Hearing response

Q: What is sound intensity (\(I\)) and how is it calculated?

Sound intensity is the rate of sound energy transmitted per unit area perpendicular to the wave's direction, calculated as \(I = \frac{P}{A}\) and measured in \(\text{W}\cdot\text{m}^{-2}\).

Q: What are the values of the human threshold of hearing (\(I_0\)) and the threshold of pain?

The threshold of hearing at \(1\text{ kHz}\) is \(I_0 = 1.0 \times 10^{-12}\text{ W}\cdot\text{m}^{-2}\), and the threshold of pain is approximately \(1\text{ W}\cdot\text{m}^{-2}\) (\(120\text{ dB}\)).

Q: Why is a logarithmic scale used for sound intensity level?

Because the human ear responds logarithmically to sound intensity across a vast range from \(10^{-12}\text{ W}\cdot\text{m}^{-2}\) to \(1\text{ W}\cdot\text{m}^{-2}\).

Q: What is the formula for sound Intensity Level (\(\text{IL}\)) in decibels?

\(\text{IL} = 10 \log_{10}\left(\frac{I}{I_0}\right)\), where \(I_0 = 1.0 \times 10^{-12}\text{ W}\cdot\text{m}^{-2}\) and \(\text{IL}\) is in \(\text{dB}\).

Q: How does the Intensity Level change when sound intensity is doubled or increased tenfold?

Doubling intensity adds \(\approx 3\text{ dB}\), a 10-fold increase adds \(10\text{ dB}\), and a 100-fold increase adds \(20\text{ dB}\).

Q: What is the audible frequency range for human hearing and at what frequency is the ear most sensitive?

The human audible range is \(20\text{ Hz}\) to \(20\text{,}000\text{ Hz}\), with peak sensitivity around \(2\text{ kHz}\) to \(4\text{ kHz}\) (\(\approx 3\text{ kHz}\)).

Q: What does a downward dip in an equal loudness contour signify?

It indicates a region of higher auditory sensitivity where a lower Intensity Level (\(\text{dB}\)) produces the same perceived loudness.

Q: How is the phon scale defined to measure subjective loudness?

The loudness in phons of any sound equals the Intensity Level in \(\text{dB}\) of an equally loud pure tone at \(1000\text{ Hz}\).

Q: What is the purpose of the \(\text{dBA}\) scale?

It uses an electronic frequency filter on sound meters to mimic the frequency-dependent sensitivity of the human ear for realistic noise assessment.

Life and Health Sciences 0008 · CCEA A Level · 8 min read · Updated: 31 Aug 2026

Chapter Study Notes: Hearing Response

Welcome to your study notes for Hearing Response in Unit A2 4: Sound and Light! Have you ever wondered why whispering in a quiet library sounds clear, but shouting at a rock concert doesn't sound a million times louder? The human ear is a remarkable biological detector that processes sounds spanning an enormous range of intensities and frequencies. In this chapter, we explore how sound intensity is measured, why we use a logarithmic decibel scale, how our ears respond differently to different frequencies, and how sound meters are adjusted to protect human hearing.

Don't worry if the maths or graphs seem challenging at first—we will break down every single formula and curve step by step!

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1. Sound Intensity and Auditory Thresholds

Before we look at how the brain perceives sound, we need to understand the physical property of sound traveling through the air.

What is Sound Intensity?

Sound Intensity (\(I\)) is defined as the rate at which sound energy is transmitted per unit area perpendicular to the direction in which the sound wave is traveling. Because rate of energy is power (in Watts, \(\text{W}\)), intensity is calculated as:

\(I = \frac{P}{A}\)

Where:
• \(I\) = Sound Intensity, measured in Watts per square metre (\(\text{W}\cdot\text{m}^{-2}\) or \(\text{W/m}^2\))
• \(P\) = Acoustic Power emitted by the source, measured in Watts (\(\text{W}\))
• \(A\) = Surface area through which the sound passes, measured in square metres (\(\text{m}^2\))

Everyday Analogy: Think of a garden hose with a spray nozzle. The water flow represents power (\(P\)). If you spray the water over a tiny patch of soil, the water is intense. If you spray the same amount of water over an entire lawn, the water hitting each square metre is much less intense!

Key Auditory Thresholds

The human ear can detect an astonishingly broad range of sound intensities:

1. Threshold of Hearing (\(I_0\)):
This is the minimum sound intensity detectable by an average, healthy human ear at a standard reference frequency of \(1\text{ kHz}\) (\(1000\text{ Hz}\)).
\(I_0 = 1.0 \times 10^{-12}\text{ W}\cdot\text{m}^{-2}\)

2. Threshold of Pain:
This is the upper limit of sound intensity where the sensation of hearing turns into physical discomfort or pain.
\(\text{Threshold of Pain} \approx 1\text{ W}\cdot\text{m}^{-2}\) (which corresponds to \(120\text{ dB}\)).

Did you know? The threshold of pain (\(1\text{ W}\cdot\text{m}^{-2}\)) is one trillion (\(10^{12}\)) times more intense than the threshold of hearing (\(10^{-12}\text{ W}\cdot\text{m}^{-2}\))! If our ears responded in a simple linear way, loud sounds would utterly overwhelm our nervous system.

Key Takeaway for Section 1: Sound intensity \(I\) is measured in \(\text{W}\cdot\text{m}^{-2}\). The human ear can detect sounds as faint as \(I_0 = 1.0 \times 10^{-12}\text{ W}\cdot\text{m}^{-2}\) and tolerate sounds up to \(1\text{ W}\cdot\text{m}^{-2}\).

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2. The Decibel Scale & Intensity Level (\(\text{IL}\))

The Logarithmic Nature of the Human Ear

Because the range of sound intensities from \(10^{-12}\text{ W}\cdot\text{m}^{-2}\) to \(1\text{ W}\cdot\text{m}^{-2}\) is so vast, our ears do not respond linearly. Instead, the ear responds logarithmically to changes in sound intensity. When a sound intensity increases tenfold, our brain perceives it as a steady step up in loudness, not a ten-times explosion in volume.

The Intensity Level Formula

To reflect this logarithmic behaviour, scientists use the Intensity Level (\(\text{IL}\)), measured in decibels (\(\text{dB}\)):

\(\text{Intensity Level (IL)} = 10 \log_{10}\left(\frac{I}{I_0}\right)\)

Where:
• \(\text{IL}\) = Intensity Level in decibels (\(\text{dB}\))
• \(I\) = The measured sound intensity in \(\text{W}\cdot\text{m}^{-2}\)
• \(I_0\) = The reference threshold of hearing (\(1.0 \times 10^{-12}\text{ W}\cdot\text{m}^{-2}\))

Crucial Quantitative Relationships (Rules of Thumb)

You can quickly check your exam calculations using these logarithmic rules:

• Doubling the intensity (\(2 \times I\)): Increases the Intensity Level by approximately \(+3\text{ dB}\).
\(\Delta\text{IL} = 10 \log_{10}(2) \approx 3.01\text{ dB}\)
• A 10-fold increase (\(10 \times I\)): Increases the Intensity Level by \(+10\text{ dB}\).
\(\Delta\text{IL} = 10 \log_{10}(10) = 10\text{ dB}\)
• A 100-fold increase (\(100 \times I\)): Increases the Intensity Level by \(+20\text{ dB}\).
\(\Delta\text{IL} = 10 \log_{10}(100) = 20\text{ dB}\)

Step-by-Step Calculation Example

Question: A noisy machine produces a sound intensity of \(1.0 \times 10^{-5}\text{ W}\cdot\text{m}^{-2}\). Calculate the sound intensity level in \(\text{dB}\).

Step 1: Write down the known values.
\(I = 1.0 \times 10^{-5}\text{ W}\cdot\text{m}^{-2}\)
\(I_0 = 1.0 \times 10^{-12}\text{ W}\cdot\text{m}^{-2}\)

Step 2: Substitute values into the decibel formula.
\(\text{IL} = 10 \log_{10}\left(\frac{1.0 \times 10^{-5}}{1.0 \times 10^{-12}}\right)\)

Step 3: Simplify the fraction inside the logarithm.
\(\frac{1.0 \times 10^{-5}}{1.0 \times 10^{-12}} = 10^{(-5 - (-12))} = 10^{7}\)

Step 4: Calculate the logarithm and multiply by 10.
\(\log_{10}(10^7) = 7\)
\(\text{IL} = 10 \times 7 = 70\text{ dB}\)

Common Examiner Pitfall: Adding Decibels Linearly

Never add decibels together like normal numbers!
If one machine generates \(50\text{ dB}\) and an identical second machine is turned on, the total sound intensity doubles (\(2 \times I\)). That adds \(+3\text{ dB}\), giving a total of \(53\text{ dB}\), NOT \(100\text{ dB}\)! A sound level of \(100\text{ dB}\) would be \(100\text{,}000\) times more intense!

Key Takeaway for Section 2: The decibel scale is logarithmic. Doubling sound intensity adds \(\approx 3\text{ dB}\), a \(10\times\) increase adds \(10\text{ dB}\), and a \(100\times\) increase adds \(20\text{ dB}\).

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3. Frequency Range, Equal Loudness Contours, and the Phon Scale

Human Audible Frequency Range

The human ear does not hear all frequencies equally. The normal range of human hearing extends from approximately \(20\text{ Hz}\) to \(20\text{,}000\text{ Hz}\) (or \(20\text{ kHz}\)).

Peak Sensitivity: The human ear is naturally most sensitive in the frequency band around \(2\text{ kHz}\) to \(4\text{ kHz}\) (commonly cited as approximately \(3\text{ kHz}\)). This sensitivity occurs because of the natural acoustic resonance of the auditory canal (ear canal), which amplifies frequencies in this specific range.

Equal Loudness Contours (Curves)

An Equal Loudness Contour is a graph plotting Sound Intensity Level (\(\text{dB}\)) against Frequency (\(\text{Hz}\)) for tones that are perceived by a human listener to have the exact same loudness.

Key features to recognize on an Equal Loudness Graph:
• The Characteristic Dip around \(3\text{ kHz}\): The curves dip downward between \(2\text{ kHz}\) and \(4\text{ kHz}\). A lower Intensity Level (\(\text{dB}\)) is required here to produce the same sensation of loudness because the ear is at its maximum sensitivity.
• Steep Rise at Low Frequencies: Below \(500\text{ Hz}\), the curves curve steeply upward. A much higher Intensity Level (\(\text{dB}\)) is needed for a low-pitch bass tone to sound as loud as a mid-frequency tone.
• Rise at Very High Frequencies: Above \(10\text{ kHz}\), the curves rise again, showing reduced auditory sensitivity.
• Flattening at High Sound Levels: At high intensity levels (close to the threshold of pain), the curves become flatter, meaning the ear's response is more uniform across frequencies at very loud volumes.

Watch Out in Exams: A downward dip in an equal loudness curve means HIGHER sensitivity (less physical energy needed to hear it), NOT worse hearing!

The Phon Scale (Subjective Loudness)

While Intensity Level in \(\text{dB}\) is an objective physical measurement, loudness is a subjective sensation in the human brain. To quantify loudness, scientists use the Phon scale.

Definition of the Phon:
The loudness of any sound in phons is numerically equal to the Intensity Level in \(\text{dB}\) of an equally loud pure tone at the reference frequency of \(1000\text{ Hz}\) (\(1\text{ kHz}\)).

Example: If a \(100\text{ Hz}\) bass tone needs an intensity level of \(62\text{ dB}\) to sound equally loud as a \(40\text{ dB}\) tone at \(1000\text{ Hz}\), then the \(100\text{ Hz}\) tone has a loudness of \(40\text{ phons}\).

Memory Trick: At exactly \(1\text{ kHz}\) (\(1000\text{ Hz}\)), \(\text{phons} = \text{dB}\). Everywhere else on that curve, the sound has the exact same phon value, even though the \(\text{dB}\) value changes!

Key Takeaway for Section 3: Equal loudness curves dip at \(\approx 3\text{ kHz}\) due to ear canal resonance. The phon scale measures subjective loudness, calibrated to equal the \(\text{dB}\) level at \(1000\text{ Hz}\).

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4. The \(\text{dBA}\) Scale (Adjusted Decibels)

Why Do We Need the \(\text{dBA}\) Scale?

Standard sound level meters measure pure physical intensity level in unweighted \(\text{dB}\). However, if an unweighted sound meter records a loud \(50\text{ Hz}\) rumble at \(70\text{ dB}\), a human might barely notice it, whereas a \(70\text{ dB}\) sound at \(3\text{ kHz}\) would sound very loud and could potentially damage hearing over time.

To assess real noise hazards in workplaces and communities, meters use an electronically filtered scale called the \(\text{dBA}\) (A-weighted decibel) scale.

Characteristics and Calibration of \(\text{dBA}\)

The \(\text{dBA}\) filter modifies raw sound measurements to match human ear sensitivity across frequencies:
• At \(1\text{ kHz}\) (\(1000\text{ Hz}\)): The filter is calibrated so that \(\text{dBA} = \text{dB}\). There is zero adjustment at the reference frequency.
• Around \(3\text{ kHz}\): The \(\text{dBA}\) response peaks (values are slightly boosted) to mirror the ear's heightened sensitivity.
• At Low Frequencies (< \(500\text{ Hz}\)) and Very High Frequencies: The \(\text{dBA}\) filter heavily attenuates (reduces/subtracts) the readings because the human ear is relatively insensitive to these frequencies.

Summary of \(\text{dBA}\) in Practice: A sound level meter set to \(\text{dBA}\) will register low numbers for low-frequency rumbles and high numbers for mid-frequency tones, accurately reflecting how loud and potentially damaging the noise is to a human worker.

Key Takeaway for Section 4: The \(\text{dBA}\) scale is an electronically weighted scale designed to mimic human hearing. It attenuates low and high frequencies and matches the standard \(\text{dB}\) scale at \(1\text{ kHz}\).

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5. Chapter Summary & Quick Reference Guide

Distinguishing the Four Core Terms

• Sound Intensity (\(I\)): Objective rate of energy per unit area, measured in \(\text{W}\cdot\text{m}^{-2}\).
• Intensity Level (\(\text{IL}\)): Objective logarithmic ratio compared to \(I_0\), measured in \(\text{dB}\).
• Loudness: Subjective human perception, measured in phons (equal to \(\text{dB}\) at \(1\text{ kHz}\)).
• \(\text{dBA}\) Reading: Filtered meter reading simulating the ear's frequency-dependent sensitivity.

Quick Check Formula & Benchmark Sheet

• Intensity formula: \(I = \frac{P}{A}\)
• Decibel formula: \(\text{IL} = 10 \log_{10}\left(\frac{I}{I_0}\right)\)
• Threshold of Hearing: \(I_0 = 1.0 \times 10^{-12}\text{ W}\cdot\text{m}^{-2}\)
• Threshold of Pain: \(\approx 1\text{ W}\cdot\text{m}^{-2} = 120\text{ dB}\)
• Human Hearing Range: \(20\text{ Hz}\) to \(20\text{,}000\text{ Hz}\)
• Most Sensitive Frequency: \(\approx 3\text{ kHz}\) (\(2\text{ kHz}\) to \(4\text{ kHz}\))
• Reference Frequency for Phons and \(\text{dBA}\): \(1000\text{ Hz}\) (\(1\text{ kHz}\))

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is sound intensity (\(I\)) and how is it calculated?

Sound intensity is the rate of sound energy transmitted per unit area perpendicular to the wave's direction, calculated as \(I = \frac{P}{A}\) and measured in \(\text{W}\cdot\text{m}^{-2}\).

What are the values of the human threshold of hearing (\(I_0\)) and the threshold of pain?

The threshold of hearing at \(1\text{ kHz}\) is \(I_0 = 1.0 \times 10^{-12}\text{ W}\cdot\text{m}^{-2}\), and the threshold of pain is approximately \(1\text{ W}\cdot\text{m}^{-2}\) (\(120\text{ dB}\)).

Why is a logarithmic scale used for sound intensity level?

Because the human ear responds logarithmically to sound intensity across a vast range from \(10^{-12}\text{ W}\cdot\text{m}^{-2}\) to \(1\text{ W}\cdot\text{m}^{-2}\).

What is the formula for sound Intensity Level (\(\text{IL}\)) in decibels?

\(\text{IL} = 10 \log_{10}\left(\frac{I}{I_0}\right)\), where \(I_0 = 1.0 \times 10^{-12}\text{ W}\cdot\text{m}^{-2}\) and \(\text{IL}\) is in \(\text{dB}\).

How does the Intensity Level change when sound intensity is doubled or increased tenfold?

Doubling intensity adds \(\approx 3\text{ dB}\), a 10-fold increase adds \(10\text{ dB}\), and a 100-fold increase adds \(20\text{ dB}\).

What is the audible frequency range for human hearing and at what frequency is the ear most sensitive?

The human audible range is \(20\text{ Hz}\) to \(20\text{,}000\text{ Hz}\), with peak sensitivity around \(2\text{ kHz}\) to \(4\text{ kHz}\) (\(\approx 3\text{ kHz}\)).

What does a downward dip in an equal loudness contour signify?

It indicates a region of higher auditory sensitivity where a lower Intensity Level (\(\text{dB}\)) produces the same perceived loudness.

How is the phon scale defined to measure subjective loudness?

The loudness in phons of any sound equals the Intensity Level in \(\text{dB}\) of an equally loud pure tone at \(1000\text{ Hz}\).

What is the purpose of the \(\text{dBA}\) scale?

It uses an electronic frequency filter on sound meters to mimic the frequency-dependent sensitivity of the human ear for realistic noise assessment.

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