Maekawa diffraction

A noise barrier does not zero out noise: it reduces it. The physical reason is diffraction — the phenomenon by which a wave bends around an obstacle instead of being fully blocked. This Open Lab tool isolates, in didactic form, the calculation of diffraction over a single edge, showing how attenuation depends on geometry and frequency, and comparing on the same configuration the three calculation methods offered by the Barrier calculator.

The geometry of the problem

The model considers a source S, a diffracting edge (the top of the barrier) and a receiver R. The sound that reaches R passing over the edge travels a longer path than the direct path S→R. This path difference δ is the key quantity:

$$\delta =\sqrt{d_s^2+h^2}+\sqrt{d_r^2+h^2}-(d_s+d_r)$$

where h is the excess of the edge above the source–receiver line of sight and d_s, d_r are the distances from the projections of S and R onto that same line. The domain should be stated: this symmetric form, with the same h in both radicals, holds for the reduced geometry in which S and R are brought onto a common reference line — which is what the explorer builds. In a general geometry, with S and R at arbitrary heights, one uses the actual geometric paths: δ = d(S,E) + d(E,R) − d(S,R), which does not necessarily reduce to this expression.

A caveat on the sign, because h appears squared in the formula: δ(+h) = δ(−h), so the formula alone does not distinguish an edge that screens from one below the line of sight. The rule “edge below the line of sight → no attenuation” is a logical condition applied before the formula, not a consequence of it: the explorer works with positive h and builds the screened geometry directly.

The Fresnel number

The path difference alone is not enough: what matters is its ratio to the wavelength of the sound. Hence the Fresnel number N, a dimensionless parameter:

$$\lambda =\frac{c}{f}(c=343\text{m/s},20°\text{C})N=\frac{2\delta }{\lambda }$$

N expresses how many half-wavelengths of extra path the wave must travel passing over the barrier. The larger N, the more effective the attenuation. This explains a counterintuitive but fundamental fact of acoustics: a barrier is far more effective at high frequencies (small λ → large N) than at low ones (large λ → small N). The deep rumble of a truck remains far from negligible at low frequencies — the table below gives 7.1 dB at 63 Hz, which is not a small effect: significantly less, not absent. The low rumble of a truck penetrates behind a barrier far more than the high-pitched hiss of tyres.

The Maekawa formula

In 1968, Z. Maekawa published, from systematic measurements on screens, the relation between Fresnel number and single-edge attenuation — in the form of an experimental curve, not an analytic expression. The formula the tool uses is the analytic approximation of that curve which entered common use (a parametrisation attributed to Tatge):

$$\begin{gathered} A=10 \\ , \\ lo{g}_{10}(3+20 \\ ,N) \end{gathered}$$

It is the curve plotted in the explorer’s Att(N) chart: it rises rapidly for the first values of N, then slows — but it does not saturate on its own. The function has no ceiling: at N = 100 it would give about 33 dB. The 25 dB limit is a clamp of the implementation, aligned with the cap ISO 9613-2 places on its own multiple-diffraction term, and should be read as such.

Do not attribute to it a physical meaning it does not have: there is no law by which a barrier’s insertion loss cannot exceed 25 dB. Real effectiveness is limited by lateral diffraction, transmission through the panel, reflections, finite length, ground, spectrum and meteorology — none of which fixes a threshold at that precise value. And the ISO caps (20 dB single edge, 25 dB multiple) limit the D_z term, not the barrier’s overall attenuation.

An example: the same barrier, eight bands

Consider a source d_s = 10 m from the barrier, a receiver d_r = 20 m on the other side, and an edge exceeding the line of sight by h = 2 m. The path difference is:

$$\delta =\sqrt{10^2+2^2}+\sqrt{20^2+2^2}-30\approx 0.30\text{m}$$

The geometry is fixed, but attenuation changes with frequency because λ — and therefore N — changes. With the Maekawa formula:

The same barrier, with the same geometry, cuts 7.1 dB at 63 Hz and 24.5 dB at 8 kHz: seventeen decibels between the two ends of the spectrum. The δ/λ column says why: at 63 Hz the path difference is a twentieth of a wavelength — the wave barely notices the obstacle — while at 8 kHz it is seven wavelengths. It is why, behind a motorway barrier, what you keep hearing of a vehicle is mostly the low rumble of the engine and little of the high hiss of the tyres. The design consequence is that a barrier sized against an overall dB(A) figure can disappoint when the source spectrum sits low.

How much taller does the barrier need to be

The question that follows the first calculation is: how much taller to gain a few decibels? The answer is not linear, because δ grows less than proportionally with h. On the same geometry (d_s = 10 m, d_r = 20 m, 500 Hz):

The last column gives about 5 dB per doubling of the useful height over the range shown — and it is a range, not a rule. The value grows (2.6, 4.4, 5.4, 5.6 dB) towards a computable asymptote: for h small compared with the distances δ ≈ (h²/2)(1/d_s + 1/d_r), so δ goes as h² and doubling h quadruples the Fresnel number; in the regime where the 20N term dominates the 3 in the formula the gain tends to 10·log₁₀(4) = 6.02 dB per doubling. Below that regime it is less, above it the clamp takes over. Turned around, from the client’s point of view: starting at h = 2 m, gaining 5 dB takes 3.8 m, nearly double the height. This is the arithmetic that makes tall barriers so expensive, and that pushes towards other answers — moving the barrier closer to the source, or working on the source itself.

Mind what “useful h” means: it is the rise above the line of sight, not the height of the barrier above ground. With source and receiver both elevated, a 4 m barrier can have one metre of useful rise.

The near receiver is better protected than the far one

A counter-intuitive result, and the reason to calculate rather than reason by intuition. Holding the barrier (h = 2 m) and the source (d_s = 10 m) fixed, and moving only the receiver:

Going from 5 to 80 m behind the barrier, the attenuation the barrier provides drops by 3.7 dB. The geometric reason is that moving R away makes the over-the-edge path resemble the direct one more and more, so δ shrinks — but it does not tend to zero, as one might assume. For d_r → ∞ the term √(d_r²+h²) − d_r vanishes and δ → √(d_s²+h²) − d_s, which in this geometry is 0.198 m: the attenuation tends to an asymptote of about 11.6 dB at 500 Hz, not to zero. The column of values shows it: from 40 to 80 m it falls by only 0.4 dB. Physically the far receiver is still better off, because it gains geometric divergence — but the barrier’s own merit shrinks. When reading an insertion-loss map, the band of maximum effectiveness is the one immediately downstream, not the far field.

The five methods compared

The explorer shares its calculation engine (@lib/physics) with the Barrier calculator and the Acoustic map, and applies three formulations to the same geometry:

• ISO 9613-2:2024 — the current edition, the Barrier calculator default: Dz = 10·log₁₀[1 + (2 + (C₂/λ)·C₃·z)·Kmet], with a z_min threshold below which attenuation is zero and a Kmet meteorological correction; 20 dB cap for a single edge.
• Legacy 1996 mode — based on the previous edition’s formulation: Dz = 10·log₁₀(3 + (C₂/λ)·C₃·z·Kmet); capped at 20 dB. It should be called legacy rather than “ISO 9613-2:1996”: the code declares that it omits a term of the canonical Kmet, and it is kept unchanged to preserve comparability with the tool’s historic results. It compares versions of the tool, not the two editions of the standard.
• CNOSSOS-EU:2015 — the European common method (Directive (EU) 2015/996, Annex II §VI.4.4.b): Δ_dif = 10·log₁₀(3 + (40/λ)·C″·δ), with the threshold (40/λ)·C″·δ ≥ −2 below which attenuation is zero, C″ = 1 for a single edge, 25 dB cap. It is not a variant of the ISO editions: the coefficient is 40/λ rather than 20/λ, there is no K_met factor, and there are two propagation conditions — homogeneous, with straight rays, and favourable, with rays curved into arcs of radius Γ = max(1000, 8·d). Under favourable conditions the ray clears the edge higher up, δ shrinks and the barrier performs worse: this is why a barrier is less effective downwind, and in the model it comes from the geometry rather than from a correction term.
• Maekawa 1968 — the historic empirical formula described above; 25 dB cap, no meteorological correction.

Between CNOSSOS and Maekawa there is a coincidence worth knowing before reading it as an error: for a single edge under homogeneous conditions the two give the same number, because 3 + 20·N and 3 + 40δ/λ are the same expression written with N = 2δ/λ. They part company when there are two edges, where C″ enters, and under favourable conditions.

The complete formulas, with the C₂/C₃ constants and the definition of Kmet, are documented in the “Diffraction formulas” section of the Barrier calculator. On the behaviour with a clear line of sight a clarification is due, because the three methods are not equivalent. Maekawa and the 1996 form give 0 dB for N ≤ 0. ISO 9613-2:2024 does not: it sets D_z = 0 only for z ≤ z_min, with z_min = −2λ/(C₂C₃) negative, and gives the path difference a negative sign when the line of sight passes above the edge. There is therefore a band z_min < z < 0 in which the line of sight is clear and the attenuation is still positive — it avoids an artificial discontinuity at the shadow-zone boundary. This tool does not cover it: the geometry upstream produces only δ ≥ 0 and the dispatcher truncates δ ≤ 0 to zero. That is a simplification of the explorer, not behaviour of the standard, and implementing it would need a signed path difference throughout the calculation chain.

The differing caps matter, but what they limit needs stating precisely. The implementation’s Maekawa clamp sits at 25 dB, the ISO cap at 20 dB for a single edge: in saturating geometries — tall barriers, high frequencies — the Maekawa diffraction term therefore reaches 5 dB higher. In the spectrum table above the 8 kHz band touches 24.5 dB with Maekawa, while under ISO it would stop at 20.

Those 5 dB, though, are the distance between two saturated terms, not between two predictions: one cannot conclude that an ISO and a Maekawa prediction differ by 5 dB. It is also necessary to keep apart three quantities which this page, being an explorer of the geometric term alone, compares in raw form: D_z is the diffraction term; A_bar is the barrier attenuation, which for over-the-edge diffraction with A_gr > 0 equals D_z − A_gr (§7.4); insertion loss is the difference between the levels with and without the barrier, and includes the ground effect, any lateral paths combined under §7.4.4, and the energetic sum over the spectrum. The barrier calculator works on A_bar and IL; here we look at D_z. On an overall dB(A) figure, where the high bands carry weight, the choice of method shows up in the result.

Terms

Diffraction — the bending of a wave around an obstacle; the reason sound is still heard behind a barrier. Knife-edge — an idealised model of an infinitely thin edge. It is a good approximation for a relatively thin, long barrier with a simple edge; much less so for thick barriers, earth bunds and berms, T- or Y-tops, cylindrical profiles or absorbing edges, which are designed precisely to do better than a knife edge. Fresnel number — the dimensionless parameter linking path difference and wavelength.

Limitations

• One or two edges. The explorer solves single and double diffraction, the latter with the edge-to-edge distance e — hence C₃ and C″ — and the 25 dB cap. Only edges that actually protrude 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, otherwise the path difference is inflated and with it the attenuation. Geometries with more than two edges, or buildings, stay out; so does lateral diffraction around the sides, which ISO 9613-2:2024 covers in §7.4.3 and which is implemented in the Barrier calculator.
• Point source. For extended line sources (roads, railways) the δ geometry varies along the source.
• Quiescent atmosphere for Maekawa, which has no meteorological correction: wind and temperature gradients can reduce the effective attenuation by several dB in the far field. It does not hold for ISO 2024, which does have Kmet, and which is in fact set up for conditions favourable to propagation (downwind or an equivalent inversion): the three methods do not share the same atmospheric assumption, and seeing them on the same curve is convenient but does not make them homogeneous.
• Only diffraction over the top edge is considered; transmission through the panel is neglected. How legitimate that is depends on the device’s performance, which for road barriers is characterised by the EN 1793 series (airborne sound insulation, absorption, intrinsic diffraction) and not by R_w, a quantity typical of building elements: a 3 m geometric barrier and a real 3 m barrier are not the same thing.
• The obstacle is assumed already eligible to be treated as a screen. ISO 9613-2 sets requirements for an object to screen — sufficient surface mass, a substantially closed surface without significant gaps or openings, adequate extent relative to the wavelength. This explorer is purely geometric and checks none of them: it gives the diffraction term of an edge, taking for granted that the edge belongs to something that screens.

A demonstrative tool: it does not replace a professional assessment. Note that “certified software” is not a category Italian regulation provides for these calculations: what counts is that the model be adequate to the required methodology and verifiable, and that a competent technician be responsible for the work.

Standards & scientific references

• ISO 9613-2:2024 — Attenuation of sound during propagation outdoors, §7.4.1 (screening). Current edition, Barrier calculator default.
• ISO 9613-2:1996 — previous edition, §7.4; available as a comparison mode.
• ISO 9613-2:2024/DAmd 1 — draft amendment, out for DIS ballot: it restores D_z and K_met to their 1996 form and leaves z_min, lateral diffraction and the caps in place. It is not implemented — a draft does not go into a calculation engine — and the formulas side by side are in ISO 9613-2:2024/DAmd 1: diffraction back to 1996.
• Maekawa, Z. (1968) — “Noise Reduction by Screens”, Applied Acoustics, vol. 1, pp. 157-173. DOI 10.1016/0003-682X(68)90020-0.
• Directive (EU) 2015/996 — common noise assessment methods (CNOSSOS-EU), Annex II §VI.4.4.b: diffraction, homogeneous and favourable conditions. Text on EUR-Lex.

What the tool puts in your hands

Geometry is entered as absolute heights above ground level — source, barrier and receiver — plus the horizontal distances: it is how you measure on site, and the form in which ISO 9613-2 asks for it. The excess h of the edge over the source-receiver line — the parameter of the classical formulation — is not an input but a result, together with δ and N.

The section is drawn to scale, with numbered axes: the horizontal scale is set by the source-receiver distance alone, and changing a height does not move it; the vertical one steps up through round values, the first that sits two metres above the tallest element. Source, barrier tops and receiver can be dragged in the drawing.

The multi-band mode shows the eight octave bands together for every method, and at the bottom the A-weighted insertion loss on a source spectrum chosen from six presets: that is the number a barrier is sized with, and it is not the average of the dB in the table. No level at the receiver is computed: only the diffraction term lives here, and an Lp without divergence, ground effect and air absorption would be wrong — that calculation belongs to the Barrier calculator.

Results leave as text on the clipboard or as a PNG image, with the drawing and the numerical summary built from the same values.

The tool is available as a runnable application at /en/tools/maekawa/.