Decibel Scale and Noise Limits: Understanding Log Problems

[maccaman]

Here is the question as it follows:

The decibel scale, defined as

$$L = 10 log (\frac{I}{I_0})$$

where L is the Loudness (or comparative intensity) in decibels (dB),
$$I_0$$ is the reference level (10^-12 watts per m^2)
I is the actual intensity of the sound measured (watts per m^2).

now here's the question part:
An owner of an industrial plant next to a residential suburb agreed to limit noise to 75dB with a variation of up to 15% at a distance of 100m from the factory gates. Residents subsequently complained that sounds often reached 85dB. The manager replied, "It's not far over the limit - it's under the 15% variation agreed." Discuss whether or not the mangers statement is justified.

Dont quite know what there getting at, any help would be greatly appreciated.

 

[HallsofIvy]

85- 75= 10 so the variation is 10. 10/75= 0.133333.. which is less than 0.15. Yes, the variation is less than 15% of the noise limit. I am interpreting the statement "75dB with a variation of up to 15%" literally: 15% of 75dB.

Of course, 75 decibels means 10 log(I/I₀)= 75 or I= I₀10^7.5= 10^-1210[7.5]= 10^-4.5 watts/m² and
85 decibels means 10 log(I/I₀)= 85 or I= I₀10^8.5= 10^-1210[8.5]= 10^-3.5 watts/m² so the actual difference in intensity is 0.000285 so the percentage variation is 0.000285/10^-4.5= 9 (900%) which is far more than 15%.

But, the original agreement was about decibels, not intensity. The manager's statement is justified. (and that second paragraph is irrelevant.)

 

[M98Ranger]

The manager's statement is not entirely justified. While it is true that the noise levels are within the agreed upon variation of 15%, it is important to consider the actual decibel level that the residents are being exposed to. 85dB is considered a high level of noise and can be harmful to human health if exposed to for a prolonged period of time. The agreed upon limit of 75dB is meant to protect the residents from excessive noise pollution and the manager should take responsibility for ensuring that the noise levels are not consistently reaching 85dB.

Additionally, the decibel scale is logarithmic, meaning that even a small change in decibel level can have a significant impact on the perceived loudness of the sound. For example, a 10dB increase in noise level is perceived as a doubling of loudness. Therefore, even though the noise levels may only be 10dB over the agreed limit, it can still have a significant impact on the residents' quality of life.

Furthermore, the agreed limit of 75dB is meant to be a maximum limit, not an average or regular level of noise. The manager should take measures to reduce the noise levels below 75dB to ensure that the residents are not consistently exposed to high levels of noise.

In conclusion, while the manager's statement may be technically correct, it is not a sufficient justification for the consistently high noise levels reaching 85dB. The well-being of the residents should be the top priority and steps should be taken to reduce the noise levels to within the agreed upon limit of 75dB.