Cochlea (2D) — travelling wave on the basilar membrane

• Sweep the frequency: the peak moves from the base (high pitch) to the apex (low pitch), about 3–4 mm per octave in the middle of the range.
• Set a second tone close to the first: the two peaks overlap, which is masking.
• Lower hair-cell health to 0: the peak becomes lower and broad, hearing thresholds rise by tens of dB.
• Raise the level to 100 dB: the active gain saturates, so tuning broadens.

Place-frequency map (Greenwood 1990, human): F(x) = 165.4 (10^{2.1 x} − 0.88) Hz, with x the fractional distance from the apex over a 35 mm duct, so F runs from about 20 Hz to 20.7 kHz.

Each point of the basilar membrane behaves like a damped resonator with characteristic frequency CF(x). With r = f / CF the response is 1 / |1 − r² + i r/Q|. A travelling wave enters at the stapes, slows as r approaches 1, peaks near r ≈ 1 and dies quickly beyond; its phase gain follows k(x) = K₀ r / √(1 − r² + i r/Q) (long-wave WKB form).

Outer hair cells add active gain: the model raises Q from 2.5 (passive) to 12 (healthy) and the gain saturates with level, Q = 2.5 + 9.5 h / (1 + 10^(L−50)/20), which gives compression and level-dependent broadening. This is a teaching model: real cochlear mechanics include fluid coupling, nonlinear amplification and cochlear delays. K₀ and the Q values are illustrative, not fitted to a specific ear. Not a diagnostic tool.