV across the whole sweep) and the P.368 GRWAVE curve (magenta, plunging through the noise floor around 650 km) is the signal the globe diffraction physics removes — the load-bearing discriminator in the BotB test. #### Telefunken July 1939 over-sea tests in microvolts ![[ITU_Calc_sn_vs_grwave_uv_telefunken.png]] Same as the dB Telefunken sweep higher up, re-plotted on a log y axis in microvolts at the 50 Ω receiver input. TX 72 m, RX 4,000 m, 31.5 MHz, 3 kW, 26 dBi, seawater. Radio horizon at 296 km. Primary source [[1939_BArch_RL19-6-40_230Q8_App2_Telefunken_Range_Tests|BArch RL 19-6/40 ref. 230Q8 App. 2]]. The 400 and 500 km test points clear the 0.078 μV physics noise floor on both Sommerfeld-Norton FE (green) and P.368 GRWAVE (magenta). The 700, 800, and 1000 km test points drop below it on the GRWAVE curve, despite the primary source documenting reliable reception at every test distance — which is the primary-source leg of the falsification against the GE family. ### Dan comparison at 2 μV During the 13 Feb 2025 *Drunk Debunk Show* presentation at 2:36:36, DC Dan typed **1 μV (−107 dBm)** into his link-budget tool as the FuBl 1 receiver sensitivity and stated on air that the real figure is "actually one to two microvolts for the fu1". The 1 μV number is uncited and is not traceable to any primary source. The figure below is the master bar chart with one extra reference line added at **2 μV**, using the upper end of Dan's stated range as the most generous defensible assumption from his own argument. It is plotted here only as a comparison against the physics noise floor, not as a primary-sourced receiver spec. #### Master bar chart with 2 μV Dan-comparison line ![[ITU_Calc_sn_vs_grwave_fubl2_master_bargraph.png]] Same thirteen-path μV master bar chart as above, with one extra light-orange dashed reference line at 2 μV matching Dan's on-air sensitivity input. Every Stollberg → Midlands path on the GRWAVE (magenta) column sits below both the 2 μV Dan line and the 0.078 μV physics noise floor by one to four orders of magnitude, so the arbitrary 2 μV constraint does not close the Stollberg gap — the noise floor, not the receiver spec, is the load-bearing bound. Every Kleve → Midlands path and every Telefunken 400 and 500 km path clears both reference lines on both the Sommerfeld-Norton FE column (green) and the GRWAVE column (magenta). ### Verdict against the physics noise floor Setting aside every assumed receiver spec and testing only against the 0.078 μV thermal plus galactic physics noise floor, the result is: - **Kleve → Midlands paths (Spalding, Retford, Derby, Birmingham, Liverpool)**: PASS on both ITU-R globe standards (P.526-16 Fock and P.368 GRWAVE). The Kleve director beam works on the globe, consistent with the 21 June 1940 Bufton detection flight and the year-long 1940 to 1941 operational bombing campaign against the Midlands. - **Stollberg → Midlands paths (Beeston, Derby, Birmingham, Liverpool)**: FAIL on both ITU-R globe standards. Every confirmed target is below the 0.078 μV physics noise floor by one to four orders of magnitude on both P.526 Fock and P.368 GRWAVE. - **Telefunken 400 km and 500 km over-sea tests**: PASS on both ITU-R globe standards. - **Telefunken 700, 800, 1000 km over-sea tests**: FAIL on both ITU globe standards, below the physics noise floor, despite the primary-source documentation of reliable reception. - **Sommerfeld-North flat-Earth baseline**: PASSES every confirmed path by orders of magnitude. **The globe falsification is independent of the receiver specification.** A signal below the receiver's own thermal plus galactic input noise cannot be detected by any receiver with any noise figure, and no assumption about receiver sensitivity, generous or strict, primary-sourced or uncited, closes that gap. The falsification lives in the physics noise floor. --- ## See Also - [[Knickebein_Propagation_Null]] — main null hypothesis document (uses ITU P.526) - [[X_Y_Gerat_Propagation_Null]] — companion null for X-Gerät and Y-Gerät - [[DC_Dan]] — critique dossier, see Critique 6 for the ground-wave rescue argument - [[1943_DLuft_T4058_FuBl_2_Geraete_Handbuch]] — FuBl 2 receiver service manual - [[1939_BArch_RL19-6-40_230Q8_App2_Telefunken_Range_Tests]] — Telefunken July 1939 over-sea range tests - [[2024_Doerenberg_Knickebein_Reference]] — Dörenberg compilation - [[Balanis_Antenna_Theory]] — Balanis for the aperture-to-directivity formula