The cochlea's basilar membrane is not a single sensor — it is a spatially ordered array of places, each mechanically tuned to its own characteristic frequency (CF), with high CFs near the stiff base (oval window) and low CFs near the floppy apex. A pressure wave entering at the base travels down the membrane, and — because it takes finite time to reach each place — every point starts responding only after the wavefront arrives there. This simulator treats each place as an independently driven, damped harmonic oscillator tuned to its own CF, switched on only once the traveling wave has had time to reach it:
Fres(x) = F_max · (F_min/F_max)^x [place → CF, x=0 base … x=1 apex]
y_tt(x,t) = -ω₀(x)² y(x,t) - 2ζ_eff·ω₀(x)·y_t(x,t) + F(x,t)
F(x,t) = A₀·sin(ω(t-τ(x))) for t ≥ τ(x), else 0 [τ(x) = travel-time delay]
Because every place obeys the exact, closed-form driven-damped-oscillator equation, its steady-state response amplitude follows the textbook resonance curve A(ω)=F₀/√[(ω₀²-ω²)²+(2ζω₀ω)²] — maximal where the local CF matches the stimulus frequency. Sweeping the whole array from base to apex reproduces the tonotopic place code exactly, while the per-place delay τ(x) makes the response visibly build up as a genuine traveling wave sweeping across the membrane, rather than appearing everywhere at once.
- Tone 1 / Tone 2 frequency — one or two simultaneous pure tones; because the underlying equation is linear, the two-tone response at any place is exactly the sum of each tone's response alone.
- Sound level — sets the drive amplitude A₀ = 10^((dB-60)/20), and — matching a real, measured cochlear nonlinearity — also broadens ζ_eff at high level, since real outer-hair-cell tuning genuinely loses sharpness as intensity rises.
- Tuning sharpness ζ — the baseline damping ratio; a small ζ gives a narrow, high-Q resonance that resolves nearby frequencies into separate peaks, a large ζ blurs them together.
- Base→apex travel time — the total delay τ(1) for the wavefront to reach the apex; watch the response literally sweep across the canvas from left to right as this increases, exactly like a real cochlear traveling wave measured with laser interferometry.
Simplifications: real cochlear places are coupled through the fluid (this model treats them as independent, foregoing the coupled-wave impedance transformation that also shapes the traveling wave's exact envelope), the base→apex delay is treated as a fixed function of place rather than frequency-dependent group delay, and both the absolute frequency range and timescale are compressed for a legible real-time animation. This complements — rather than duplicates — the static population-tuning-curve snapshot in the companion 3D model, which has no dynamics or wave-travel delay at all.