Distance law

The distance law or distance law describes the decrease in a physical quantity with increasing distance to the source or the transmitter. The prerequisites are a point source (approximately: small expansion of the source compared to the distance), which emits isotropically - i.e. not directed - and a free field without reflective borders. Thus, the law of distance can only be used approximately for practical applications.

In the case of e.g. B. lasers or parabolic antennas , other influences must be taken into account, such as the divergence angle or the antenna diagram .

For energy quantities - the 1 / r² law

Among the energy terms include not only radiation intensities such. B. X-rays , radioactivity , solar radiation (visible light radiation) or other electromagnetic waves that propagate in all directions, including the sound intensity .

Distance law for energy quantities

The energy  E emanating from a source that radiates uniformly into three-dimensional space is distributed over a spherical surface that increases proportionally with the square of the distance  r from the source. The radiation intensity  I , i.e. H. the "power per area" ( P / A ), therefore decreases with 1 / r ² :

${\displaystyle {\begin{aligned}I&\propto {\frac {1}{r^{2}}}\\\Rightarrow {\frac {I_{1}}{I_{2}}}&={\frac {{r_{2}}^{2}}{{r_{1}}^{2}}}\\\Leftrightarrow I_{2}&=I_{1}\cdot \left({\frac {r_{1}}{r_{2}}}\right)^{2}\end{aligned}}}$

When the distance is doubled, the intensity falls to a quarter of the initial value. This corresponds to a level decrease of 6 dB _RMS .

In general, the level decrease can be calculated as follows:

${\displaystyle {\begin{alignedat}{2}\Delta L[dB_{RMS}]={L_{2}}-{L_{1}}&=-10\cdot \lg {\frac {{r_{2}}^{2}}{{r_{1}}^{2}}}=&&-20\cdot \lg {\frac {r_{2}}{r_{1}}}\\\Leftrightarrow L_{2}[dB_{RMS}]&=L_{1}&&-20\cdot \lg {\frac {r_{2}}{r_{1}}}\end{alignedat}}}$

Here, lg is the logarithm to base 10.

The above applies to many radiation sources known from everyday life. Prerequisite that it does not radiate uniformly into the three-dimensional space or only as a rough approximation.

For rms values ​​of linear field sizes - the 1 / r law

The linear field sizes include B. the acoustic field quantities such as sound pressure , sound velocity and sound displacement as sound field quantities . The effective values ​​of these quantities decrease inversely proportional to the distance from the sound source, i.e. with 1 / r :

${\displaystyle {\begin{aligned}{\tilde {p}}&\propto {\frac {1}{r}}\\\Rightarrow {\frac {{\tilde {p}}_{1}}{{\tilde {p}}_{2}}}&={\frac {r_{2}}{r_{1}}}\\\Leftrightarrow {\tilde {p}}_{2}&={\tilde {p}}_{1}\cdot {\frac {r_{1}}{r_{2}}}\end{aligned}}}$

When the distance is doubled, the values ​​fall to half of the initial value. As with the square sizes, this corresponds to a level decrease of 6 dB. Here, too, the following applies to the change in level of the rms values ​​in dB _SPL :

${\displaystyle {\begin{alignedat}{2}\Delta L[dB_{SPL}]={L_{2}}-{L_{1}}&=&&-20\cdot \lg {\frac {r_{2}}{r_{1}}}\\\Leftrightarrow L_{2}[dB_{SPL}]&=L_{1}&&-20\cdot \lg {\frac {r_{2}}{r_{1}}}\end{alignedat}}}$

Changes in level can therefore be specified without knowing whether the measured variable is a square or linear variable. However , this knowledge must be available to determine the physical units .

Web links

• Decrease in level of sound pressure and sound intensity with distance (PDF file; 76 kB)
• Attenuation of the sound level with distance
• How does the sound decrease with distance?