---
title: "Sound pressure level and power level calculations"
subject: "Environment and Pollution Control"
chapter: 6
key: "semester_7/environment_and_pollution_control:spl-pwl-numerical"
source: https://www.bctengineer.com/q/semester_7/environment_and_pollution_control/spl-pwl-numerical
site: BCT Engineering
---

# Sound pressure level and power level calculations

Environment and Pollution Control · Chapter 6: Noise pollution

- Asked in: year not given, 4 marks
- Video: [Sound Power, Intensity, Pressure Level](https://www.youtube.com/watch?v=p2RLn6q8Z20) — 0순위_NVH
- 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/spl-pwl-numerical
- Whole chapter: https://www.bctengineer.com/md/semester_7/environment_and_pollution_control/chapter-6

## Question

(a) Define sound pressure level (SPL) and sound power level (PWL). (b) A sound has an rms pressure of 0.2 Pa. Find its SPL. (c) A machine radiates 0.01 W of sound power. Find its PWL, and its SPL at 10 m on open flat ground (hemispherical radiation). Reference values: p_ref = 20 micro-Pa and W_ref = 10^-12 W.

## Short answer

(b) SPL = 20 log(0.2/20×10⁻⁶) = 80 dB. (c) PWL = 10 log(0.01/10⁻¹²) = 100 dB. For hemispherical spreading, SPL at 10 m = 100 – 20 log(10) – 8 = 72 dB. Results: 80 dB SPL, 100 dB PWL, 72 dB at 10 m.

## Answer

#### (a) Definitions
- **SPL:** $L_p = 10\log_{10}\left(\dfrac{p^2}{p_{ref}^2}\right) = 20\log_{10}\left(\dfrac{p}{p_{ref}}\right)$ dB, with $p_{ref} = 20\ \mu$Pa (threshold of hearing at 1 kHz). It depends on the distance from the source.
- **PWL:** $L_w = 10\log_{10}\left(\dfrac{W}{W_{ref}}\right)$ dB, with $W_{ref} = 10^{-12}$ W. It is the property of the source and is independent of distance and surroundings.

#### (b) SPL
$$
L_p = 20\log_{10}\left(\frac{0.2}{20\times10^{-6}}\right) = 20\log_{10}(10^4) = 80\ \text{dB}
$$

#### (c) PWL and SPL at 10 m
$$
L_w = 10\log_{10}\left(\frac{0.01}{10^{-12}}\right) = 10\log_{10}(10^{10}) = 100\ \text{dB}
$$
For hemispherical spreading, intensity $I = W/(2\pi r^2)$, so
$$
L_p = L_w - 20\log_{10} r - 8 = 100 - 20\log_{10}(10) - 8 = 100 - 20 - 8 = 72\ \text{dB}
$$
(The constant is $10\log_{10}2\pi = 8$.)

**Answer: (b) SPL = 80 dB; (c) PWL = 100 dB; SPL at 10 m = 72 dB.**

## Check yourself

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

1. Which statement about PWL is correct?
   A. It depends on the distance from the source
   B. It is a property of the source, independent of distance and surroundings
   C. It is defined with p_ref = 20 micro-Pa
   D. It is always equal to SPL
   Answer: B. PWL = 10 log(W/W_ref) describes the source itself; SPL depends on distance. (Key point: SPL vs PWL)

2. A sound has rms pressure 0.2 Pa and p_ref = 20 micro-Pa. What is the SPL?
   A. 40 dB
   B. 60 dB
   C. 80 dB
   D. 100 dB
   Answer: C. Lp = 20 log(0.2 / 20e-6) = 20 log(10^4) = 80 dB. (Key point: SPL calculation)

3. A machine radiates 0.01 W with W_ref = 10^-12 W. What is its PWL?
   A. 100 dB
   B. 80 dB
   C. 120 dB
   D. 10 dB
   Answer: A. Lw = 10 log(0.01/10^-12) = 10 log(10^10) = 100 dB. (Key point: PWL calculation)

4. For hemispherical radiation Lp = Lw - 20 log r - 8. What is Lp at 10 m for Lw = 100 dB?
   A. 80 dB
   B. 92 dB
   C. 72 dB
   D. 60 dB
   Answer: C. 100 - 20 log(10) - 8 = 100 - 20 - 8 = 72 dB; the 8 comes from 10 log(2 pi). (Key point: Hemispherical spreading)

## Related in Environment and Pollution Control

- [Adding sound levels from several sources](https://www.bctengineer.com/question.md?key=semester_7%2Fenvironment_and_pollution_control%3Acombining-noise-levels-numerical) — Environment and Pollution Control, chapter 6
- [Maximum ground-level concentration from a stack](https://www.bctengineer.com/question.md?key=semester_7%2Fenvironment_and_pollution_control%3Amaximum-ground-level-concentration) — Environment and Pollution Control, chapter 2
- [Gaussian plume model and ground concentration](https://www.bctengineer.com/question.md?key=semester_7%2Fenvironment_and_pollution_control%3Agaussian-plume-ground-concentration) — Environment and Pollution Control, chapter 2

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