Combining Noise Levels: Calculation

Sound levels in decibels (dB) represent sound intensity on a logarithmic scale. To find the resultant level of multiple noise sources, you cannot simply add the dB values. Instead, you must convert them to sound intensity or power, sum these values, and then convert back to dB.

The standard formula for combining two sound levels, $L_1$ and $L_2$, is:

$ L_{total} = 10 \log_{10} \left( 10^{L_1/10} + 10^{L_2/10} \right) $

Applying the Noise Formula

We are given two noise levels:

• $L_1 = 80$ dB
• $L_2 = 50$ dB

Step 1: Calculate the proportional sound power (or intensity) corresponding to each level.

• For $L_1 = 80$ dB: The value is $10^{80/10} = 10^8$
• For $L_2 = 50$ dB: The value is $10^{50/10} = 10^5$

Step 2: Sum these proportional values.

$ \text{Sum} = 10^8 + 10^5 = 100,000,000 + 100,000 = 100,100,000 $

Step 3: Convert the sum back to the decibel scale using the formula.

$ L_{total} = 10 \log_{10} (100,100,000) $

To calculate this:

$ L_{total} \approx 10 \times \log_{10}(1.001 \times 10^8) $

$ L_{total} \approx 10 \times ( \log_{10}(1.001) + \log_{10}(10^8) ) $

$ L_{total} \approx 10 \times (0.00043 + 8) $

$ L_{total} \approx 10 \times 8.00043 $

$ L_{total} \approx 80.0 \text{ dB} $

Result Analysis

The calculation shows that the resultant noise level is approximately 80.0 dB. When combining noise levels, the dominant source (the one with the higher dB value) largely determines the total level. The contribution of the 50 dB source is minimal compared to the 80 dB source.