Maekawa explorer

How do you use the tool?

Explore the diffraction of sound over one or two edges. You enter the absolute heights of source, barrier and receiver plus the horizontal distances; the tool shows how attenuation varies, comparing five methods on the same geometry: ISO 9613-2:2024, ISO 9613-2:1996, CNOSSOS-EU:2015 under homogeneous and favourable conditions, and Maekawa 1968. The section is drawn to scale with numbered axes, and source, tops and receiver can be dragged.

Interactive exploration of the diffraction of a sound wave around a single edge (knife-edge). It shows how attenuation depends on geometry (barrier height, source-barrier-receiver distances) and on frequency, comparing on the same geometry the three diffraction methods of the Barrier calculator: ISO 9613-2:2024, ISO 9613-2:1996 and Maekawa 1968.

Geometry is entered as absolute heights above ground level — source, barrier and receiver height, plus the horizontal distances. The excess h of the edge over the source-receiver line is not an input: it is a result, together with the path difference δ and the Fresnel number N. Source, barrier tops and receiver can be dragged directly in the diagram, which carries two numbered axes, each with its own scale: the horizontal one is set by the source-receiver distance and does not move when a height changes, the vertical one steps up through round values, always keeping five metres above the tallest element.

With a double barrier, the C3 term of ISO 9613-2 comes into play — it depends on the distance e between the two edges — and the attenuation limit becomes 25 dB instead of the 20 dB of a single edge. Only edges that actually protrude count: if one of the two barriers stays below the chord joining the source to the top of the other, the sound does not see it and the calculation drops it — adding its leg would inflate the path difference, and with it the attenuation. Maekawa 1968 has no C3 term: for double diffraction it receives the total path difference, and remains the coarsest of the three approximations.

CNOSSOS-EU:2015 — the European common method (Directive (EU) 2015/996, Annex II §VI.4.4.b) is not a variant of the ISO editions: the coefficient is 40/λ instead of 20/λ, there is no meteorological factor K_met, and there are two propagation conditions. Under homogeneous conditions the rays are straight; under favourable conditions they are circular arcs of radius Γ = max(1000, 8·d), so the ray clears the edge higher up, δ shrinks and the barrier performs worse — the well-known result that a barrier is less effective downwind, obtained here from the geometry rather than from a correction term. The difference shows up over long distances: at 50 m the two values coincide, at 500 m with an 8 m barrier the tool gives 11.2 dB under homogeneous conditions against 4.8 under favourable ones.

One note that prevents a false alarm: for a single edge under homogeneous conditions, CNOSSOS coincides with Maekawa 1968, because 10·log10(3 + 40δ/λ) is the same formula written with N = 2δ/λ. The two part company with multiple diffraction, where C″ enters, and under favourable conditions. Of CNOSSOS, only the Δ_dif term is computed here: combining it with the ground effect requires the full propagation model, and that lives in Barrier calculator and Acoustic map.

Results leave the tool in two ways: copy to the clipboard as text, and save as PNG — the drawing with the numerical summary underneath. Text and image are built from the same values, so they cannot tell two different stories.

It is the same calculation engine (Barrier calculator, Acoustic map), isolated here in didactic form.

Which formulas describe the diffraction?

The path difference is the length of the diffracted path minus the direct one, on the real heights: δ = √(d_s²+(h_b−h_s)²) + √(d_r²+(h_b−h_r)²) − √((d_s+d_r)²+(h_r−h_s)²). With two edges the leg between the tops is added as well. The wavelength is λ = c/f (c = 343 m/s) and the Fresnel number N = 2δ/λ. The methods apply different formulas to δ: ISO 9613-2:2024 and 1996 (20 dB cap for a single edge, 25 for a double one), CNOSSOS-EU:2015 with 10·log₁₀(3 + (40/λ)·C″·δ) and a 25 dB cap, Maekawa 1968 with 10·log₁₀(3 + 20·N) and a 25 dB cap. For δ ≤ 0 all give 0 dB.

Path difference:

with h_s, h_b and h_r the heights of source, barrier top and receiver above ground level, and d_s, d_r the horizontal source-barrier and barrier-receiver distances. The excess h of the edge over the source-receiver line — the parameter of the classical formulation — follows from the same geometry: h = h_b − [h_s + (h_r − h_s)·d_s/(d_s+d_r)].

With two barriers the leg between the two tops is added as well, and the path difference becomes:

where e is the distance between the two edges and d = d_s + e + d_r the total horizontal distance. Only protruding edges enter the calculation: if one barrier stays below the chord joining the source to the top of the other, its leg must not be added.

Wavelength and Fresnel number:

The five methods apply different attenuation formulas to δ:

• ISO 9613-2:2024 — Dz = 10·log₁₀[1 + (2 + (C₂/λ)·C₃·z)·Kmet], with a z_min threshold and the Kmet weather correction; cap 20 dB (single edge).
• 1996 (legacy) — Dz = 10·log₁₀(3 + (C₂/λ)·C₃·z·Kmet); cap 20 dB. Based on the 1996 formulation but not canonical: it omits a term of Kmet, and is kept for comparison with the tool's historic results.
• CNOSSOS-EU:2015 — Δ_dif = 10·log₁₀(3 + (40/λ)·C″·δ) when (40/λ)·C″·δ ≥ −2, otherwise 0; C″ = 1 for a single edge and [1+(5λ/e)²]/[1/3+(5λ/e)²] for multiple edges; 25 dB cap. Under homogeneous conditions the rays are straight; under favourable conditions they are arcs of radius Γ = max(1000, 8·d) and δ is computed on the arcs. Directive (EU) 2015/996, Annex II §VI.4.4.b.
• Maekawa 1968 — A = 10·log₁₀(3 + 20·N); 25 dB cap. For a single edge under homogeneous conditions it coincides with CNOSSOS: 3 + 20·N and 3 + 40δ/λ are the same expression.

The complete formulas (C₂/C₃ constants, K_met) are in the “Diffraction formulas” section of the Barrier calculator. For N ≤ 0 (clear line of sight) all methods give 0 dB.

Top-edge diffractors

A straight barrier acts on a single edge (knife-edge): the diffracted sound makes one diffraction pass at its top. By modifying the top profile — adding a T, Y, cylindrical, or absorptive cap — the edge becomes multiple: the sound makes two or more diffraction passes, each introducing additional attenuation.

Rigorous ISO 9613-2 multi-edge formulas

The double diffraction of a barrier with a cap (T, Y, cylindrical profile) is formally codified in ISO 9613-2:1996 §7.4 (with minor extensions in the 2024 edition). The rigorous calculation applies:

where:

• δ — path difference between diffracted and direct sound (m)
• d_ss, d_sr — source-first-edge and last-edge-receiver distances
• e — distance between the two cap diffraction edges (m)
• d — direct source-receiver distance (m)
• λ — wavelength at the considered frequency
• C₂ = 20 (includes reflected ground effect) or 40 (ground modelled separately)
• C₃ = [1+(5λ/e)²]/[1/3+(5λ/e)²] for double diffraction (C₃ = 1 for single)
• K_met — meteorological correction (1.0 in neutral conditions)

ISO 9613-2 caps the maximum attenuation at 25 dB for double diffraction (vs 20 dB for single-edge).

What the tool computes and what it does not

The cap geometry is computed. The edges the cap introduces are added to the barrier’s and go through the same filter of protruding edges; from there come δ and the distance e between the first and last edge — exactly what C₃ (ISO) and C″ (CNOSSOS) need. The four profiles enter as follows:

• T — two edges at the same height as the top, separated by the cap width.
• Y — two edges at the tips of arms at 45°, hence higher than the top by w/2·sin45° and w/2·√2 apart.
• Cylinder — a circular profile with its axis on the top, sampled at eleven points: it protrudes by w/2, and the diffracted path follows the arc between the two tangency points. Keeping the axis on the top avoids confusing the profile’s effect with a plain increase in height.
• Absorptive cap — the T geometry, plus the absorption term below.

δ therefore responds to the real width: on the default geometry a 1 m T cap takes the 2024 edition from 11.1 to 11.3 dB, a 2 m one to 11.9; a 2 m Y to 13.1; a Ø 2 m cylinder to 13.2. These are rigid-edge diffraction figures, and they are lower than the increments the literature measures on real caps (2-5 dB(A)): that difference is the part which is not geometry — absorption, surface reactance, diffusers — and must not be credited to the profile.

Absorption cannot be derived from rigid-edge diffraction: it remains an empirical term, and it is declared as such. The values are only the absorbency part measured in the literature on absorptive caps, not the cap’s total effect, and they scale linearly with the α coefficient you set, referred to the samples in those measurements (α ≈ 0.8). At α = 0 the result goes back to exactly the rigid T. The term is interpolated continuously between band centres: quantising it per band produced a staircase curve.

Transmission through the panel. A real barrier is not a perfect obstacle: the sound passing through the panel arrives together with the diffracted one, and the two add in energy. The effective attenuation is therefore A = −10·log₁₀(10^(−A_dif/10) + 10^(−R/10)), where R is the panel’s insulation: no top profile can go below that ceiling. ISO 9613-2 requires transmission to be negligible, i.e. R at least 10 dB above the diffraction attenuation; when that condition fails the tool says so next to the results instead of showing a number that cannot be achieved.

Limitations:

• The absorption term does not depend on the cap geometry: it only scales with α. A wide and a narrow absorptive cap receive the same term, while their geometric part is computed separately.
• Multiple reflections between barrier and source (the canyon effect with two parallel barriers) and the interaction with the ground are not modelled, though both matter when computing a level: only diffraction lives here.
• Surface reactance, quadratic-residue diffusers and active profiles are not modelled.
• Expert reports and real design work need software implementing the whole propagation model, not the screening term alone.

Normative and bibliographic references

• ISO 9613-2:1996 §7.4 — Original formulation for single and double diffraction.
• ISO 9613-2:2024 §7.4 — Current edition, refines D_z and K_met formulas for low barriers and long distances.
• FHWA (Federal Highway Administration, USA) — "Highway Noise Barriers" Design Guide §3, Acoustical Considerations. T-top and Y-top values.
• Watts G.R., Morgan P.A. (1996). "A parametric investigation of the performance of multiple edge highway noise barriers." Applied Acoustics.
• Ishizuka T., Fujiwara K. (2004). "Performance of profiled single noise barriers covered with quadratic residue diffusers." Applied Acoustics.
• Watts G.R., Crombie D.H., Hothersall D.C. (1994). "Acoustic performance of new designs of traffic noise barriers." Journal of Sound and Vibration.

Terms

Diffraction — the phenomenon by which a wave bends around an obstacle instead of being fully blocked. It is the physical reason why sound is still heard behind a noise barrier.

Knife-edge — idealised model of an infinitely thin edge. It is a good approximation for thin walls and for the top of real barriers with respect to the single upper edge.

Fresnel number — a dimensionless parameter expressing how many half-wavelengths of extra path the wave must travel passing over the barrier. The larger N, the more effective the attenuation.

What are the tool’s limitations?

The model covers one or two edges: geometries with more than two edges, or buildings, are more poorly approximated. It assumes a point source, and computes the diffraction term only — not geometrical divergence, ground effect and air absorption, which are needed for a level at the receiver and live in the Barrier calculator. Of the ISO family it uses the K_met meteorological factor; of CNOSSOS the two propagation conditions, without the coupling with the ground effect that the method prescribes. It covers only diffraction over the top, not transmission through the barrier nor lateral diffraction. A demonstrative tool, not for expert use.

• Single edge. Real geometries with several edges (top + side, multiple barriers, buildings) are more poorly approximated; ISO 9613-2 extends the model with multi-screen methods that this tool does not implement.
• Point source assumed. For extended line sources (roads, railways) the δ geometry varies along the source.
• Quiescent atmosphere. Weather effects (wind-speed gradient, thermal gradient) can reduce the effective attenuation by several dB in the far field.
• Diffraction over the top edge only; transmission through the barrier is not considered (R_w assumed sufficient, typically > 25 dB).

Standards & references

• ISO 9613-2:2024 — Attenuation of sound during propagation outdoors, §7.4.1 (screening). Current edition, Barrier calculator default.
• 1996 (legacy) — based on the previous edition, §7.4, but not canonical (incomplete Kmet): it compares versions of the tool, not the two editions of the standard.
• Maekawa, Z. (1968) — “Noise Reduction by Screens”, Applied Acoustics, vol. 1, pp. 157-173. DOI 10.1016/0003-682X(68)90020-0.

Learn more: Maekawa diffraction wiki →