# Solid angle measure and the sound radiation of exterior components

Again and again, we receive questions about the solid angle measure and its use. This includes even very experienced users and therefore we considered it necessary to write a letter of recommendation.

/1/ *Building acoustics – Calculation of the acoustic properties of buildings from the component properties – Part 4: Sound transmission from rooms to the outside (ISO 12354-4:2017); German version EN ISO 12354-4:2017*

**What is the solid angle measure?**

As an emitter, propagation software in noise immission protection basically only knows point sound sources (area and line sound sources are deconstructed as such) with a defined sound power level. In principle, it is assumed that the sound power is always emitted omni-directionally (i.e. uniformly in all directions). In ISO 9613-2, the calculated immission level

is calculated based on the sound power level by subtracting a damping term A and possibly adding a directional correction. The damping term A includes (besides terms for air absorption, ground effects and shielding) damping due to geometric propagation. This results from the fact that the energy is evenly distributed on a spherical surface.

At a distance of 1 m from the point source, the attenuation at a discrete point is 11 dB due to geometric propagation since the energy is distributed evenly over a sphere with a radius of r = 1 m and the surface O_{K}:

O_{K} = 4πr^{2} with r = 1→A = 10 log(4π1^{2} ) = 11 dB

If the source has a directional effect or radiates the entire energy only into a room, ISO provides a directional effect correction D_{C}, which increases or reduces the calculated immission level by the deviation between omni-directional and directional sound radiation. ISO 9613-2 includes the reflections on the ground and surfaces (e.g. house facades)

and therefore, in the first approach, there is no need to worry about whether the energy components that reach the immission point via the reflecting surfaces are taken into account in the calculation since these components are calculated by the forecasting software.

The directivity correction therefore only plays a role if the source itself is directed (e.g. loudspeaker), which is defined via the directivity function in the IMMI software. However, it becomes more complicated if the radiating surface is part of a building or if the sound source is structurally connected to the building.

**DIN EN 12354-4 – Sound transmission from rooms to the outside (input variable is an indoor level)**

A typical task in noise immission protection is the calculation of the noise impact of a commercial enterprise on surrounding residents. The radiation of noise from a workshop can make a significant contribution to the sound scape via

- outer walls,
- open windows or
- ventilation openings.

For these and similar cases, IMMI can define area sound sources for the facades and integrated partial surfaces (e.g. for windows or ventilation openings) for which an effective sound power level is calculated from the input variable indoor level in the room using

the standard DIN EN 12354-4 /1/.

The standard distinguishes between two types of radiating surfaces (see /1/, Annex D):

- planar emitter
- opening

The planar emitter is to be understood as a radiating exterior component (e.g. wall, window, door, etc.) and the opening as an open surface (e.g. open doors, but also ventilation openings with internal silencers). According to /1/, each radiating surface or opening is divided into individual replacement point sound sources with a sound power level resulting from:

- indoor level
- diffusivity term
- sound reduction index (for openings with R = 0)
- directional correction

The directional correction D_{C} results (in analogy to ISO 9613-2) from the directional efficiency D_{I} and the solid angle measure D_{Ω}.

**DIN EN 12354-4 – Directional correction?**

When considering whether a radiating surface must be provided with a solid angle measure, a distinction must be made between a planar emitter and an opening.

Planar emitter:

According to /1/, Annex D.1, the sound radiation is only emitted into a hemisphere (half-space in front of the facade), but since in practice the directivity correction D_{C} fluctuates between ± 5 dB, “an average value of D_{C} = 0 dB for radiation angles between 0° and 90° relative to the area standards may be assumed” (/1/, Annex D.1). The solid angle measure of 3 dB is already included (see /1/, Annex G.2.2, Table G.3, footnote b).

Opening:

According to /1/, Annex D.2, an opening radiates mainly perpendicular to the opening surface. This results in a directivity which varies in rough approximation between D_{I}= 2 dB and D_{I} = 10 dB. The following note is also considered: “For an opening in one plane or close to one (i.e. less than one wavelength apart) or more reflecting surfaces, the influence of these planes can be taken into account with directional correction by looking at the solid angle into which sound can be radiated. (/1/, Annex D.2). However, no specific recommendation

is defined. Only in Appendix G (calculation example) a value of 3 dB for the directivity correction D_{C} is given for a soundproofed ventilation opening, which already contains

the solid angle measure D_{Ω} (see /1/, Appendix G.2.2, Table G.6, footnote a).

**What to do if the input variable is a sound power level?**

In some cases it may also make sense to assign a sound power level directly to a radiating component (e.g. based on measurements) or to place a structurally connected unit with a specified sound power level shortly before the outer wall. In such cases, a solid angle measure of D_{Ω} = 3 dB should always be assigned. Otherwise, the calculated level

would be 3 dB too low, as ISO 9613-2 assumes that the sound power is radiated into a full room. In the reference settings, IMMI always defines a free space of 1 m before reflecting surfaces for sources and immission locations (this can be changed in the calculation parameters, but we recommend the default setting). This means that no reflections are taken into account in the calculation at a distance of up to 1 m from the walls.

In principle, this free field can also be set to 0 m, but a reflection does not lead to the 3 dB, since each reflection is accompanied by an absorption loss. This also makes sense for sources located in front of the surface (and therefore they should be modeled at a

distance > 1 m from the wall), but structurally connected surfaces should include the correct solid angle measure of 3 dB in the calculation approach.

**Summary**

For sound-emitting components, a spatial angle measure must be specified depending on the input variable and the type of radiating surface. If the input variable is an existing indoor level which is radiated via an external component, proceed as follows on the basis of the DIN EN 12354-4 standard when assigning a solid angle measure:

- If the radiating surface is an outdoor component (wall, roof, closed door,...), no directional correction is required and therefore no solid angle measure needs to be used for calculations according to ISO 9613-2 (D
_{C}= 0, already includes a solid

angle measure of 3 dB). - If the radiating surface is an opening (open door, ventilation opening,...), a directivity correction of D
_{C}= 3 dB must be applied and therefore a solid angle measure of 3 dB should be set for calculations according to ISO 9613-2 (D_{C}= 3, contains a solid

angle measure of 3 dB).

If the input variable is a sound power level, a solid angle measure of D_{Ω} = 3 dB must always be used. Make sure that the source is placed less than 1 m in front of the facade (which is guaranteed with the “Construct opening” function), because otherwise additional reflections from the facade, which are already represented by the defined solid angle measure, are included in the calculation.

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