---
title: "Adding sound levels from several sources"
subject: "Environment and Pollution Control"
chapter: 6
key: "semester_7/environment_and_pollution_control:combining-noise-levels-numerical"
source: https://www.bctengineer.com/q/semester_7/environment_and_pollution_control/combining-noise-levels-numerical
site: BCT Engineering
---

# Adding sound levels from several sources

Environment and Pollution Control · Chapter 6: Noise pollution

- Asked in: year not given, 4 marks
- Video: [NOISE POLLUTION - Addition of sound level](https://www.youtube.com/watch?v=TNg-5YOhZM8) — JE Center By Ravi Kant Verma
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/environment_and_pollution_control/combining-noise-levels-numerical
- Whole chapter: https://www.bctengineer.com/md/semester_7/environment_and_pollution_control/chapter-6

## Question

Three machines in a workshop produce sound pressure levels of 85 dB, 88 dB and 80 dB at a worker's position when operating individually. Find the combined SPL when all work together. What is the level if the 80 dB machine is switched off? Also, why can two equal sources never raise the level by more than 3 dB?

## Short answer

Combined SPL = 10 log(10⁸·⁵ + 10⁸·⁸ + 10⁸) = 90.2 dB. With only the 85 and 88 dB machines, level is 89.8 dB. Two equal sources add 3 dB (10 log 2) because doubling energy adds 3 dB, which is the maximum possible increase; adding lower-level sources gives a smaller rise.

## Answer

Sound levels are logarithmic and cannot be added directly. Add the **intensities (or mean-square pressures)**:
$$
L_{total} = 10\log_{10}\left(\sum_i 10^{L_i/10}\right)
$$

#### All three machines
$$
\begin{aligned}
10^{8.5} &= 3.162\times10^{8} \\
10^{8.8} &= 6.310\times10^{8} \\
10^{8.0} &= 1.000\times10^{8} \\
\text{Sum} &= 10.472\times10^{8}
\end{aligned}
$$
$$
L_{total} = 10\log_{10}(10.472\times10^{8}) = 10\times9.0200 = 90.2\ \text{dB}
$$

#### Without the 80 dB machine
$$
L = 10\log_{10}\left[(3.162+6.310)\times10^{8}\right] = 10\log_{10}(9.472\times10^{8}) = 89.8\ \text{dB}
$$
Switching off the quietest source reduces the total only by 0.4 dB.

#### Two equal sources
$$
L_{total} = 10\log_{10}(2\times10^{L/10}) = L + 10\log_{10}2 = L + 3\ \text{dB}
$$
Doubling the energy adds 3 dB. If the two levels differ, the increase is smaller (for a difference of 10 dB it is only about 0.4 dB), so 3 dB is the maximum. Thus $n$ equal sources give $L + 10\log_{10}n$.

**Answer: Combined level = 90.2 dB; without the 80 dB machine = 89.8 dB.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How should decibel levels of several sources be combined?
   A. Add the dB values directly
   B. Convert to intensities, sum them, then convert back with 10 log
   C. Take the average of the dB values
   D. Take the largest dB value only
   Answer: B. Levels are logarithmic, so L_total = 10 log(sum of 10^(Li/10)). (Key point: Method of adding levels)

2. What is the combined SPL of 85, 88 and 80 dB sources?
   A. 253 dB
   B. 88 dB
   C. 93 dB
   D. 90.2 dB
   Answer: D. Sum = (3.162 + 6.310 + 1.000) x 10^8 = 10.472 x 10^8, giving 10 log = 90.2 dB. (Key point: Combined level)

3. What happens when the quietest (80 dB) machine is switched off?
   A. Level falls to 85 dB
   B. Level falls by 10 dB
   C. Level falls only about 0.4 dB to 89.8 dB
   D. Level does not change at all
   Answer: C. The 80 dB source contributes little energy, so the total drops from 90.2 to 89.8 dB. (Key point: Switching off quiet source)

4. Why can two sources never raise the level by more than 3 dB?
   A. Because 10 log 2 = 3 dB is the largest increase, reached when levels are equal
   B. Because dB cannot exceed 3
   C. Because the ear cannot detect more
   D. Because equal sources cancel partly
   Answer: A. Two equal sources double the energy, adding 10 log 2 = 3 dB; unequal levels give a smaller rise. (Key point: Two equal sources)

## Related in Environment and Pollution Control

- [Sound pressure level and power level calculations](https://www.bctengineer.com/question.md?key=semester_7%2Fenvironment_and_pollution_control%3Aspl-pwl-numerical) — Environment and Pollution Control, chapter 6
- [Typical noise levels and effects of noise](https://www.bctengineer.com/question.md?key=semester_7%2Fenvironment_and_pollution_control%3Anoise-levels-places-effects) — Environment and Pollution Control, chapter 6

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