Will two separate 50-dB sounds together constitute a 100-dB sound? Explain.
The correct answer and explanation is :
No, two separate 50-dB sounds together will not constitute a 100-dB sound. Instead, the combined sound level will be slightly higher than 50 dB, but far less than 100 dB.
Explanation:
Decibels (dB) are measured on a logarithmic scale, meaning that sound levels do not add arithmetically. Instead, when two sounds of equal intensity are combined, the total intensity increases by a specific logarithmic formula:
[
L_{\text{total}} = 10 \log_{10} (10^{L_1/10} + 10^{L_2/10})
]
For two sounds of 50 dB each:
[
L_{\text{total}} = 10 \log_{10} (10^{50/10} + 10^{50/10})
]
[
= 10 \log_{10} (10^5 + 10^5)
]
[
= 10 \log_{10} (2 \times 10^5)
]
[
= 10 [\log_{10} (2) + \log_{10} (10^5)]
]
[
= 10 [0.301 + 5]
]
[
= 10 \times 5.301
]
[
\approx 53.01 \text{ dB}
]
Thus, two 50-dB sounds together result in approximately 53 dB, not 100 dB.
Why Doesn’t It Double?
- The decibel scale is logarithmic, meaning each 10-dB increase represents a tenfold increase in intensity.
- A 100-dB sound is 10 billion times more intense than a 50-dB sound, which cannot be achieved by merely adding two equal 50-dB sources.
- When two identical sound sources combine, the total sound level increases by only about 3 dB, not double.
So, even if you add multiple 50-dB sounds, you will never reach 100 dB—you need a sound that is inherently 100 dB in intensity.