Chapter: Sound Measurement
Welcome to Sound Measurement! In this chapter of Unit A2 4 (Sound and Light), you will explore how we quantify sound energy, how human biology influences our perception of loudness, and how sound levels are calculated and monitored in medical and workplace settings. Don't worry if the physics and maths seem a bit daunting at first — we will break down every single formula and concept step by step!
---1. Sound Intensity and Auditory Thresholds
What is Sound Intensity?
When an object vibrates, it transmits energy through the surrounding medium as a longitudinal wave. Sound Intensity (\(I\)) is defined as the rate of sound energy flow per unit area perpendicular to the direction of propagation.
Because energy per unit time is power (measured in Watts, \(\text{W}\)), intensity is mathematically defined as power divided by area:
\(I = \frac{P}{A}\)
The standard scientific unit for sound intensity is Watts per square metre (\(\text{W m}^{-2}\)).
The Range of Human Hearing
The human ear is an astonishingly sensitive biological detector. It can detect extraordinarily quiet sounds, but it can also tolerate sound waves carrying millions of times more energy.
• Threshold of Hearing (\(I_0\)): This is the minimum reference sound intensity that a normal human ear can detect at a standard test frequency of \(1\text{ kHz}\) (\(1000\text{ Hz}\)). Its standard value is:
\(I_0 = 1.0 \times 10^{-12} \text{ W m}^{-2}\)
• Threshold of Pain: This is the upper limit where sound intensity becomes physically painful and can rapidly cause acute acoustic trauma or permanent hearing damage. This occurs at an intensity of approximately \(1 \text{ W m}^{-2}\) to \(10 \text{ W m}^{-2}\) (which corresponds to an intensity level of roughly \(120\text{--}130\text{ dB}\)).
Key Takeaway
Sound intensity (\(I\)) is measured in \(\text{W m}^{-2}\). The quietest sound we can hear at \(1\text{ kHz}\) is \(I_0 = 1.0 \times 10^{-12} \text{ W m}^{-2}\), and physical pain occurs between \(1 \text{ W m}^{-2}\) and \(10 \text{ W m}^{-2}\).
---2. The Decibel Scale (Sound Intensity Level)
Why Do We Need a Logarithmic Scale?
Human perception of loudness is logarithmic rather than linear. This means that if the physical intensity of a sound is multiplied by \(10\), our ears perceive it as roughly doubling in loudness, rather than becoming ten times louder.
Furthermore, because the intensity range from the threshold of hearing (\(1.0 \times 10^{-12}\text{ W m}^{-2}\)) to the threshold of pain (\(1\text{ W m}^{-2}\)) spans a factor of \(1,000,000,000,000\) (\(10^{12}\)), dealing with raw numbers is impractical. To compress this enormous range into manageable numbers, we use the Sound Intensity Level, measured in decibels (\(\text{dB}\)).
The Sound Intensity Level Formula
To calculate the sound intensity level in decibels (\(\text{dB}\)), use the formula:
\(\text{Sound Intensity Level (dB)} = 10 \log_{10}\left(\frac{I}{I_0}\right)\)
Where:
• \(I\) = sound intensity being measured (\(\text{W m}^{-2}\))
• \(I_0\) = standard reference threshold of hearing (\(1.0 \times 10^{-12} \text{ W m}^{-2}\))
• \(\log_{10}\) = logarithm to base 10 (use the log button on your calculator, not ln!)
Step-by-Step Calculation Examples
Example 1: Threshold of Hearing
What is the decibel level of a sound at the threshold of hearing (\(I = 1.0 \times 10^{-12} \text{ W m}^{-2}\))?
\(\text{Level} = 10 \log_{10}\left(\frac{1.0 \times 10^{-12}}{1.0 \times 10^{-12}}\right) = 10 \log_{10}(1) = 10 \times 0 = 0\text{ dB}\)
Example 2: Sound at the Threshold of Pain
What is the decibel level for a sound with intensity \(I = 1.0 \text{ W m}^{-2}\)?
\(\text{Level} = 10 \log_{10}\left(\frac{1.0}{1.0 \times 10^{-12}}\right) = 10 \log_{10}(10^{12}) = 10 \times 12 = 120\text{ dB}\)
Adding Sound Levels: The \(+3\text{ dB}\) Rule
Because decibels are logarithmic, you cannot add decibel values using standard addition.
Analogy / Rule: If you have a machine producing a sound level of \(60\text{ dB}\), and you turn on an identical second machine, the total sound intensity doubles (\(2I\)).
\(\text{New Level} = 10 \log_{10}\left(\frac{2I}{I_0}\right) = 10 \log_{10}\left(\frac{I}{I_0}\right) + 10 \log_{10}(2) \approx 60\text{ dB} + 3.01\text{ dB} = 63\text{ dB}\)
Important: Doubling the sound intensity always adds approximately \(+3\text{ dB}\) to the sound intensity level. Two \(60\text{ dB}\) sound sources combined produce \(63\text{ dB}\), never \(120\text{ dB}\)!
Key Takeaway
The decibel scale matches human logarithmic hearing. The formula is \(\text{dB} = 10 \log_{10}\left(\frac{I}{I_0}\right)\). Doubling the intensity results in a \(+3\text{ dB}\) increase.
---3. The Inverse Square Law for Sound Radiation
Point Sources and Spherical Spreading
If an acoustic point source emits sound power (\(P\)) uniformly in all directions into open space, the sound wave spreads out over the surface of an expanding sphere. The surface area of a sphere of radius \(r\) is \(A = 4\pi r^2\).
Therefore, the sound intensity at a distance \(r\) from the source is given by:
\(I = \frac{P}{4\pi r^2}\)
The Inverse Square Relationship
This equation shows that sound intensity is inversely proportional to the square of the distance from the source:
\(I \propto \frac{1}{r^2}\)
• If you double the distance (\(2r\)), the intensity decreases by a factor of \(2^2 = 4\).
• A fourfold decrease in intensity corresponds to a drop of approximately \(6\text{ dB}\):
\(10 \log_{10}\left(\frac{1}{4}\right) \approx -6.02\text{ dB}\)
Key Takeaway
Sound from a point source obeys the Inverse Square Law (\(I \propto \frac{1}{r^2}\)). Doubling your distance from the source quarters the intensity and reduces the sound level by approximately \(6\text{ dB}\).
---4. Frequency Response and Decibel Weighting (dB vs. dBA)
Human Ear Frequency Sensitivity
The human ear does not respond equally to all frequencies of sound:
• The ear is most sensitive to frequencies in the range of \(1\text{ kHz}\) to \(4\text{ kHz}\) (the frequency region most important for understanding human speech).
• The ear is much less sensitive to very low frequencies (bass) and very high frequencies (treble).
Unweighted Decibels (\(\text{dB}\)) vs. A-Weighted Decibels (\(\text{dBA}\))
Because of this biological variation, a simple physical measurement of sound intensity does not accurately reflect how loud or damaging a noise feels to a human.
• \(\text{dB}\) Scale (Unweighted): A purely physical measurement of sound intensity/energy that treats all frequencies equally.
• \(\text{dBA}\) Scale (A-Weighted Decibels): A sound level measurement adjusted with an electronic filter (the "A-weighting" curve) that matches the non-linear frequency sensitivity of the human ear.
Health and Safety Applications
In occupational health, audiology, and environmental monitoring, sound level meters are calibrated to the \(\text{dBA}\) scale. This ensures that noise measurements accurately reflect perceived loudness and the actual risk of hearing impairment or damage to workers.
Key Takeaway
The \(\text{dB}\) scale measures raw physical intensity, while the \(\text{dBA}\) scale incorporates a filter matching the human ear's sensitivity (peak sensitivity between \(1\text{ kHz}\) and \(4\text{ kHz}\)), making it the standard for health and safety regulations.
---5. Common Pitfalls & Examiner Tips
1. Using the Wrong Logarithm Button: Always use \(\log_{10}\) (base 10) on your calculator. Using natural log (\(\ln\)) will produce completely wrong answers.
2. Forgetting the Factor of 10: The formula is \(10 \log_{10}(I/I_0)\). Forgetting the \(10\) calculates bels, not decibels.
3. Inverting the Fraction: Remember that \(I\) goes on top and \(I_0\) goes on the bottom: \(\frac{I}{I_0}\). Putting \(I_0\) on top results in incorrect negative decibel values for loud sounds.
4. Linear Decibel Addition: Never add decibels directly (e.g., \(70\text{ dB} + 70\text{ dB} \neq 140\text{ dB}\)). Doubling intensity adds \(+3\text{ dB}\), giving \(73\text{ dB}\).
5. Explaining \(\text{dBA}\) Incorrectly: Do not describe \(\text{dBA}\) as a "different power unit". It is a frequency-filtered decibel measurement that mirrors human ear sensitivity.
---Quick Revision Checklist
• Can you state the definition and unit of sound intensity (\(\text{W m}^{-2}\))?
• Do you know the value of \(I_0\) (\(1.0 \times 10^{-12} \text{ W m}^{-2}\) at \(1\text{ kHz}\))?
• Can you use \(\text{dB} = 10 \log_{10}\left(\frac{I}{I_0}\right)\) to calculate sound levels?
• Can you explain why doubling sound intensity leads to a \(+3\text{ dB}\) rise?
• Can you apply \(I = \frac{P}{4\pi r^2}\) to explain the inverse square law?
• Can you distinguish clearly between \(\text{dB}\) and \(\text{dBA}\)?