Source: https://jmrplens.github.io/phonometry/environment/propagation/ground-barriers/ # Spherical ground effect and advanced barriers (Attenborough / Salomons / Bies) The [ISO 9613-2 general method](https://jmrplens.github.io/phonometry/environment/propagation/outdoor-propagation/) folds the ground and barrier terms into tabulated, energy-based corrections. This page covers the underlying wave acoustics in `phonometry.environment.propagation.ground_barriers`: the **spherical-wave reflection coefficient** of a finite-impedance ground (Weyl-Van der Pol) and the **wave-theoretic diffraction** of a screen, both in a homogeneous (non-refracting, non-turbulent) atmosphere. These are the physical core of the Nord2000 / CNOSSOS ground and barrier models, and they show the frequency-dependent interference structure the octave-band $A_\mathrm{gr}$/$D_z$ terms smooth away. ## 1. Spherical-wave ground effect (Weyl-Van der Pol) The sound field of a point source above a locally reacting ground is the sum of a direct wave and a reflected wave weighted by the **spherical-wave reflection coefficient** $Q$ (Attenborough Eq. 2.40a; Salomons Eq. 3.2): $$ p = \frac{e^{ikR_1}}{4\pi R_1} + Q\,\frac{e^{ikR_2}}{4\pi R_2}, $$ with $R_1$ the source-receiver distance and $R_2$ the image-source distance. The coefficient (Attenborough Eq. 2.40c; Salomons Eq. D.58) corrects the plane-wave coefficient $R_\mathrm{p}$ for the curvature of the wavefront: $$ Q = R_\mathrm{p} + (1 - R_\mathrm{p})\,F(w), \qquad R_\mathrm{p} = \frac{Z\cos\theta - 1}{Z\cos\theta + 1}, $$ $$ F(w) = 1 + i\sqrt{\pi}\,w\,e^{-w^2}\operatorname{erfc}(-iw), \qquad w = \sqrt{\tfrac{i k R_2}{2}}\left(\cos\theta + \tfrac{1}{Z}\right). $$ Here $Z$ is the ground surface impedance normalized by $\rho c$, $\theta$ is the angle of incidence from the ground normal ($\cos\theta = (h_\mathrm{s} + h_\mathrm{r})/R_2$), and the boundary-loss factor $F(w)$ is written through the scaled complementary error function $e^{-w^2}\operatorname{erfc}(-iw)$, i.e. the Faddeeva function `scipy.special.wofz`. The second term of $Q$ is the *ground wave* that keeps the field finite at grazing incidence, where $R_\mathrm{p} \to -1$ and a plane-wave model would predict silence (Salomons Eq. D.59, D.60, D.57). The relative sound level (the *excess attenuation*, dB re free field) is (Salomons Eq. 3.4): $$ \Delta L = 20\log_{10}\!\left|\,1 + Q\,\frac{R_1}{R_2}\,e^{i k (R_2 - R_1)}\,\right|. $$ ```python import numpy as np from phonometry import ground_effect bands = np.array([63., 125., 250., 500., 1000., 2000., 4000., 8000.]) # Grassland (effective flow resistivity sigma = 200 kPa.s/m^2), source 1 m and # receiver 1.5 m high, 50 m apart. The impedance comes from the Delany-Bazley # porous model of phonometry.materials (a semi-infinite ground). res = ground_effect(bands, 1.0, 1.5, 50.0, flow_resistivity=2e5) print(res.excess_attenuation) # the ground dip (dB re free field) print(res.reflection_coefficient) # complex Q per band res.plot() # excess attenuation vs frequency ``` The ground impedance is either derived from an effective `flow_resistivity` (via the `delany_bazley` or `miki` model of [`phonometry.materials`](https://jmrplens.github.io/phonometry/materials/absorbers/porous-absorbers/), which model a semi-infinite porous ground) or supplied directly as a normalized complex `impedance` (a scalar, per-band array, or a `PorousMediumResult`). A plain `impedance` value is taken in the $e^{-i\omega t}$ convention of Salomons, in which a passive ground has $\operatorname{Im}(Z) > 0$; the porous models of `phonometry.materials` work in the opposite $e^{+j\omega t}$ convention ($\operatorname{Im}(Z) < 0$), so anything obtained from them (a `flow_resistivity` or a `PorousMediumResult`) is conjugated internally before it enters the Weyl-Van der Pol formulas. Excess attenuation (level re free field) against frequency on a log axis for four ground types. Fresh snow (10 kPa) dips deepest and lowest in frequency, near minus 18 dB around 150 Hz; forest floor (50 kPa) reaches about minus 15 dB near 290 Hz and grassland (200 kPa) about minus 12 dB near 540 Hz; asphalt (20000 kPa) hugs the plus 6 dB hard-ground enhancement limit until a deep dip near 2.4 kHz. A dotted line marks the plus 6 dB hard-ground limit and a solid line the 0 dB free field
Show the code for this figure ```python import matplotlib.pyplot as plt import numpy as np from phonometry import ground_effect freqs = np.geomspace(50.0, 4000.0, 400) grounds = [ ("Fresh snow (10 kPa)", 10e3, "#2ca02c"), ("Forest floor (50 kPa)", 50e3, "#9467bd"), ("Grassland (200 kPa)", 200e3, "#1f77b4"), ("Asphalt (20000 kPa)", 20000e3, "#d62728"), ] fig, ax = plt.subplots(figsize=(11, 6.4)) for label, sigma, color in grounds: res = ground_effect(freqs, 1.0, 1.5, 50.0, flow_resistivity=sigma) ax.plot(freqs, res.excess_attenuation, color=color, label=label) ax.axhline(6.0, color="k", ls=":", label="Hard-ground limit (+6 dB)") ax.axhline(0.0, color="k", lw=0.8) ax.set_xscale("log") ax.set_xlabel("Frequency [Hz]") ax.set_ylabel("Level re free field [dB]") ax.legend() plt.show() ```
**Limits reproduced by the implementation** (each a pinned test or conformance anchor): an acoustically hard ground ($|Z| \to \infty$) gives $R_\mathrm{p} \to 1$, $|w| \to 0$, $F \to 1$ and $Q \to 1$, so $\Delta L$ reaches +6 dB in phase; the effective flow resistivity $\sigma \to \infty$ tends to that hard ground; grazing incidence ($h_\mathrm{s}, h_\mathrm{r} \to 0$) gives $R_\mathrm{p} \to -1$; and the ground effect is reciprocal under an exchange of source and receiver heights. ## 2. Advanced barrier diffraction Three levels of screening beyond the ISO 9613-2 $D_z$ term are provided by `barrier_insertion_loss` and its building blocks. **Kurze-Anderson closed form.** The insertion loss of a thin screen as a function of the Fresnel number $N = (2/\lambda)(A + B - d)$ (Bies Eq. 5.134, with $A$ and $B$ the two segments of the shortest source-edge-receiver path and $d$ the straight distance) is (Bies Eq. 5.138; Kurze & Anderson 1971): $$ \Delta = 5 + 20\log_{10}\!\left(\frac{\sqrt{2\pi N}}{\tanh\sqrt{2\pi N}}\right)\;\text{dB}, $$ which tends to 5 dB at the shadow boundary $N \to 0$ and approximates Maekawa's point-source curve within about 1.5 dB. The clip below is that formula as a field. It is the [2D FDTD solver](https://jmrplens.github.io/phonometry/simulation/fdtd-simulation/) run twice on one 12 × 7 m half-space over rigid ground with a thin rigid screen 2.5 m tall, once at 100 Hz and once at 500 Hz, each with a barrier-free reference run over the same ground so the annotated insertion loss is a true one. The geometry fixes the path difference at 1.06 m for the receiver it marks, so the Fresnel number is $N = 0.62$ at 100 Hz and $N = 3.1$ at 500 Hz — the same screen, a factor of five apart in $N$ purely because $\lambda$ changed — and the field shows what that buys: about 8 dB against about 17 dB. Two things are worth watching for. The edge of the lit region running down from the top of the screen is the shadow boundary, the $N \to 0$ locus where the formula bottoms out at 5 dB; and inside the shadow the field is a cylindrical wave centred on the top of the screen, which is what "the edge acts as a secondary source" looks like. One caveat: the ground in the clip is perfectly rigid, so it shows diffraction alone and none of the finite-impedance ground effect of section 1 — the coherent four-path model below adds that, and its curve swings tens of decibels where this one is smooth. Animation: a point source behind a thin 2.5 metre rigid barrier on reflecting ground, simulated at 100 Hz and 500 Hz side by side; the long wavelength diffracts over the edge and fills the shadow zone, the short wavelength is cast into a deep clean shadow [Watch the high-resolution video (WebM)](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_fdtd_barrier.webm) The thin-screen methods share the same three geometric quantities: the two diffracted segments over the edge and the straight path they replace. Drawn on the 4 m screen of the snippets, they differ by just 0.15 m. Section of the barrier geometry: a loudspeaker source 1 m above the ground, a thin 4 m screen at 50 m and a microphone receiver 1.5 m high at 100 m, the blocked direct path of 100.00 m drawn dashed through the screen and the diffracted path bent over the edge in two segments A = 50.09 m and B = 50.06 m, with the resulting path difference of 0.15 m giving a Fresnel number of 0.44 and a Kurze-Anderson insertion loss of 10.0 dB at 500 Hz, rising to 15.5 dB at 2 kHz ```python from phonometry import barrier_insertion_loss, kurze_anderson_attenuation kurze_anderson_attenuation(0.0) # 5.0 dB at the shadow boundary # A 4 m barrier 50 m from a 1 m source, receiver 1.5 m high at 100 m. il = barrier_insertion_loss(bands, 1.0, 50.0, 4.0, 100.0, 1.5, method="kurze_anderson") il.plot() # insertion loss vs frequency ``` **Exact rigid half-plane.** With `method="exact"` the wave-theoretic insertion loss of a rigid thin screen is used: the compact Fresnel-integral form of the MacDonald / Hadden & Pierce solution (Attenborough Eqs. 9.19-9.20), built from the auxiliary Fresnel functions. It gives 6 dB at the shadow boundary (the field is exactly halved, the flat-wedge limit) and tracks Kurze-Anderson through the shadow zone. **Thick barriers.** A `thickness` (top width $e$) lengthens the diffracted path to $A + e + B$, the double-edge Fresnel number $N = (2/\lambda)(A + B + e - d)$ of Bies Eq. 5.157, so a thick barrier or a soil mound attenuates monotonically more than the thin screen of the same height. **Coherent barrier on the ground.** With a `ground_impedance` (or a `ground_flow_resistivity`) the four source-image / receiver-image diffracted paths are combined coherently, each ground reflection weighted by the spherical-wave coefficient $Q$ above (Attenborough Ch. 9; Bies Sec. 5.3.5). This exposes the ground-barrier interference structure that a purely energetic sum of $A_\mathrm{gr}$ and $D_z$ cannot. As a first-order simplification a single $Q$ (over the overall source-receiver geometry) weights every bounce rather than a separate coefficient per image path; the model is coherent and reciprocal but not a full boundary-element solution. ```python il = barrier_insertion_loss(bands, 1.0, 50.0, 4.0, 100.0, 1.5, method="exact", ground_flow_resistivity=2e5) il.ground # True: the four-path coherent ground model was applied il.plot() ``` The three models side by side on the same geometry tell the whole story: the Kurze-Anderson fit and the exact half-plane track each other to within about 1.5 dB across two decades of Fresnel number, while the coherent ground model swings tens of decibels around them, up where the barrier removes the ground-effect dip of the unscreened path, down where the four diffracted paths interfere destructively. Barrier insertion loss against frequency on a log axis for a 4 m screen between a 1 m source at 50 m and a 1.5 m receiver at 100 m. The dashed Kurze-Anderson curve and the solid exact rigid half-plane curve rise together from about 6 dB at 50 Hz to 19 dB at 5 kHz, never more than about 1.5 dB apart; the coherent four-path ground curve oscillates around them, peaking above 45 dB near 230 Hz where the unscreened ground dip is removed and falling below minus 10 dB near 550 Hz, with the dotted Kurze-Anderson 5 dB grazing-limit line underneath
Show the code for this figure ```python import matplotlib.pyplot as plt import numpy as np from phonometry import barrier_insertion_loss # The 4 m barrier of the snippets above, on a fine frequency grid. freqs = np.geomspace(50.0, 5000.0, 240) il_ka = barrier_insertion_loss(freqs, 1.0, 50.0, 4.0, 100.0, 1.5, method="kurze_anderson") il_ex = barrier_insertion_loss(freqs, 1.0, 50.0, 4.0, 100.0, 1.5, method="exact") il_gr = barrier_insertion_loss(freqs, 1.0, 50.0, 4.0, 100.0, 1.5, method="exact", ground_flow_resistivity=2e5) fig, ax = plt.subplots(figsize=(11, 6.4)) ax.semilogx(freqs, il_ka.insertion_loss, "--", label="Kurze-Anderson (thin screen)") ax.semilogx(freqs, il_ex.insertion_loss, label="Exact rigid half-plane") ax.semilogx(freqs, il_gr.insertion_loss, label="Exact + coherent ground (four paths)") ax.axhline(5.0, color="k", ls=":", label="Kurze-Anderson grazing limit (5 dB)") ax.set(xlabel="Frequency [Hz]", ylabel="Insertion loss [dB]") ax.legend() plt.show() ```
All three curves share the same base geometry, and `.plot_geometry()` draws it to scale: at these road-traffic distances the 4 m screen is a sliver. For the thin-screen curves the 0.15 m path difference combines with wavelength in the Fresnel number; the coherent-ground curve also depends on ground impedance and image-path interference. To-scale section of the barrier geometry of the insertion-loss curves: a source star 1 m above the hatched ground, a thin 4 m screen at 50 m, and a receiver triangle 1.5 m high at 100 m, with the dashed direct path cut by the screen, the solid diffracted path bent over its top edge, the path difference of 0.15 m annotated and the 50 m and 100 m distances dimensioned *Drawn to scale the screen almost vanishes: the diffracted path over the top is only 0.15 m longer than the blocked direct path, and that difference is the geometric input of the Fresnel number $N = (2/\lambda)(A + B - d)$ that the thin-screen methods above key on.*
Show the code for this figure ```python import matplotlib.pyplot as plt import numpy as np from phonometry import barrier_insertion_loss freqs = np.geomspace(50.0, 5000.0, 240) il = barrier_insertion_loss(freqs, 1.0, 50.0, 4.0, 100.0, 1.5) # One line: the section to scale, with the path-length difference annotated. il.plot_geometry() plt.show() ```
## Relation to ISO 9613-2 The tabulated $A_\mathrm{gr}$ and $D_z$ of the [ISO 9613-2 method](https://jmrplens.github.io/phonometry/environment/propagation/outdoor-propagation/) are octave-band, energy-based engineering fits; `ground_effect` and `barrier_insertion_loss` are their narrowband wave-acoustic counterparts. Over hard ground both agree on the +6 dB enhancement and the 5 dB grazing barrier floor, but only the wave models resolve the interference dips that move with geometry, frequency and ground impedance, which is why they are the natural infrastructure for the meteorological schemes of Nord2000 and CNOSSOS. ## What this guide covers **Covered.** The Weyl-Van der Pol spherical-wave ground reflection coefficient (`ground_effect`; Attenborough Eq. 2.40a/c, Salomons Eq. 3.2/D.58, with the Faddeeva-function boundary-loss factor), its hard-ground +6 dB, grazing-incidence and reciprocity limits pinned as tests; and wave-theoretic barrier diffraction in `barrier_insertion_loss` — the Kurze-Anderson closed form (`kurze_anderson_attenuation`, Bies Eq. 5.138), the exact rigid half-plane (MacDonald / Hadden & Pierce, Attenborough Eqs. 9.19-9.20), thick barriers through the double-edge Fresnel number (Bies Eq. 5.157) and the coherent four-path barrier-on-ground model weighted by the `ground_effect` coefficient. **Not covered.** Both models assume a homogeneous, non-refracting, non-turbulent atmosphere; a vertical sound-speed gradient is the subject of [atmospheric refraction](https://jmrplens.github.io/phonometry/environment/propagation/atmospheric-refraction/) instead. The coherent barrier-on-ground model weights all four diffracted paths with a single reflection coefficient $Q$ computed over the overall source-receiver geometry, not with a separate coefficient per image path — so it is coherent and reciprocal, but it is not a boundary-element solution. ## See also - [Outdoor Sound Propagation](https://jmrplens.github.io/phonometry/environment/propagation/outdoor-propagation/): the octave-band ISO 9613-2 fits these models sit underneath, and the rule an obstacle has to satisfy to be a barrier at all. - [Atmospheric refraction: rays and the GFPE](https://jmrplens.github.io/phonometry/environment/propagation/atmospheric-refraction/): the vertical sound-speed gradient both models here assume away. - [Porous absorbers](https://jmrplens.github.io/phonometry/materials/absorbers/porous-absorbers/): the Delany-Bazley and Miki models behind `flow_resistivity`, and their fit range. - Theory: [Outdoor propagation](https://jmrplens.github.io/phonometry/reference/theory/environment-transport/#outdoor-propagation-general-method-iso-9613-2): the ground and barrier terms of ISO 9613-2 in the context of the whole attenuation sum. - API reference: [`environment.propagation.ground_barriers`](https://jmrplens.github.io/phonometry/reference/api/environment/ground-barriers/). ## References - Attenborough, K., & Van Renterghem, T. (2021). *Predicting Outdoor Sound* (2nd ed.). CRC Press. ISBN 978-1-138-30655-2. [doi:10.1201/9780429470141](https://doi.org/10.1201/9780429470141). Chapter 2 (spherical-wave reflection over an impedance ground, the Weyl-Van der Pol equation and the boundary-loss factor) and Chapter 9 (outdoor noise barriers, the MacDonald and Hadden & Pierce diffraction solutions). - Salomons, E. M. (2001). *Computational Atmospheric Acoustics*. Kluwer Academic. ISBN 978-1-4020-0390-5. [doi:10.1007/978-94-010-0660-6](https://doi.org/10.1007/978-94-010-0660-6). Chapter 3 and Appendix D (the two-ray field, the plane- and spherical-wave reflection coefficients, and the numerical distance). - Bies, D. A., Hansen, C. H., & Howard, C. Q. (2017). *Engineering Noise Control* (5th ed.). CRC Press. ISBN 978-1-4987-2405-0. Sections 5.2.3 (spherical-wave ground reflection) and 5.3.5-5.3.7 (Fresnel number, Kurze-Anderson, thin- and thick-barrier diffraction and terrain shielding). - Kurze, U. J., & Anderson, G. S. (1971). Sound attenuation by barriers. *Applied Acoustics*, 4(1), 35-53. [doi:10.1016/0003-682X(71)90024-7](https://doi.org/10.1016/0003-682X(71)90024-7). - Hadden, W. J., & Pierce, A. D. (1981). Sound diffraction around screens and wedges for arbitrary point source locations. *Journal of the Acoustical Society of America*, 69(5), 1266-1276. [doi:10.1121/1.385809](https://doi.org/10.1121/1.385809).