# ITU-R Recommendation P.526-16 (2025) — Propagation by diffraction **Citation:** International Telecommunication Union. *Recommendation ITU-R P.526-16: Propagation by diffraction*. Geneva: ITU Radiocommunication Sector, 2025. Question ITU-R 202/3. Approval history: 1978, 1982, 1986, 1990, 1992, 1994, 1995, 1997, 1999, 2001, 2003, 2005, 2007, 2009, 2013, 2019, 2025. 49 pages. **Series:** P (Radiowave propagation). **Archive:** ITU Publications, `itu.int/pub/R-REC-P.526`. ## Table of contents ![[2025_ITU-R_P526-16_Diffraction_p02_TOC.png]] The Recommendation is structured in five major sections. The one relevant for over-the-horizon VHF paths on a spherical Earth is: > **§3 Diffraction over a spherical Earth** > §3.1 Diffraction loss for over-the-horizon paths > §3.2 Diffraction loss for any distance at 10 MHz and above The 31.5 MHz Knickebein paths fall squarely inside the §3 scope, and because 31.5 MHz is above 10 MHz the §3.2 general method applies at any distance. ## §3 Diffraction over a spherical Earth — the residue-series intro ![[2025_ITU-R_P526-16_Diffraction_p06_section3_intro_residue_series.png]] Page 6 of the Recommendation opens §3 with the following verbatim statement: > "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." Three points follow from this paragraph: 1. The classical residue series is the canonical solution for diffraction of radio waves around a smooth spherical Earth. The series is derived by separation of variables in spherical coordinates and solving the resulting wave equation with the boundary condition that the field vanishes at infinity and satisfies the impedance boundary condition on the Earth's surface. Each term in the series corresponds to a creeping-wave mode that propagates around the sphere with a complex eigenvalue $\tau_n$. The imaginary part of $\tau_n$ controls the exponential decay of the mode with angular distance past the geometric horizon. 2. The reference "LFMF-SmoothEarth" is the canonical numerical implementation, maintained by ITU-R Study Group 3 and available from the ITU directly. It computes the full residue series with arbitrary numbers of modes. 3. ITU-R P.368 presents tabulated outputs of LFMF-SmoothEarth for ground-wave propagation at lower frequencies and close-to-ground antennas. P.526 and P.368 are complementary: P.526 gives the field-strength equations and approximations usable by hand or short numerical code, P.368 gives the canonical calculator. ## §3.1.1.1 Influence of the electrical characteristics of the surface — the surface admittance K ![[2025_ITU-R_P526-16_Diffraction_p07_surface_admittance_K.png]] Page 7 introduces the normalised surface admittance $K$, which is the dimensionless parameter that controls how much the Earth's ground constants (effective permittivity $\varepsilon$ and conductivity $\sigma$) affect the diffraction loss. For **horizontal polarisation** (practical units): $K_H = 0.36 \, (a_e f)^{-1/3} \, [(\varepsilon - 1)^2 + (18000 \, \sigma / f)^2]^{-1/4} \quad \text{(Eq. 11a)}$ For **vertical polarisation** (practical units): $K_V = K_H \, [\varepsilon^2 + (18000 \, \sigma / f)^2]^{1/2} \quad \text{(Eq. 12a)}$ where: - $a_e$ = effective Earth radius (km) - $\varepsilon$ = effective relative permittivity - $\sigma$ = effective conductivity (S/m) - $f$ = frequency (MHz) ## §3.1.1.1 K nomogram ![[2025_ITU-R_P526-16_Diffraction_p08_K_nomogram.png]] Page 8 (Figure 2) shows the $K$ nomogram for both polarisations across sea, wet ground, medium dry ground, and very dry ground, as a function of frequency from 10 kHz to 10 GHz. The figure includes the critical note: > "If $K$ is less than 0.001, the electrical characteristics of the Earth are not important. For values of $K$ greater than 0.001 and less than 1, the appropriate formulae given in §3.1.1.2 can be used. When $K$ has a value greater than about 1, the diffraction field strength calculated using the method of §3.1.1.2 differs from the results given by the 'LFMF-SmoothEarth', and the difference increases rapidly as $K$ increases. 'LFMF-SmoothEarth' should be used for $K$ greater than 1. This only occurs for vertical polarization, at frequencies below 10 MHz over sea, or below 200 kHz over land. In all other cases the method of §3.1.1.2 is valid." For Knickebein at 31.5 MHz vertical polarisation over sea, $K < 1$, so the §3.1.1.2 first-term residue approximation is valid. ## §3.1.1.2 Diffraction field strength equations 13 to 17 ![[2025_ITU-R_P526-16_Diffraction_p09_eq13_diffraction_field_strength.png]] Page 9 gives the core field-strength equation: $20 \log_{10} \frac{E}{E_0} = F(X) + G(Y_1) + G(Y_2) \quad \text{dB} \quad \text{(Eq. 13)}$ This is the **first-term residue approximation** from the classical residue series. $E / E_0$ is the diffraction field strength relative to the free-space field strength. The quantity on the left side is generally negative, and its magnitude is the Fock smooth-Earth diffraction loss added on top of the free-space path loss. The equation splits the loss into three additive terms: - $F(X)$: the distance term, a function of the normalised path length $X$. - $G(Y_1)$: the height-gain term at the transmitter, a function of the normalised transmitter antenna height $Y_1$. - $G(Y_2)$: the height-gain term at the receiver, a function of the normalised receiver antenna height $Y_2$. The normalisations are given in practical units by: $X = 2.188 \, \beta \, f^{1/3} \, a_e^{-2/3} \, d \quad \text{(Eq. 14a)}$ $Y = 9.575 \times 10^{-3} \, \beta \, f^{2/3} \, a_e^{-1/3} \, h \quad \text{(Eq. 15a)}$ where: - $d$ = path length (km) - $a_e$ = equivalent Earth radius (km) - $h$ = antenna height (m) - $f$ = frequency (MHz) $\beta$ is a parameter allowing for ground type and polarisation. For horizontal polarisation at all frequencies, and for vertical polarisation above 20 MHz over land or 300 MHz over sea, $\beta$ may be taken as 1. Otherwise $\beta$ is computed from $K$ via: $\beta = \frac{1 + 1.6 K^2 + 0.67 K^4}{1 + 4.5 K^2 + 1.53 K^4 + K^2} \quad \text{(Eq. 16)}$ The distance term $F(X)$ is given by two ranges: $F(X) = 11 + 10 \log_{10}(X) - 17.6 \, X \quad \text{for } X \geq 1.6 \quad \text{(Eq. 17a)}$ $F(X) = -20 \log_{10}(X) - 5.6488 \, X^{1.425} \quad \text{for } X < 1.6 \quad \text{(Eq. 17b)}$ For distances past the radio horizon (the operational domain for Knickebein → UK paths), $X > 1.6$ and the $-17.6 X$ term dominates. That $-17.6 X$ term is the **exponential decay of the Fock creeping-wave mode with distance**: each unit increase in the normalised distance $X$ subtracts 17.6 dB from the field strength. This is the shadow-zone exponential decay that defines the smooth-Earth diffraction regime. ## §3.1.1.2 Height gain equations 18 ![[2025_ITU-R_P526-16_Diffraction_p10_eq18_height_gain.png]] Page 10 gives the height-gain term: $G(Y) \approx 17.6 \, (B - 1.1)^{1/2} - 5 \log_{10}(B - 1.1) - 8 \quad \text{for } B > 2 \quad \text{(Eq. 18)}$ $G(Y) \approx 20 \log_{10}(B + 0.1 B^3) \quad \text{for } B \leq 2 \quad \text{(Eq. 18a)}$ where $B = \beta Y$ (Eq. 18b). The height-gain term recovers part of the shadow-zone loss when one or both antennas are elevated above the Earth's surface. For a Knickebein aircraft receiver at 6,000 m altitude, $Y_2$ is large and $G(Y_2)$ can contribute 70 to 90 dB of positive height gain back against the $F(X)$ distance decay. This is the only mechanism by which a beyond-horizon path becomes usable at VHF: the aircraft altitude adds enough positive height-gain dB to partially offset the exponential distance decay. The Recommendation includes a floor on $G(Y)$: > "If $G(Y) < 2 + 20 \log_{10} K$, set $G(Y)$ to the value $2 + 20 \log_{10} K$." This lower-bound clamp prevents the height-gain term from driving the signal below the minimum field strength supportable by the surface-wave mode at the given $K$. ## Accuracy of Eq. 13 — first-term residue approximation The Recommendation notes on 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 $X$, $Y_1$ and $Y_2$ that are constrained by the formula: $X - (\beta Y_1)^{1/2} \Delta(Y_1, K) - (\beta Y_2)^{1/2} \Delta(Y_2, K) > X_{lim}$ (Eq. 19)" For short over-the-horizon paths at VHF with elevated antennas (the Knickebein operational regime) Eq. 13 meets the Eq. 19 validity test and gives the diffraction loss to within 2 dB of the full residue series. For longer paths or different frequency regimes, LFMF-SmoothEarth implements the full multi-mode residue series and is the authoritative calculator. ## See also - [[1945_Fock_Diffraction_Radio_Waves_Earth]] - [[1988_Shatz_Spherical_Earth_Diffraction_SEKE]] - [[1937_Eckersley_Ultra_Short_Wave_Refraction_Diffraction]] - [[2021_ITU-R_P833-10_Vegetation]] - [[2025_ITU-R_P2001-6_Wide_Range_Propagation]] - [[Knickebein_Propagation_Null]] - [[GRWAVE_P368_BotB]]