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How to use
The default preset is a singing pipe: the gauze sits at a quarter of the pipe, the delay is about a fifth of the period, and a whisper grows into a loud tone within a second. Use the presets to see the Rayleigh criterion at work: gauze at the middle or in the upper half stays silent, a wrong delay kills the sound, and with long delays the pipe can jump to its second or third harmonic.
- Move the gauze: the third graph shows which positions let each mode grow (above zero).
- Raise the heater until the level stops growing (saturation).
- Clap gives the decaying tones a push; Fast-forward 1 s skips ahead; Sound plays the pressure at the middle of the pipe (needs 1x time).
Drag to orbit, wheel or pinch to zoom, or press Guided tour. Quality switches High/Low (also ?q=low).
The science
A Rijke tube is a heat engine with no moving parts. A pipe open at both ends has acoustic modes with pressure p' = sum P_j sin(j pi x/L) and velocity u' = sum U_j cos(j pi x/L) and frequencies f_j = j c/(2L), c = sqrt(gamma R T). The model keeps six modes and writes the linearised 1-D acoustic equations with a compact heat source Q'(t) (W/m2) at the gauze position x_h = xi L:
rho dU_j/dt = -(j pi/L) P_j - 2 zeta_j w_j rho U_jdP_j/dt = rho c^2 (j pi/L) U_j + (2(gamma-1)/L) Q' sin(j pi xi)
The gauze heat release follows Heckl's (King's-law) form, delayed by the thermal response time tau of the wires: Q'(t) = q0 [ sqrt|1/3 + u'(x_h, t-tau)/u_bar| - sqrt(1/3) ]. For small amplitude this is Q' = K u'(t-tau) with K = q0/(2 sqrt(1/3) u_bar); the square root and the absolute value saturate the response at large amplitude, which creates the limit cycle.
Rayleigh criterion. The acoustic energy gains (gamma-1)/(gamma p0) <p' Q'>: heat added while the pressure is high feeds the sound, heat added while it is low damps it. For weak coupling the growth rate of mode j is
sigma_j = -zeta_j w_j + (gamma-1) K sin(2 j pi xi) sin(w_j tau) / (2 rho c L)
The factor sin(2 j pi xi) is positive only in the lower half of the pipe for mode 1 (and best at xi = 1/4), where velocity and pressure are both large; at the middle (a velocity node) and above it the heater cannot drive mode 1. The delay enters through sin(w_j tau): the heat must arrive in the right part of the cycle. That is why the same heater can make a mode grow, decay, or hand over to a higher harmonic.
Honest simplifications. Linear acoustics with a compact heat source; the temperature jump and the extra flow resistance of the gauze, the mean-flow Mach number, radiation at the open ends and boundary-layer losses are folded into a single damping ratio zeta_j = zeta_1 sqrt(j) (end reflection is perfectly open). The mean flow u_bar is a parameter, not computed from buoyancy. Heat release is a one-parameter delayed law, not a CFD of the wires. The displayed air motion is magnified (real displacements are tenths of a millimetre).
In-page checks. (1) The time-domain mode-1 growth rate of the linearised model against the formula above (heater at xi = 1/4). (2) The same with the heater moved to xi = 3/4, where the rate is negative. (3) The limit-cycle frequency against the pipe pitch c/(2L). (4) The Rayleigh balance on the limit cycle: the mean energy input (gamma-1) Q' p'/(rho c^2) equals the mean dissipation, a check on both the integrator and the saturating law.