The ear as a sound detection system
Sound is a longitudinal pressure wave in air. The ear has three parts.
- Outer ear: the pinna collects sound. The ear canal (about 2.5 cm long) guides it to the eardrum (tympanic membrane). The canal resonates near 3 kHz, which helps us hear that range.
- Middle ear: three small bones, the ossicles (hammer, anvil, stirrup), link the eardrum to the oval window of the cochlea. They act as a lever that multiplies the force about 1.3 times.
- Inner ear: the cochlea is a snail-shaped tube filled with fluid. Its basilar membrane carries hair cells. When the membrane moves, the hair cells bend and make electrical signals in the auditory nerve.
Why the pressure goes up
Pressure = force ÷ area. The eardrum (about 55 mm²) is about 17 times bigger in area than the oval window (about 3.2 mm²). The same force on a smaller area gives a bigger pressure. With the lever gain of 1.3: pressure gain ≈ 17 × 1.3 ≈ 22. This impedance matching lets sound pass from air into fluid. Without it, about 99.9% of the sound would be reflected at the air–fluid boundary.
Place theory of pitch
The basilar membrane is narrow and stiff at the base (near the oval window) and wide and floppy at the apex. High frequencies make the base vibrate most. Low frequencies travel further and make the apex vibrate most. The brain knows the pitch from which hair cells are firing.
Sensitivity and frequency response
The threshold of hearing is the smallest intensity we can just hear. At 1 kHz it is I₀ = 1.0 × 10⁻¹² W m⁻². The threshold of pain is about 1 W m⁻², a trillion times more. Because the range is so large, we use a log scale.
Intensity level: L = 10 log₁₀(I ÷ I₀), in decibels (dB).
- Each ×10 in intensity adds 10 dB.
- Each ×2 in intensity adds about 3 dB.
The ear is not equally sensitive at all frequencies. A healthy young ear hears about 20 Hz to 20 kHz and is most sensitive at about 2–5 kHz. A graph of threshold against frequency (log scale) is U-shaped with the lowest point there.
Loudness, the phon and the dBA scale
Loudness is how strong the sound seems to the listener. It depends on intensity and frequency. Curves of equal loudness join sounds that seem equally loud; their unit is the phon (a sound has a loudness of n phon if it seems as loud as an n dB sound at 1 kHz). The dBA scale weights each frequency like the ear does, so sound meters give readings that match what we hear. Doubling the loudness we feel needs roughly a 10 dB rise.
Defects of hearing
A hearing test (audiogram) measures the threshold at several frequencies. Hearing loss shows as the threshold curve moving up (you need a louder sound to hear).
- Age-related loss (presbycusis): hair cells at the base wear out first, so the threshold rises most at high frequencies. Older people find it hard to follow speech consonants like s, f and th.
- Noise-induced loss: long exposure to loud sound (above about 85 dBA for hours) damages hair cells. A typical sign is a dip near 4 kHz. Loss can be temporary (ringing after a concert) or permanent.
- Conductive loss: sound cannot reach the cochlea well (wax, fluid in the middle ear, stiff ossicles). It is often treatable.
Hearing aids amplify chosen frequencies; cochlear implants send electrical signals straight to the nerve.
Try it: test your own hearing range
With an adult's help, play a free online tone generator at a low volume through earphones. Slowly raise the frequency from 1 kHz up to 20 kHz. Note the highest tone you can still hear. Ask an older family member to try. Compare: who stops hearing first? Then use the 3D free-play step: set 12 kHz and 20 dB and see which ears hear it.
Key formulas and definitions
- Intensity I = P ÷ A (W m⁻²)
- Intensity level L = 10 log₁₀(I ÷ I₀) dB, I₀ = 1.0 × 10⁻¹² W m⁻²
- Pressure gain ≈ (A_eardrum ÷ A_oval window) × lever ratio ≈ 17 × 1.3 ≈ 22
- Change in level ΔL = 10 log₁₀(I₂ ÷ I₁)
- Hearing range ≈ 20 Hz – 20 kHz; most sensitive 2–5 kHz
Worked examples
1. A sound has intensity 1.0 × 10⁻⁶ W m⁻². Find its intensity level.
L = 10 log₁₀(10⁻⁶ ÷ 10⁻¹²) = 10 log₁₀(10⁶) = 10 × 6 = 60 dB.
2. A drill gives 90 dB. What is its intensity?
90 = 10 log₁₀(I/I₀) → I/I₀ = 10⁹ → I = 10⁹ × 10⁻¹² = 1.0 × 10⁻³ W m⁻².
3. Two identical machines each give 80 dB. What level do both together give?
Intensity doubles. ΔL = 10 log₁₀2 = 3.0 dB. Total = 83 dB (not 160 dB).
4. A force of 0.020 N acts on an eardrum of area 55 mm². The ossicles multiply force by 1.3 and the oval window is 3.2 mm². Find the pressure at the eardrum and at the oval window.
At eardrum: p = 0.020 ÷ (55 × 10⁻⁶) = 364 Pa. Force at oval window = 1.3 × 0.020 = 0.026 N. p = 0.026 ÷ (3.2 × 10⁻⁶) = 8125 Pa ≈ 8.1 kPa. Gain ≈ 22.
5. A speaker gives 2.0 W of sound spread evenly in all directions. Find the intensity level 10 m away.
I = P ÷ 4πr² = 2.0 ÷ (4π × 100) = 1.59 × 10⁻³ W m⁻². L = 10 log₁₀(1.59 × 10⁹) = 92 dB.
6. A worker's audiogram shows thresholds of 10 dB at 1 kHz, 45 dB at 4 kHz and 20 dB at 8 kHz. What does this suggest?
The threshold is raised most at 4 kHz with better hearing on both sides: a 4 kHz dip, the classic sign of noise-induced hearing loss.
Common mistakes
- Adding decibels like ordinary numbers: two 80 dB sources give 83 dB, not 160 dB.
- Saying 0 dB means no sound. It means the intensity equals I₀, the threshold at 1 kHz.
- Mixing up base and apex: high frequencies are detected at the base (near the oval window), not at the apex.
- Thinking the ossicles mainly increase pressure by the lever. Most of the gain (about 17 times) comes from the area ratio of eardrum to oval window.