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Hearing is the second of the special senses treated in the AQA medical-physics option, and it presents a striking contrast with vision. Where the eye is an imaging system that converts a 2-D irradiance pattern into a neural code, the ear is a frequency analyser that converts a single 1-D pressure waveform into a continuous neural representation of frequency content. The dynamic range of normal hearing — from a threshold of about 10⁻¹² W m⁻² up to a pain threshold of around 1 W m⁻² — spans twelve orders of magnitude in acoustic intensity. No camera or microphone built by humans matches this range, and the logarithmic decibel scale is the natural language for talking about it. This lesson reconstructs the ear as a three-stage acoustic system, defines the decibel and the dBA weighting, and interprets the audiogram — the central diagnostic tool of clinical audiology.
Spec mapping: This lesson sits under AQA 7408 section 3.10.2 (Physics of the ear). It covers the basic anatomy of the outer, middle and inner ear; the role of the ossicles in impedance matching between air and cochlear fluid; the threshold of hearing (I_0 ≈ 10⁻¹² W m⁻² at 1 kHz) and the threshold of feeling/pain; sound-intensity level expressed in decibels as L = 10 log_{10}(I/I_0); the frequency-dependent sensitivity of the ear with peak sensitivity in the 1-4 kHz range; the dBA weighting curve as a model of perceived loudness; the audiogram as a frequency-versus-threshold-shift plot and the distinction between conductive and sensorineural hearing loss. (Refer to the official AQA specification document for exact wording.)
Synoptic links:
- Section 3.4 (waves): sound is a longitudinal mechanical wave; intensity I = (1/2)ρcω²s_0² and pressure-amplitude/intensity relations link directly to the wave-equation work from earlier in the course. The decibel scale is just a logarithmic restatement of intensity.
- Section 3.10.4 (ultrasound): the same acoustic-impedance ideas (Z = ρc; reflection coefficient (Z_2 - Z_1)² / (Z_2 + Z_1)²) that explain impedance matching across the middle ear also explain why a coupling gel is essential in diagnostic ultrasound. Both rely on the same physics.
- Section 3.6 (oscillations / resonance): the basilar membrane in the cochlea behaves like a graded mechanical resonator — different positions along its length have different natural frequencies. The place theory of pitch is, at heart, a resonance argument.
The ear is naturally divided into three stages, each with a clear acoustical role.
Outer ear — comprises the pinna (the visible cartilage flap) and the external auditory meatus (ear canal). Funnels acoustic energy onto the tympanic membrane (eardrum). The pinna provides some directional cueing through frequency-dependent diffraction.
Middle ear — a small air-filled cavity containing the three smallest bones in the body: the malleus, incus and stapes (collectively, the ossicles). The stapes' footplate rests on the oval window of the cochlea. The middle ear's role is impedance matching.
Inner ear — the cochlea, a fluid-filled spiral cavity containing the basilar membrane with its array of hair cells (mechanically sensitive neural transducers). Mechanical vibrations in the cochlear fluid generate travelling waves on the basilar membrane; hair cells along the membrane respond to displacement and fire neural signals encoding both frequency and intensity.
Air has acoustic impedance Z_air ≈ 430 kg m⁻² s⁻¹. The fluid filling the cochlea (perilymph) has Z_fluid ≈ 1.5 × 10⁶ kg m⁻² s⁻¹ — about 3,500 times larger. If a sound wave in air met cochlear fluid directly through a flat membrane, the intensity reflection coefficient would be
α_r = (Z_2 - Z_1)² / (Z_2 + Z_1)² ≈ (1.5 × 10⁶ / 1.5 × 10⁶)² → close to 1.
so very little acoustic power would couple into the fluid; essentially everything would reflect back into the air. The middle-ear ossicles fix this in two ways:
Together these provide a net acoustic-pressure gain of about a factor of 22 — close to the optimum for matching air to perilymph and a beautifully evolved piece of acoustic engineering.
By long-standing convention, the threshold of hearing of a young, healthy adult listening to a pure 1 kHz tone is defined as
I_0 = 1.0 × 10⁻¹² W m⁻²
This is the reference intensity used in the decibel scale. The corresponding pressure amplitude is about 2 × 10⁻⁵ Pa — about 5 × 10⁻¹¹ atmospheres. The eardrum is displaced by less than the diameter of a hydrogen atom at the threshold of audibility, which is one of those facts that earns a small place in everyone's mental library.
At the loud end:
Threshold of feeling / pain ≈ 1 W m⁻² (varies a little between sources; the AQA spec uses 1 W m⁻² as the canonical figure).
The dynamic range of normal hearing therefore spans 10⁻¹² to 10⁰ W m⁻² — a factor of 10¹² in acoustic intensity. Such a range is hopelessly clumsy on a linear scale; the decibel is the natural unit.
Sound-intensity level L is defined relative to I_0:
L = 10 log_{10}(I / I_0) measured in decibels (dB).
A doubling of intensity adds 10 log_{10}(2) ≈ 3 dB. A tenfold increase in intensity adds 10 dB. A factor of 100 in intensity adds 20 dB. These three benchmarks are worth memorising.
| Source | Approximate intensity (W m⁻²) | Level (dB) |
|---|---|---|
| Threshold of hearing | 10⁻¹² | 0 |
| Quiet whisper at 1 m | 10⁻¹⁰ | 20 |
| Normal conversation at 1 m | 10⁻⁶ | 60 |
| Busy traffic | 10⁻⁴ | 80 |
| Pneumatic drill at 1 m | 10⁻² | 100 |
| Threshold of feeling/pain | 1 | 120 |
The figures above are illustrative textbook values; actual levels vary considerably with environment.
Two identical machines in a workshop each radiate 70 dB at the operator's position. What is the combined sound-intensity level?
Two identical intensities sum to twice the intensity. 70 dB corresponds to I = 10⁻⁵ W m⁻²; doubling gives 2 × 10⁻⁵ W m⁻²; the new level is
L = 10 log_{10}(2 × 10⁻⁵ / 10⁻¹²) = 10 × (log_{10}(2) + 7) = 10 × (0.301 + 7) = 73 dB.
So two equal-intensity sources increase the level by 3 dB — never 6, never doubled. The logarithmic scale flattens the addition.
The ear is not equally sensitive at all frequencies. At low frequencies (below about 200 Hz) and high frequencies (above about 8 kHz) much higher intensities are needed to reach the threshold of audibility. Peak sensitivity sits in the 1-4 kHz band, which is also (not coincidentally) where most of the spectral content of human speech lies.
A 100 Hz tone at 40 dB is audible but feels quiet; a 3 kHz tone at the same 40 dB feels loud. To capture this perceptual reality in measurements of environmental and occupational noise, sound-level meters apply a frequency-weighting curve called dBA, which de-emphasises low and very high frequencies in proportion to the ear's reduced sensitivity at those frequencies.
dBA is the A-weighted sound level — the sound-intensity level after applying the A-weighting filter that mimics the inverse of the ear's threshold-of-hearing curve at around 40 phons. Workplace noise limits and environmental noise regulations are almost always specified in dBA.
A typical example: 90 dB at 100 Hz becomes about 71 dBA after weighting; 90 dB at 3 kHz becomes about 91 dBA. Noise dose calculations for hearing-protection purposes use dBA throughout.
The audiogram is the standard clinical test of hearing. The patient listens through headphones to pure tones at a series of test frequencies (usually 250 Hz, 500 Hz, 1 kHz, 2 kHz, 4 kHz, 8 kHz; sometimes more). At each frequency the intensity is increased from below threshold until the patient just signals that they hear it. The result is plotted as threshold elevation in dB hearing level (dB HL) on the y-axis (with 0 dB HL = "normal young-adult threshold at that frequency"; positive values = hearing loss; the axis is conventionally inverted so that worse hearing sits lower on the plot) against frequency on the x-axis (logarithmic scale).
A young adult with normal hearing scores 0-10 dB HL at all frequencies. A patient with hearing loss shows positive values; the pattern of loss across frequencies is the diagnostic key.
| Audiogram pattern | Likely cause |
|---|---|
| Flat loss across frequencies (e.g. 30 dB HL throughout) | Conductive loss — outer- or middle-ear problem (impacted wax, otosclerosis) |
| High-frequency-dominant loss (sloping down at 2-8 kHz) | Age-related sensorineural loss (presbycusis); noise-induced loss |
| Notch at 4 kHz with recovery at 8 kHz | Noise-induced hearing loss from broadband industrial noise |
| Low-frequency loss with recovery at high frequencies | Ménière's disease (less common) |
Conductive hearing loss arises from a problem in the outer or middle ear that prevents sound reaching the cochlea efficiently — wax impaction, fluid in the middle ear, perforated eardrum, otosclerosis (fixation of the stapes). It can usually be corrected medically or surgically; hearing aids work well because the cochlea itself is healthy.
Sensorineural hearing loss arises from damage to the cochlear hair cells or to the auditory nerve. It is generally not reversible. Hearing aids help by amplifying frequencies in the impaired range, but cannot restore the resolution lost when hair cells die. Cochlear implants are the option of last resort for profound loss.
The two are distinguished clinically by comparing air conduction (sound through the headphones, traversing outer-middle-inner ear) with bone conduction (sound delivered via a small vibrator behind the ear, bypassing the outer and middle ear and stimulating the cochlea directly). A conductive loss shows depressed air-conduction thresholds but normal bone-conduction thresholds. A sensorineural loss shows both depressed equally — the cochlea itself is the limiting factor.
A patient's audiogram shows a 35 dB HL threshold elevation at 4 kHz and 30 dB HL at 8 kHz, with thresholds at or near 0 dB HL below 1 kHz. Air-conduction and bone-conduction results agree. Interpret the pattern.
The pattern is a high-frequency-dominant loss with a 4 kHz notch. The agreement between air and bone conduction rules out a conductive component, so this is sensorineural. The 4 kHz notch is the textbook signature of noise-induced hearing loss — broadband industrial or recreational noise damages hair cells preferentially in the 3-6 kHz region of the cochlea because of the mechanical-resonance properties of the ear canal and middle ear. Management: cease the noise exposure, fit hearing protection, consider hearing aids tuned to the affected band.
graph LR
A["Headphones<br/>250 Hz - 8 kHz tones"] --> B["Patient<br/>presses button<br/>at threshold"]
B --> C["Audiometer logs<br/>threshold (dB HL)<br/>per frequency"]
C --> D["Audiogram plot:<br/>freq x-axis<br/>threshold y-axis (inverted)"]
D --> E{"Pattern?"}
E -->|Flat loss<br/>BC normal| F["Conductive loss"]
E -->|Sloping high-freq<br/>or 4kHz notch| G["Sensorineural<br/>(presbycusis or NIHL)"]
style F fill:#f39c12,color:#fff
style G fill:#e74c3c,color:#fff
A jet engine produces a free-field sound intensity of I₁ = 10.0 W m⁻² at the runway edge, taken to be 5.0 m from the engine. Treat the engine as a point source radiating uniformly (a crude but adequate first-order model for runway distances much larger than the engine dimensions).
(a) Sound-intensity level at the runway edge.
L = 10 log₁₀(I / I₀), with I₀ = 1.0 × 10⁻¹² W m⁻².
L₁ = 10 log₁₀(10.0 / 10⁻¹²) = 10 log₁₀(10¹³) = 130 dB.
(b) Level at 100 m using inverse-square attenuation.
For a point source in free space, intensity falls as I ∝ 1/r². So
I₂ = I₁ × (r₁ / r₂)² = 10.0 × (5.0 / 100)² = 10.0 × 2.5 × 10⁻³ = 2.5 × 10⁻² W m⁻².
L₂ = 10 log₁₀(2.5 × 10⁻² / 10⁻¹²) = 10 log₁₀(2.5 × 10¹⁰) = 10 × 10.40 = 104 dB.
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