Calculating the Resultant Sound Level: Adding 70 dB and 80 dB Decibel Levels

When it comes to combining sound levels measured in decibels (dB), it's crucial to understand that simply adding the numbers doesn't provide an accurate result due to the logarithmic nature of the decibel scale. This article will guide you through the correct method, including the calculations and the principles behind sound level addition.

Understanding the Decibel Scale

The decibel scale is logarithmic, which means that a difference of 10 dB represents a tenfold increase in sound intensity. Converting between decibels and sound intensity (in watts per square meter, W/m2) involves using the formula:

I 10L/10 where I is the intensity and L is the sound level in decibels.

Adding Sound Levels Through Intensity Calculation

To calculate the resultant sound level when 70 dB and 80 dB sounds are combined, follow these steps:

Convert each sound level to its corresponding intensity: 70 dB: I_{70} 10^{70/10} 10^7 10,000,000 Convert 80 dB to intensity: I_{80} 10^{80/10} 10^8 100,000,000 Add the intensities: I_{total} I_{70} I_{80} 10,000,000 100,000,000 110,000,000 Convert back to the resultant decibel level: L_{total} 10 ? log_{10}I_{total} 10 ? log_{10}110,000,000 ≈ 10 ? 8.043 ≈ 80.43 dB

Therefore, the resultant sound level when 70 dB and 80 dB sounds are combined is approximately 80.43 dB.

Alternative Method: Adding Amplitude for Pure Tones

If the tones are pure, undistorted, and have the same frequency and phase, an alternative method can be used to find the resultant sound level by adding the amplitudes. Here’s how to do it:

Convert each sound level to amplitude: For 70 dB: A 20 ? 10^{-6} ? √2 ? 10^{70/20} ≈ 0.08944 Pa Convert 80 dB to amplitude: A 20 ? 10^{-6} ? √2 ? 10^{80/20} ≈ 0.2828 Pa Add the amplitudes: A_{total} ≈ 0.08944 0.2828 ≈ 0.37224 Pa Convert back to decibels: L 20 ? (log_{10} frac{0.37224}{20 ? 10^{-6} ? √2}) ≈ 82.39 dB SPL

Thus, the resultant sound level when combining 70 dB and 80 dB pure tones is approximately 82.39 dB SPL.

Conclusion

When dealing with sound levels, it's crucial to understand the correct method for addition. The logarithmic nature of the decibel scale means that simple addition of values doesn't yield accurate results. Using the intensity or amplitude method provides a more accurate measure. This article has demonstrated both methods for adding two sound levels – 70 dB and 80 dB – and verified the results.