Rijke Tube — Thermoacoustic Instability

Rijke tube (1859): a vertical tube open at both ends with a heated gauze. It spontaneously emits sound when the heater is in the lower half. The Rayleigh criterion: instability occurs when heat is added in phase with pressure — ∫ p'(x,t)·Q'(x,t) dt > 0.

For the n-th acoustic mode (pressure antinodes at closed ends, nodes at open ends), the growth rate is: σ_n = (γ_Rayleigh/2) · sin(nπx_h/L) − γ_damp. The heater at x_h = L/4 maximally drives the first mode. Instability sets in when σ_n > 0, producing exponential amplitude growth until nonlinear saturation.