Source: https://jmrplens.github.io/phonometry/guides/ground-barriers/ # Spherical ground effect and advanced barriers (Attenborough / Salomons / Bies) The [ISO 9613-2 general method](https://jmrplens.github.io/phonometry/reference/api/environment/outdoor-propagation/) folds the ground and barrier terms into tabulated, energy-based corrections. This page covers the underlying wave acoustics in `phonometry.environmental.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 `Agr`/`Dz` 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 `R1` the source-receiver distance and `R2` the image-source distance. The coefficient (Attenborough Eq. 2.40c; Salomons Eq. D.58) corrects the plane-wave coefficient `Rp` for the curvature of the wavefront: $$ Q = R_p + (1 - R_p)\,F(w), \qquad R_p = \frac{Z\cos\theta - 1}{Z\cos\theta + 1}, $$ $$ F(w) = 1 + i\sqrt{\pi}\,w\,e^{-w^2}\operatorname{erfc}(-iw), \qquad w = \sqrt{\tfrac{i k R_2}{2}}\left(\cos\theta + \tfrac{1}{Z}\right). $$ Here `Z` is the ground surface impedance normalized by `ρc`, `θ` is the angle of incidence from the ground normal (`cos θ = (hs + hr)/R2`), and the boundary-loss factor `F(w)` is written through the scaled complementary error function `e^{-w²} 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 `Rp → -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): $$ \Delta L = 20\lg\!\left|\,1 + Q\,\frac{R_1}{R_2}\,e^{i k (R_2 - R_1)}\,\right|. $$ ```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/guides/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^{-iωt}` convention of Salomons, in which a passive ground has `Im(Z) > 0`; the porous models of `phonometry.materials` work in the opposite `e^{+jωt}` convention (`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.

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```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| → ∞`) gives `Rp → 1`, `|w| → 0`, `F → 1` and `Q → 1`, so `ΔL` reaches `+6 dB` in phase; the effective flow resistivity `σ → ∞` tends to that hard ground; grazing incidence (`hs, hr → 0`) gives `Rp → -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 `Dz` 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/λ)(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): $$ \Delta = 5 + 20\lg\!\left(\frac{\sqrt{2\pi N}}{\tanh\sqrt{2\pi N}}\right)\;\text{dB}, $$ which tends to `5 dB` at the shadow boundary `N → 0` and approximates Maekawa's point-source curve within about 1.5 dB. ```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/λ)(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 `Agr` and `Dz` 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.

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```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() ```

## Relation to ISO 9613-2 The tabulated `Agr` and `Dz` of the [ISO 9613-2 method](https://jmrplens.github.io/phonometry/reference/api/environment/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. ## 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).