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Nodal Interference

The bulk of this page was contributed by Mike Prairie of Norwich University, from his monograph [Prairie 2014]. Mike has been very generous in letting me re-publish it here. The three introduction sections, rendering of the figures, and editing of the text are by Clint Goss.

Preamble

Nodal interference is an acoustic situation that is sometimes encountered during flute design. It can occur if a location of maximum change in air pressure inside the sound chamber (“bore”, in this article) is close to a finger hole. The finger hole “interferes” with the changing pressure in the sound chamber by allowing air pressure to “leak” in and out from the area of maximum pressure change through the finger hole. This tends to have undesirable acoustic effects such as weakening the resonance of the note.

Flow and Pressure

Two important things happen to the air inside the sound chamber: it moves (“flows”) and changes pressure. The amount of flow and the amount of change in pressure is different at different places in the bore. In fact, they tend to occur at opposite places.

Nodes and Antinodes

Many natural phenomena, including a vibrating column of air inside the sound chamber of a flute, have properties that behave like a “standing wave ” — a wave (typically a sine wave) that holds its position constant.

Standing Wave: nodes and antinodes

Standing Wave: nodes and antinodes

The changes in air pressure and air flow inside the bore both behave in this way. The important thing for this article is the places of maximum change — the places where the line moves up and down the most — and the red dots where there is no change. The red dots where there is no change are called “nodes” and the locations of maximum change are called “antinodes”.

Since we are talking about both (air) pressure and (air) flow, we have four types of locations in the bore:

  • a “pressure node”, where the air pressure does not change,
  • a “pressure antinode”, where the air pressure change is maximum,
  • a “flow node”, where the air does not move,
  • a “flow antinode”, where there is the greatest change in airflow (think of lots of air “sloshing back and forth”).

One thing to note is that, a sound wave moving from the flute to our ears is not a standing wave. While the sound chamber holds a standing wave of changes in air pressure and flow, the sound wave it generates is a moving wave.

The First Register

“Nodal Interference” occurs when the flow node (or pressure antinode) of a standing wave gets too close to an open hole in the bore. A vibrating air column in the first register of a flute is acoustically half the wavelength of the note being played (the symbol for wavelength is “λ”), and this is true whether the note being played is the fundamental or any other note that is not “overblown.”

Note that on a typical Native American flute, the octave note played with the topmost playing hole open is actually in the first register. If you cover all the holes and overblow to the octave note (often by cracking the top hole slightly: Finger diagram cracked closed closed closed closed closed), the length of the vibrating air column will be a full wavelength.

With the flute playing in the first register, the ends of the air column flow in and out of the tube while the center of the air column stays still, but the acoustic pressure rises and falls. I call the ends of the air column the “flow zone” because the acoustic flow is at its largest, while the acoustic pressure stays pretty much held at zero by the outside atmosphere. Since the pressure is zero, the end of the air column is where the “pressure node” resides, but since the flow is at a maximum, it is also called the “flow antinode.” Likewise, the center of the air column is the “pressure zone” where the “pressure antinode” or “flow node” resides.

Figure 1. Nodes and antinodes in the first register

Figure 1. Nodes and antinodes in the first register. Larger image

In the second register there are more zones because the air column looks as if two half-λ air columns were stacked together, as shown in figure 2.

Figure 2. Nodes and antinodes in the second register

Figure 2. Nodes and antinodes in the second register. Larger image

Another thing to consider is that the acoustic length of an air column is not the same length as the bore between the open holes. There is always an acoustic “end correction” that makes the air column longer. The end correction is an “invisible extension” that includes a bit of air in a bubble just outside the opening, and its length depends on the size of the opening relative to the size of the cross section of the bore. If the size of the opening is small compared to the bore (like a playing hole or the TSH), the physical length of the opening (i.e., the wall thickness) plus the additional bubble of air will be multiplied by a factor of roughly the bore area divided by the hole area. For the top note in the first register, the two openings are the TSH and the top open playing hole. Their acoustic lengths are k2 for the TSH and Lp for the playing hole. Now let’s look at a flute that would work properly when playing the top note.

Figure 3. Sound chamber flow and pressure on the octave note.

Figure 3. Sound chamber flow and pressure on the octave note. Larger image

When nodal interference is not a problem, the pressure zone will remain tucked comfortably inside the bore as is seen in Figure 3. Note that the sizes of the zones are drawn only to show the basic idea — they are not to scale. In reality, the effective size may be a little smaller than drawn, but the relative strength of each type of zone will increase as it gets closer to the antinode at the center of the zone.

One type of nodal interference occurs when the TSH is too small, which makes k2 large. In this case, the air column is shifted toward the sound hole, and the pressure zone gets too close to the TSH. This has the effect of limiting the highest note the flute can normally play. Lew Paxton Price found that for a flute with a range of about 1.4 octaves, the maximum value of k2 is a little more than 80% of the node-antinode spacing of the highest note played (or 80% if the distance between the end of the acoustic air column and the middle of the nearest flow zone for the highest note) ([Price 1997]).

Figure 4. Pressure zone close to the sound hole.

Figure 4. Pressure zone close to the sound hole. Larger image

On the South side, another type of nodal interference can occur if the top hole is too high. As the playing hole is moved higher on the bore, it must be made smaller to get the correct pitch. The smaller hole increases the acoustic length, and if the hole is too high, the pressure zone gets too close to the open hole, and the note does not play. But something interesting usually happens — a higher note is played, or “when the top note is played, it jumps to a higher note.” A possible explanation for this phenomenon is as follows:

The operation of woodwind register holes can shed some light. The purpose of a register hole is basically to destroy the resonance of a particular note so that the next-register version of that note (an octave higher) can be played. So the register hole is designed to be placed over the center of the lower-register air column for a certain note, which is the location of the pressure antinode for the lower-register note. When the register hole is opened, the pressure cannot be contained in the pressure zone, so the resonance is destroyed. However, the air column for the second-register note has a pressure node in that location, so there is no pressure to leak out. That is also the flow antinode at the center of a flow zone. Since there is little or no pressure in the flow zone, it can flow back and forth under an open hole, hardly noticing that it is open.

So when the “top note jumps to a higher pitch” it may mean the top open holes are acting like a register hole, and a second-register version of a lower note is being played.

Figure 5. Pressure zone close to the top finger hole.

Figure 5. Pressure zone close to the top finger hole. Larger image

In Figure 5, the top note has the highest playing hole too high and too small so that the acoustic length Lp is long enough to draw the pressure zone too near the open hole. This results in pressure leaking from the pressure zone, which in turn weakens the resonance of the top note. The second-register note associated with the 3rd open playing hole in this case has both pressure zones safely inside the bore, and the top open playing holes are over the center flow zone, so the upper open holes appear like register holes. If the resonance of this higher-pitched note is stronger than that of the original note, the conditions are more favorable for the higher pitch, so the flute will jump to the higher note.

The minimum playing hole size is another concept that is a result of nodal interference, and it affects playing of the highest notes that are played in the second octave. In this case, the acoustic length Lp of the highest open playing hole is constrained to about 80% of node-antinode spacing for the highest note, just as the max k2. The acoustic length Lp is roughly proportional to the ratio of the square of the bore diameter (D2) and the playing hole diameter (P2), so Lp gets larger as P gets smaller for a given bore diameter. Let’s say you are making an F#4 flute and would like it to play the fourth above the octave (the B5 with fingering Finger diagram closed closed closed closed open open). That range is 17 semitones, or 1.4 octaves. The wavelength of the B5 is 13.72 inches, which puts the node-antinode spacing at 3.43 inches, and 80% of that is about 2.77 inches. If the bore diameter is 78 inches and the wall thickness is 316 inches, to get the acoustic length to be less than 2.77 inches, the playing hole diameter must be at least 0.338 inches (about 1132″).

Figure 6. Second register minimum finger hole size.

Figure 6. Second register minimum finger hole size. Larger image

If, on the other hand, you were happy with playing only the minor third above the octave (one hole open at the bottom, 1.25-octave range), the note would be A5, the wavelength would be 15.4 inches, the max k2 and Lp would be about 3.2 inches, and the minimum playing hole size would work out to be about 516 inches.

Going the other direction and extending the range to the C#6 fifth above the octave (Finger diagram closed closed closed open open open, 1.58-octave range), the wavelength would be 12.22 inches, the max k2 and Lp would be about 2.4 inches, and the minimum playing hole size would be about 38 inches. Note that the highest open hole dominates the minimum-playing-hole-size effect. So in the 1.58-octave flute, hole #3 would need to be at least 38 to play the C#6, but you could likely get by with hole #2 being closer to 1132 to play the B5, and the bottom hole could be as small as 516 for the A5. But then again, if you wanted all three holes to be the same size, you would have to go with 38 for all of them to make sure the C#6 was playable.

An Artifact

From this article, what might be said about the playability of this artifact?

Plains Courting Flute, late 19th Century.

Plains Courting Flute, late 19th Century. Larger image

This flute is in the collection of the National Music Museum, NMM 873. It is listed as a:

Courting flute, Plains Indians, Northern Plains Region, late 19th century. End-blown, external duct flute, stained dark brown. Branch split, hollowed and assembled using resin to fill gaps. Single metal chamfer, held in place by unadorned wooden block saddle, allowing limited amount of air to pass from upper to lower air holes.
Arne B. Larson Collection, 1979.

 
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