Calculating Simultaneous Aeroplane Noise

The sound level ($\beta$) in decibels (dB) measures sound intensity ($I$) relative to a reference intensity ($I_0$):

$ \beta = 10 \log_{10}\left(\frac{I}{I_0}\right) $

When multiple identical sound sources combine, their intensities add up. For $N$ aeroplanes, each producing intensity $I$, the total intensity $I_{total}$ is:

$ I_{total} = N \times I $

The resulting sound level ($\beta_{total}$) is:

$ \beta_{total} = 10 \log_{10}\left(\frac{I_{total}}{I_0}\right) = 10 \log_{10}\left(\frac{N \times I}{I_0}\right) $

Using logarithm properties ($\log(ab) = \log(a) + \log(b)$), we get:

$ \beta_{total} = 10 (\log_{10}(N) + \log_{10}(I/I_0)) $

$ \beta_{total} = 10 \log_{10}(N) + 10 \log_{10}(I/I_0) $

Since $10 \log_{10}(I/I_0)$ is the sound level of one aeroplane ($\beta$):

$ \beta_{total} = \beta + 10 \log_{10}(N) $

Given:

• Noise level of one aeroplane, $\beta = 120$ dB.
• Number of aeroplanes, $N = 5$.

Substitute the values:

$ \beta_{total} = 120 \text{ dB} + 10 \log_{10}(5) $

Calculation:

• $ \log_{10}(5) \approx 0.69897 $
• $ 10 \log_{10}(5) \approx 10 \times 0.69897 \approx 6.9897 $ dB
• $ \beta_{total} \approx 120 \text{ dB} + 6.9897 \text{ dB} \approx 126.9897 \text{ dB} $

The total noise level is approximately 126.99 dB.