Equal Loudness Curves & Intensity Perception Revision Notes | A Level AQA Physics

1. Why Your Ears Lie to You

Play a 100 Hz bass note and a 3000 Hz whistle at exactly the same physical intensity. They will not sound equally loud. Your ears are not flat amplifiers — they are frequency-selective, and the physics behind that selectivity is what this lesson is about.

This matters enormously in medical physics. Audiologists use the patterns of human loudness perception to diagnose hearing loss, calibrate hearing aids, and interpret audiograms. To do any of that, you need two things: equal loudness curves and a logarithmic intensity scale.

2. Intensity and the Logarithmic Scale

Sound carries energy. The intensity $I$ of a sound wave is the power transmitted per unit area, measured in watts per square metre (W m$^{-2}$). The human ear can detect sounds ranging from about $1 \times 10^{-12}$ W m$^{-2}$ (the threshold of hearing) up to around $1$ W m$^{-2}$ (corresponding to approximately 120 dB), approaching the threshold of pain. That is a factor of one trillion — twelve orders of magnitude.

A linear scale is useless here. If you plotted intensity on a ruler from 0 to 1 W m$^{-2}$, the entire range of quiet conversation would be crammed into the first invisible sliver near zero. The ear itself compresses this enormous range into a manageable perception of loudness, and the decibel scale mirrors that compression mathematically.

Notice what the logarithm does: every time the intensity increases by a factor of 10, the intensity level increases by exactly 10 dB. A factor of 100 gives 20 dB; a factor of 1000 gives 30 dB. The enormous physical range of $10^{12}$ maps neatly onto a 0–120 dB scale that the brain can process.

Worked example: Calculating Intensity Level in Decibels

A sound has an intensity of $1 \times 10^{-6}$ W m$^{-2}$. Calculate its intensity level in dB.

1. $L = 10 \log_{10}\left(\frac{I}{I_0}\right)$ — State the formula first — this earns the method mark [M1].
2. $L = 10 \log_{10}\left(\frac{1 \times 10^{-6}}{1 \times 10^{-12}}\right)$ — Both values are in W m$^{-2}$, so units cancel cleanly. [M1]
3. $\frac{1 \times 10^{-6}}{1 \times 10^{-12}} = 10^{-6-(-12)} = 10^{6}$ — Subtracting exponents: $-6 - (-12) = -6 + 12 = 6$. The ratio is $10^6$.
4. $L = 10 \times \log_{10}(10^{6}) = 10 \times 6 = 60$ — $\log_{10}(10^6) = 6$ exactly. Multiply by 10.
5. $L = 60 \text{ dB}$ — Always include the unit dB. [A1] This is roughly the level of normal conversation.

Answer: $L = 60$ dB

Here is a useful pattern to memorise: every 10 dB increase corresponds to a tenfold increase in intensity. So 70 dB is ten times more intense than 60 dB, and 80 dB is one hundred times more intense than 60 dB. This relationship comes directly from the definition of the logarithm.

3. Why a Logarithmic Scale Reflects Human Perception

The logarithmic scale is not an arbitrary mathematical convenience — it reflects how the ear actually works. The ear responds to ratios of intensity, not absolute differences. Equal multiplicative increases in intensity produce equal steps in perceived loudness, regardless of the starting level. This is a general principle of sensory physiology known as the Weber-Fechner relationship.

In practical terms, this means that equal steps on the decibel scale correspond to equal steps in perceived loudness — at least approximately. A change from 40 dB to 50 dB feels roughly as significant as a change from 80 dB to 90 dB, even though the second jump involves a far larger absolute increase in physical intensity. The logarithmic scale encodes the ear's compressive response into the measurement system itself.

4. Equal Loudness Curves: What They Show

Now that the decibel scale is established, the next question is: does 60 dB always sound equally loud, regardless of frequency? The answer is no — and equal loudness curves make this visible.

Picture a graph with frequency on the horizontal axis (ranging from about 20 Hz to 20 kHz, plotted on a logarithmic frequency scale) and sound pressure level in dB SPL on the vertical axis. Each curve on this graph connects all the frequency-intensity combinations that a listener judges to be equally loud. These are the equal loudness curves, originally determined experimentally by Fletcher and Munson in 1933 and later refined into the ISO 226 standard.

Each curve is labelled in phons. A sound has a loudness of $N$ phons if it sounds equally loud as a 1 kHz pure tone at $N$ dB SPL. The 1 kHz reference point is where every equal loudness curve crosses its phon value — so the 40-phon curve passes through 40 dB at 1 kHz, the 70-phon curve passes through 70 dB at 1 kHz, and so on.

The shape of the curves reveals the ear's frequency sensitivity directly. At low frequencies (below about 200 Hz), the curves rise steeply — meaning you need a much higher physical intensity to achieve the same perceived loudness as at 1 kHz. At around 3–4 kHz, the curves dip to a minimum, showing that the ear is most sensitive in this region. This dip corresponds to the resonant frequency of the ear canal, which naturally amplifies sounds in this range.

5. Reading and Interpreting Equal Loudness Curves

Being able to extract information from equal loudness curves is a core exam skill. There are three types of reading you need to practise: finding the dB SPL required at a given frequency for a given phon level, comparing the sensitivity of the ear at different frequencies, and explaining what the overall shape of the curves tells us about perception.

• The curves are not flat — a flat curve would mean equal sensitivity at all frequencies, which is not the case for the human ear.
• At low frequencies (below ~200 Hz), the curves rise steeply, indicating reduced sensitivity: more intensity is needed to achieve the same perceived loudness.
• Near 3–4 kHz, the curves dip to their lowest point, indicating peak sensitivity: less intensity is needed here than at 1 kHz for the same perceived loudness.
• At very high frequencies (above ~10 kHz), sensitivity falls again and the curves rise.
• As loudness level increases (higher phon values), the curves become flatter — the ear's frequency response becomes more uniform at high intensities.

That last point is particularly important and often tested. At low phon levels (quiet sounds), the low-frequency region of the curve is very high — the ear barely detects bass frequencies at low volumes. At high phon levels (loud sounds), the curve flattens out, meaning bass frequencies are perceived much more proportionally. This is why music played quietly sounds 'thin' and lacking in bass compared to the same music played loudly at the same frequency balance.

6. Worked Interpretation: Using the Curves in Context

Exam questions on equal loudness curves often ask you to read off values and then explain what those values mean physically. The step-by-step below models the reasoning chain you should use.

Worked example: Interpreting Equal Loudness Curve Data

From an equal loudness curve diagram, a student reads that a 100 Hz tone requires 70 dB SPL to lie on the 40-phon curve, while a 1 kHz tone requires 40 dB SPL to lie on the same curve. Explain what this tells us about the ear's sensitivity at these two frequencies, and calculate the ratio of intensities of the two tones.

1. Both tones lie on the 40-phon curve, so a normal listener judges them to be equally loud. — The phon value defines perceived loudness. Same phon = same perceived loudness, regardless of frequency.
2. 100 Hz tone: 70 dB SPL. 1 kHz tone: 40 dB SPL. Difference = 70 - 40 = 30 dB. — The 100 Hz tone needs 30 dB more intensity level to sound equally loud. This means the ear is less sensitive at 100 Hz than at 1 kHz. [B1]
3. $\Delta L = 10 \log_{10}\left(\frac{I_{100}}{I_{1k}}\right) = 30 \text{ dB}$ — A difference in intensity level corresponds to a ratio of intensities. Rearrange to find the ratio.
4. $\log_{10}\left(\frac{I_{100}}{I_{1k}}\right) = \frac{30}{10} = 3$ — Divide both sides by 10.
5. $\frac{I_{100}}{I_{1k}} = 10^{3} = 1000$ — The 100 Hz tone has 1000 times the physical intensity of the 1 kHz tone, yet both sound equally loud. [A1] This powerfully illustrates the ear's non-uniform frequency response.

Answer: The ear is less sensitive at 100 Hz than at 1 kHz. The 100 Hz tone must have 1000 times the physical intensity of the 1 kHz tone to sound equally loud.

That factor of 1000 in intensity for the same perceived loudness is striking. It underlines why a linear intensity scale would be completely inadequate for describing human hearing — and why both the decibel scale and the phon scale are necessary tools.

7. Exam Technique and Common Mistakes

Questions on this topic tend to fall into three categories: calculation of intensity level in dB, description or explanation of what equal loudness curves show, and evaluation of why a logarithmic scale is necessary. Each requires a slightly different approach.

For calculation questions, the most common error is forgetting that $I_0 = 1 \times 10^{-12}$ W m$^{-2}$ and substituting a different reference value, or failing to include the factor of 10 in front of the logarithm. Always write the formula in full before substituting — this secures the method mark even if arithmetic slips occur later.

For 'explain' questions about the logarithmic scale, the mark scheme typically requires two linked ideas: first, that the range of intensities detectable by the ear spans many orders of magnitude (state the range, e.g. $10^{-12}$ to $1$ W m$^{-2}$); second, that the ear responds to ratios of intensity rather than absolute differences, so a logarithmic scale reflects perceived loudness more accurately than a linear one.

Taken together, the logarithmic decibel scale and the equal loudness curves form the quantitative foundation for understanding human hearing — and both spec requirements are directly examined through calculation, description, and explanation questions at A level.