Traveling Waves In A Flute

Here we develop a very simplified digital waveguide model of waves propagating in a flute. The far end of the flute from the player's mouth has an open end, so to first approximation, pressure waves reflect with a sign inversion from the far end. Flutists can shorten the effective length of the tube by opening holes along the length of the tube. The effective length then corresponds to the first open hole. In contrast with the clarinet, saxophone, etc., the end of the flute near the player's mouth, which is known as the head, behaves acoustically more like an open end than a closed end [3]. This is because a flute player only places the lower lip against the embouchure hole--he or she does NOT completely cover the hole. Consequently, pressure waves reflect with a sign inversion at the head of the flute.

The simplified model is shown in Figure 1. We use the same basic structure as with the vibrating string in the digital waveguide model laboratory assignment, although this is somewhat coincidental because here we are modeling sound pressure waves rather than structural displacement waves. The total delay of $N$ samples around the loop corresponds to the note being played. Since both terminations support inverting reflections, the fundamental frequency $f_0 = \frac{1}{NT}$ where $T$ is the digital sampling interval in seconds. To make sure that waves circulating in the waveguide decay over time, we change one of the gains from $-1$ to $-g$ where $g \approx 1$ but $g<1$. So that the higher harmonics decay more quickly than the lower harmonics, we insert a lowpass filter into the feedback loop (see Figure 1) [4].

Figure 1: Very simplified digital waveguide model of pressure waves propagating in a flute