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# Spherical ground effect and advanced barriers (Attenborough / Salomons / Bies)

The [ISO 9613-2 general method](outdoor-propagation.md) 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`](porous-absorber.md), 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.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/ground_effect_spherical_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/ground_effect_spherical.svg" alt="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" width="90%"></picture>

**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()
```

## Relation to ISO 9613-2

The tabulated `Agr` and `Dz` of the [ISO 9613-2 method](outdoor-propagation.md)
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).
