PLANNING FOUNDATIONS
2.4.2 Calculation rules

Sound level addition

The impact of several sound sources leads to an increase of the sound immission. However, sound level values must not be simply added arithmetically. This means, for example, that the sum level L of the three sound levels 

L₁ = 35 dB(A), L₂ = 40 dB(A), L₃ = 45 dB(A)

is by no means 120 dB(A)!

The levels must not be added in decibels, which is actually no physical value. They must first be transformed into sound pressures, which are then added to a sound level.

The three sound levels must therefore be added energetically according to the following formula:

First, the expression 10^0,1Li must be formed for each addend L_i. This takes the antilogarithm of the levels, i.e. that the result represents the ratio of the physical sound pressure p and the reference parameter (standardized threshold of hearing) p₀ = 2·10^-5 Pa, which can be added.

Then we take the logarithm of the sum to get the level of the sum of the physical sound pressures:

L = 10 lg (10^3.5 + 10^4.0 + 10^4.5) dB(A) = 46.5 dB(A)

The sound level addition can also be carried out in pairs for two sound level values with the help of figure 2/6. For this, we read the value (lower part) on the addition scale, below the difference of the sound levels that are to be added (upper part), and add it to the higher of the two sound levels. It is advisable here to add the smallest level first in order to gain a higher accuracy.

Sound level difference between L_r,1 and L_r,2 in dB(A)

Add reading value in dB(A) to the higher level

Fig. 2/6: Addition of sound levels

By adding only the addends L₂ and L₃ from the example above, we get:

40 dB(A) + 45 dB(A) = 46.2 dB(A)

From this we can conclude that the addend L₁ = 35 dB(A) does not influence the sum level strongly and could have been ignored in the first place. The following important rule can be deduced from the addition scale in figure 2/6.

If two sound levels differ by at least 10 dB, the lower level does practically not contribute to the sum level. We can therefore approximately conclude that (with or without the use of A-weighting):

65 dB(A) + 54 dB(A) = 65 dB(A)

43 dB(A) + 44 dB(A) + 58 dB(A) = 58 dB(A)

(since 43 dB(A) + 44 dB(A) gives at the most 47 dB(A) and this is 11 dB less than 58 dB(A))

The addition of two identical sound levels results in a sound level higher by three decibels, which means a doubling of the sound power due to the definitions given in section 2.4.1. So we can say that:

55 dB(A) + 55 dB(A) = 58 dB(A).

Note:

Many low levels can also contribute to a sound level addition despite the existence of a level which is more than 10 dB higher.

We can say that

50 dB(A) + 40 dB(A) = 50.4 dB(A)

but

50 dB(A) + 10 x 40 dB(A) = 53 dB(A)

because 10 x 40 dB(A) = 50 dB(A)

You can sump up sound levels online with the decibel calculator.

Energetic averaging

The process of averaging is analogous to the energetic addition of sound levels with the exception that after the addition of the elements 10^0,1L, we have to divide by the number of elements, and this before taking the logarithm.

In our example, the average sound level L_m is:

L₁ = 35 dB(A),    L₂ = 40 dB(A),    L₃ = 45 dB(A)

L_m = 10 lg (1/3 (10^3.5 + 10^4.0 + 10^4.5))

L_m = 42 dB(A) (rounded up)

The example shows that the energetic average level of a series of different sound levels lies closer to the higher values than it would be in the case of arithmetic averaging.

As the averaging often concerns noises varying in time, the corresponding method of calculation can be adapted by dividing by the total observation or measurement period "T" instead of the number of values and by multiplying each of the addition elements 10^0.1L by the impact period "t_i" of the sound level value L_i during the total measurement period.

The calculation steps shall be illustrated by using the example of a fictitious noise measurement during a measurement period T = 16 hours from 6 am to 10 pm. The following energetic average values are assumed for the individual hours:

6 am to 8 am:    60 dB(A)     t₁ = 2 h
8 am to 10 am:  45 dB(A)     t₂ = 2 h
10 am to 6 pm:  35 dB(A)     t₃ = 8 h
6 pm to 8 pm:    45 dB(A)     t₄ = 2 h
8 pm to 10 pm:  55 dB(A)     t₅ = 2 h

This gives an average sound level for the whole daytime from 6 am to 10 pm (T = 16 h) of

L_m = 52.4 dB(A)

This result also shows that the highest sound level values (in our example the four loudest hours) influence the result of the average sound level the most as

What is most relevant in practice is the question which time blocks with different sound levels contribute most to the total sound level. Partial (rating) levels are created for this purpose by relating the level within the particular time period to the total (rating) period. For the partial rating level in the above-mentioned example, we can say that

The following partial rating levels result for our example

The sum of these partial rating levels is L_m = 52.4 dB(A). We can see from the partial rating levels that the 60 dB(A) level over a period of exposure of 2 h is the relevant factor for the total sound level of 16 h.

The following statements can easily be deduced from the rules of energetic addition and averaging of sound levels:

• Halving (doubling) the impact period of a noise leads to a decrease (increase) of its average level by 3 dB.
• Halving (doubling) the sound power of a noise also leads to a decrease (increase) of its average sound level by 3 dB.

Decreasing sound level during sound propagation

The sound pressure level decreases with increasing distance from the source of the sound. The theoretical description of the decreasing sound level assumes that the point source of a sound emits sound waves in the form of a spherical wave to all directions of a room. This means that the sound power from a sound source spreads to an always growing spherical surface with increasing distance. As the spherical surface increases proportionally to r², the following ratio for the sound level decrease L is true if the distance between r₁ and r₂ is increased:

∆L = 10 lg (r₂/r₁)² = 20 lg (r₂/r₁)

So the sound level decreases by 6 dB each time the distance is doubled, provided that there is no loss during sound propagation and that the sound comes from a point source (fig. 2/7).

In the case of line sources (fig. 2/8), which have a wide linear expansion in comparison to the considered distance and which run in a straight line, the sound is radiated in the form of a cylindrical wave. As the surface of a cylinder increases proportionally to r, the level decrease ∆L is described by the expression

∆L = 10 lg (r₂/r₁)

In reality, however, sound level decreases during sound propagation differ from these theoretical values as factors like the sound-absorbing properties of the ground and the air or weather conditions (wind) have to be taken into account as well.while increasing the distance from r₁ to r₂. This means that, in the case of a line sound source (e.g. roads, railways, long pipelines), the sound level only decreases by 3 dB each time the distance is doubled, provided that there is no loss during sound propagation.

What is more, the sound radiation of many sound sources does not correspond to the ideal forms of a spherical or cylindrical wave as they possess a certain directivity. Other influences on the propagation of sound come from different forms of vegetation as well as from sound-shielding and sound-reflecting structures on the propagation path of the sound waves (e.g. development).

The very complex process of outdoor sound propagation is the subject of DIN ISO 9613-2, while VDI guideline 2720-1 deals with the special aspects of "Noise control by barriers outdoors". An acoustic shadow can be created with the help of soundproof obstacles, similar to the shielding of light waves.