Duct resonance

A rigid-walled duct driven at one end at 1000 Hz, with the far end rigid and closed. The standing wave that forms has a closed-form peak pressure, and it sits at a predictable place — the rigid end.

This page used to say the case carried “two independent answers: the magnitude of the peak, and where the peak is”. That was wrong, and the correction is worth more than the claim was. The peak position is not a second answer, because cos(kL) is even about 2π: the frequency f and the frequency c/L − f amplify by exactly the same factor, and put the peak in exactly the same place. Here that is 1000 Hz and 2430 Hz.

We ran the second one. Solved at 2430 Hz and judged against the 1000 Hz closed form, it returns a peak of 3.8747450 Pa — +0.000083 %, which is the number in the table below to five figures — with the peak at x = L exactly, the drive face at exactly 1.0 Pa, a clean plane wave, and convergence at a ratio of 0.085. Every quantity this page prints comes out right on completely different physics.

What separates them is the position of the pressure node, which moves by 354 %, and the RMS of the whole profile, which differs by 89 %. Those are what the automated check now compares, and it is re-run with them switched off to prove the rest cannot tell the two apart.

The peak sits at x = L, the rigid end, at every mesh — which is where the standing-wave solution predicts it for this frequency.