Spherical ground effect and advanced barriers

The ISO 9613-2 general method 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 resolve the frequency-dependent interference structure the octave-band $A_{gr}$/$D_z$ 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):

$$p=\frac{e^{ik{R}_1}}{4\pi R_1}+Q\frac{e^{ik{R}_2}}{4\pi R_2},$$

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 $R_p$ for the curvature of the wavefront:

$$Q=R_p+(1-R_p)F(w),R_p=\frac{Z\cos \theta -1}{Z\cos \theta +1},$$ $$F(w)=1+i\sqrt{\pi }w{e}^{-w^2}\mathrm{erfc}(-iw),w=\sqrt{\frac{ik{R}_2}{2}}\left(\cos \theta +\frac{1}{Z}\right).$$

Here $Z$ is the ground surface impedance normalized by $\rho c$, $\theta$ is the angle of incidence from the ground normal ($\cos \theta =(h_s+h_r)/R_2$), and the boundary-loss factor $F(w)$ is written through the scaled complementary error function $e^{-w^2}\mathrm{erfc}(-iw)$, 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 $R_p\to -1$ and a plane-wave model would predict silence (Salomons Eqs. D.57, D.59, D.60).

The relative sound level (the excess attenuation, dB re free field) is (Salomons Eq. 3.4):

$$\Delta L=20{\log }_{10}\left|1+Q\frac{R_1}{R_2}{e}^{ik(R_2-R_1)}\right|.$$

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).
# Argument order: frequencies, source_height, receiver_height, distance.
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, 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 $\mathrm{Im}(Z)>0$; the porous models of phonometry.materials work in the opposite $e^{+j\omega t}$ convention ($\mathrm{Im}(Z)<0$), so anything obtained from them (a flow_resistivity or a PorousMediumResult) is conjugated internally before it enters the Weyl-Van der Pol formulas.

Softer ground means a deeper dip at a lower frequency: the dip tracks the effective flow resistivity across more than a decade, and only asphalt stays near the +6 dB hard-ground limit across the speech range. Note the unit — the flow resistivity is a resistivity in Pa·s/m², not a pressure.

Show the code for this figure

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·s·m⁻²)", 10e3, "#2ca02c"),
    ("Forest floor (50 kPa·s·m⁻²)", 50e3, "#9467bd"),
    ("Grassland (200 kPa·s·m⁻²)", 200e3, "#1f77b4"),
    ("Asphalt (20 000 kPa·s·m⁻²)", 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 $R_p\to 1$, $|w|\to 0$, $F\to 1$ 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 ($h_s,h_r\to 0$) gives $R_p\to -1$; and the ground effect is reciprocal under an exchange of source and receiver heights.

The ground wave is easiest to see by taking the geometry towards grazing and watching $R_p$ and $Q$ come apart.

At 3 m of height the two coefficients agree to within a few per cent and the ground wave is irrelevant. By 0.02 m, $R_p$ has reached $-1$ (magnitude 0.993 at 179.6°) and a plane-wave model predicts $-40.1$ dB — near silence. The spherical coefficient reaches $|Q|=1.26$ instead and the field settles at $-11.9$ dB. That 28 dB is what the Faddeeva term $F(w)$ buys, and it is why a grazing outdoor path is audible at all.

Show the code for this figure

import matplotlib.pyplot as plt
import numpy as np

# `ground_effect` is imported by the snippet above.
heights = np.geomspace(3.0, 0.02, 60)
c0, f, d = 343.0, 500.0, 50.0
k = 2 * np.pi * f / c0
q_mag, rp_mag, dl_q, dl_p = [], [], [], []
for h in heights:
    res = ground_effect([f], h, h, d, flow_resistivity=2e5, model="miki")
    rp = res.plane_reflection_coefficient[0]
    q = res.reflection_coefficient[0]
    phase = np.exp(1j * k * (float(res.r_reflected) - float(res.r_direct)))
    ratio = float(res.r_direct) / float(res.r_reflected)
    rp_mag.append(abs(rp))
    q_mag.append(abs(q))
    dl_q.append(20 * np.log10(abs(1 + q * ratio * phase)))
    dl_p.append(20 * np.log10(abs(1 + rp * ratio * phase)))

fig, (left, right) = plt.subplots(1, 2, figsize=(11, 4.6))
left.semilogx(heights, rp_mag, label="|Rp| (plane wave)")
left.semilogx(heights, q_mag, label="|Q| (spherical wave)")
left.invert_xaxis()
left.set(xlabel="Source = receiver height [m]", ylabel="Magnitude")
left.legend()
right.semilogx(heights, dl_q, label="with Q")
right.semilogx(heights, dl_p, label="with Rp only")
right.invert_xaxis()
right.set(xlabel="Source = receiver height [m]",
          ylabel="Level re free field [dB]")
right.legend()
plt.show()

Choosing the ground impedance

Both one-parameter models are being evaluated far below their published fit range for any outdoor ground, and the PorousAbsorberWarning these calls raise says so: the range is $0.01<X<1$ with $X={\rho }_{0}f/\sigma$, and the grassland above sits at $X=3.8\times 10^{-4}$ at 63 Hz and 0.024 at 4 kHz — the asphalt curve of the figure never enters the range at all. Below the range the Delany-Bazley regression is an extrapolation whose classic failure is a negative real part of the surface impedance; Miki is the passivity-constrained refit to prefer there (see the fit-range paragraph of Porous absorbers). The choice is not cosmetic: on the case above, model="miki" moves the excess attenuation by up to 3.6 dB (250 Hz band: 0.43 dB with Delany-Bazley, −3.18 dB with Miki).

Typical effective flow resistivities, the ranges the figure above illustrates: fresh snow ≈ 10, forest floor ≈ 50, grassland and pasture ≈ 150-300, compacted soil and gravel ≈ 1000-3000, asphalt and concrete > 20 000 kPa·s/m².

For a real site, fit $\sigma$ to a short-range level-difference measurement between two microphone heights over the surface (the template method of ANSI/ASA S1.18). The fitted value is an effective, model-bound parameter: it must be quoted with the impedance model it was fitted with, and it is not the EN 29053 airflow resistivity of the same material.

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 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.

A 2D FDTD simulation of a point source behind a thin 2.5 metre rigid barrier on reflecting ground, at 100 Hz and 500 Hz side by side. The long wavelength diffracts over the edge and fills the shadow zone with an insertion loss near 8 dB, while the short wavelength is cast into a deep clean shadow of about 17 dB.

Download the animation (WebM)

A 2D FDTD simulation of a point source behind a thin 2.5 metre rigid barrier on reflecting ground, at 100 Hz and 500 Hz side by side. The long wavelength diffracts over the edge and fills the shadow zone with an insertion loss near 8 dB, while the short wavelength is cast into a deep clean shadow of about 17 dB.

Download the animation (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.

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 $e$) moves the second diffraction edge downstream and 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. The improvement is monotone but it is small, and the reason is geometric: $B$ is now measured from the far edge, so most of what the top width adds it also gives back. On the 4 m screen of this page, a 4 m top width is worth 0.08 dB at 500 Hz and 0.14 dB at 8 kHz; even 30 m of width only reaches 1.2 dB.

The path-length effect alone is worth a fraction of a decibel at road-traffic distances. ISO 9613-2 credits a well-separated double edge with a separate factor $C_3$ that rises from 1 towards 3, worth up to $10{\log }_{10}3\approx 4.8$ dB, which the wave-theoretic model here does not include: for the same 4 m top width $D_z$ goes from 7.6 dB to 9.0 dB at 500 Hz. Neither number licenses building a berm and expecting the ISO bonus from this page’s model.

Show the code for this figure

import matplotlib.pyplot as plt
import numpy as np

# `barrier_insertion_loss` is imported by the snippet above.
widths = np.linspace(0.0, 30.0, 40)
fig, ax = plt.subplots()
for freq in (250.0, 500.0, 1000.0):
    thin = float(barrier_insertion_loss(
        [freq], 1.0, 50.0, 4.0, 100.0, 1.5, method="exact").insertion_loss[0])
    gain = [0.0 if e == 0.0 else float(barrier_insertion_loss(
        [freq], 1.0, 50.0, 4.0, 100.0, 1.5, method="exact",
        thickness=e).insertion_loss[0]) - thin for e in widths]
    ax.plot(widths, gain, label=f"{freq:g} Hz")
ax.set_xlabel("Top width e [m]")
ax.set_ylabel("Gain over the thin screen [dB]")
ax.legend()
plt.show()

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 $A_{gr}$ 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.

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.

The two thin-screen models never separate by more than about 1.5 dB, so the choice between them barely matters. The coherent ground curve is a different kind of object: it swings from above +45 dB near 230 Hz, where the barrier removes the unscreened path’s own ground dip, to below −10 dB near 550 Hz, where the four diffracted paths cancel. Both extremes are narrowband and neither is a design value.

Show the code for this figure

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.

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

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()

ground_effect() and barrier_insertion_loss() parameters

Both entry points take their distances along the ground from the source, and neither takes a distance from the barrier. That is the trap worth naming: in barrier_insertion_loss(bands, 1.0, 50.0, 4.0, 100.0, 1.5) the receiver is 100 m from the source, i.e. 50 m past the screen, not 100 m past it. The two functions also spell the same option differently — model= on ground_effect, ground_model= on barrier_insertion_loss — and order their geometry differently, which is why the two positional calls on this page are not parallel.

ground_effect parameter	Type / shape	Units	Range / default	Notes
frequencies	scalar or 1D array	Hz	> 0	Narrowband: no band smoothing
source_height	float	m	≥ 0	$h_s$
receiver_height	float	m	≥ 0	$h_r$ (before the distance)
distance	float	m	> 0	Horizontal source-receiver distance
impedance	array, PorousMediumResult or None	—	None	Normalized $Z$ in the $e^{-i\omega t}$ convention
flow_resistivity	float or None	Pa·s/m²	None	Effective $\sigma$; feeds the porous model
model	"delany_bazley" | "miki"	—	"delany_bazley"	Prefer "miki" below the fit range
speed_of_sound / air_density	float	m/s, kg/m³	343.0 / 1.205

barrier_insertion_loss parameter	Type / shape	Units	Range / default	Notes
frequencies	scalar or 1D array	Hz	> 0
source_height	float	m	≥ 0	$h_s$
barrier_distance	float	m	> 0	Source to the (near) edge, along the ground
barrier_height	float	m	> 0	Edge height above the ground
receiver_distance	float	m	> barrier_distance (+ thickness)	Measured from the source
receiver_height	float	m	≥ 0	$h_r$ (after the distance)
method	"kurze_anderson" | "exact"	—	"exact"	"exact" is the rigid half-plane
thickness	float or None	m	None	Top width $e$; double edge
ground_impedance	array, PorousMediumResult or None	—	None	Selects the coherent four-path model
ground_flow_resistivity	float or None	Pa·s/m²	None	Same, from an effective $\sigma$
ground_model	"delany_bazley" | "miki"	—	"delany_bazley"	Note the name: not model=
speed_of_sound / air_density	float	m/s, kg/m³	343.0 / 1.205

A ground argument requires method="exact"; method="kurze_anderson" with a ground raises ValueError.

Choosing a model, and what none of them models

Which one to use. Kurze-Anderson for an octave-band engineering estimate, where its ±1.5 dB against the exact solution is far inside the uncertainty of everything else in the chain. The exact rigid half-plane when the behaviour near the shadow boundary matters (it gives 6 dB there against Kurze-Anderson’s 5 dB, and the 6 dB is the physically correct halved field). The coherent four-path model only when the narrowband interference structure is the question — never raw as a design value.

The coherent curve must be averaged before it means anything. It is a coherent, narrowband, fully deterministic result. Real atmospheric turbulence decorrelates the four paths, a real source has bandwidth, and a real receiver moves; all three wash out the extremes. A +45 dB narrowband insertion loss is not a barrier that removes 45 dB, it is a frequency at which the unscreened path happened to sit in its own ground dip. Band-average the curve, or read it as structure rather than as a level.

All four models are two-dimensional endless screens. A real barrier of finite length lets sound flank around its ends, and two parallel barriers either side of a road reflect between each other and lose several decibels of their nominal performance. Neither effect exists in any model on this page.

Diffraction is the screening limit, not the realised one. The insertion loss a built barrier delivers is the smaller of the diffraction result and the transmission loss of the panel itself, and the obstacle has to qualify as a barrier in the first place — surface density, a closed surface, and a horizontal extent normal to the path larger than the wavelength. Those qualifying rules are stated in the “Barrier pitfalls” section of Outdoor Sound Propagation; they apply here unchanged.

Relation to ISO 9613-2

The Kurze-Anderson insertion loss written $\Delta$ here is the same quantity ISO 9613-2 calls $D_z$, and the two diffracted segments called $A$ and $B$ here are its $d_{ss}$ and $d_{sr}$. The tabulated $A_{gr}$ and $D_z$ of the ISO 9613-2 method 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, the Faddeeva-function boundary-loss factor), with its hard-ground +6 dB, grazing-incidence and reciprocity limits pinned as tests. Wave-theoretic barrier diffraction (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 reflection coefficient.

• Not covered

  Both models assume a homogeneous, non-refracting, non-turbulent atmosphere; a vertical sound-speed gradient is the subject of the separate atmospheric refraction guide 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 rather than a separate coefficient per image path, so it is coherent and reciprocal but not a full boundary-element solution.

See also

• Outdoor Sound 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: the vertical sound-speed gradient both models here assume away.
• Porous absorbers: the Delany-Bazley and Miki models behind flow_resistivity, and their fit range.
• Theory: Outdoor propagation: the ground and barrier terms of ISO 9613-2 in the context of the whole attenuation sum.
• API reference: environment.propagation.ground_barriers.

References

• Attenborough, K., & Van Renterghem, T. (2021). Predicting outdoor sound (2nd ed.). CRC Press. https://doi.org/10.1201/9780429470806Chapter 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). ISBN 978-1-4987-4007-4.
• Bies, D. A., Hansen, C. H., & Howard, C. Q. (2017). Engineering noise control (5th ed.). CRC Press. Sections 5.2.3 (spherical-wave ground reflection) and 5.3.5-5.3.7 (Fresnel number, Kurze-Anderson, thin- and thick-barrier diffraction). ISBN 978-1-4987-2405-0.
• Hadden, W. J., & Pierce, A. D. (1981). Sound diffraction around screens and wedges for arbitrary point source locations. The Journal of the Acoustical Society of America, 69(5), 1266-1276. https://doi.org/10.1121/1.385809The exact wedge-diffraction solution whose flat-wedge (thin-screen) limit the barrier insertion loss uses.
• Kurze, U. J., & Anderson, G. S. (1971). Sound attenuation by barriers. Applied Acoustics, 4(1), 35-53. https://doi.org/10.1016/0003-682X(71)90024-7The closed-form fit to Maekawa's chart in the Fresnel number.
• Salomons, E. M. (2001). Computational atmospheric acoustics. Kluwer Academic. https://doi.org/10.1007/978-94-010-0660-6Chapter 3 and Appendix D: the two-ray field, the plane- and spherical-wave reflection coefficients, and the numerical distance (Eqs. D.57-D.60). ISBN 978-1-4020-0390-5.