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Acoustic Length of a Flute

The acoustic length of a flute is the effective length of the sound chamber of the flute when the fundamental note note is played. The acoustic length is typically longer than the physical length of the sound chamber, because of physical effects that take place at the foot end and at the sound hole of the flute. These effects can be calculated by formulas and, when added to the physical length of the sound chamber, produce a theoretical acoustic length for the flute.

This page of Flutopedia was contributed by Mike Prairie.

Acoustic Length

The acoustic length of a flute sounding the fundamental note is half the wavelength of that note.  However, the physical bore length (measured from the plug to the foot) is significantly shorter than the acoustic length. To account for this difference, two acoustic corrections at the two ends of the flute that are open to the outside air must be considered. The total acoustic length, then, is the sum of the bore length plus the acoustic corrections (sometimes called “false lengths”) at the foot (k1) and the sound hole (k2).

Formula Acoustic length = bore length + k1 + k2

k1 — The Acoustic Correction at the Foot of the Flute

The vibrating air column in the bore extends past the openings of the tube. This can be approximated by a bubble that is a half sphere with a volume given below. The bubble can be converted to a cylinder with the same volume, and the length of that cylinder is defined as k1. k1 is acoustically how far the air column extends past the end of the foot.

Formula for the volume of half a sphere

Formula for the volume of half a cylinder

Relationship of the volumes of half a sphere and a cylinder


k2 — The Acoustic End Correction at the Sound Hole

Detail of the sound hole area

In Secrets of the Flute by Lew Paxton Price ([Price 1991], page 17), k2 is described for a round blow hole, or embouchure. An adjustment for the false length above the hole is given when the air column experiences partial containment by the player’s upper lip. This is the same effect as what I describe as “bird encroachment.”  In this case, the embouchure is cylindrical, so the development of the idea is fairly straightforward.

Later, geometric conversions of the air column contained within the hole are shown which are needed (but not necessarily the whole story …) to account for the rectangular TSH and the expansion of the hole due to the cutting edge ramp.

Conversion of end correction to cylinders

The new height of the false-length cylinder on top of the blow hole is for this specific case. It includes an adjustment factor that is more easily measured than calculated, but I will go over my own concept that attempts to explain it in a moment. 

Now the corrected cylinder needs to be converted to an acoustically-equivalent cylinder with the same cross section as the main bore.  That is done with what Lew Paxton Price describes as the “magic ratio,” and is explained in the appendix.

Conversion to the cross-section of the main bore

For the Native American flute, some geometric gymnastics are required, as mentioned before. I will show the pictures, and leave the detailed calculations for the reader (!).  Note that I have not tested this (yet), but it is a consistent extension of the approach used by Price, and it shows a conceptual explanation of the bird effects.  So let’s begin:

Correction for air column in the TSH

Bird-encroachment correction

The amount of bird encroachment depends on the bird configuration.  A bird with no chimney will have a small effect, while a bird with a deep chimney will have a much larger effect and reduce the cross section of the end correction cylinder substantially. I have not worked out a sensible way to quantify the encroachment yet, but will likely end up measuring it.  To do that, I will assume I have everything else accounted for, calculate the k2 with no encroachment, then back the difference up to the point where the cross section of the top end-correction cylinder is adjusted to account for the encroachment.

I hope to eventually be able to generate some curves for different bird geometries so that the final k2 prediction will be more accurate. Once that is done, frequency-dependent effects could be added to the theory. This also extends to the encroachment of the plug on the lower bubble, which seems to be important at higher frequencies. But I digress …

Now, we convert the cylinders to new cylinders with the same cross section as the main bore with diameter DB.

Conversion of cylinders to the diameter of the main bore

I mentioned earlier that I would “leave the detailed calculations for the reader.”  This isn’t so much a cop-out than it is a practicality.  I can do the math — and probably will eventually to see how good the approach is and generate those curves I mentioned above — but such a long series of approximations and outright guesses for specific values will likely be fraught with errors.  For that reason k2 is best measured.

To measure k2, simply play the fundamental note and figure out the half wavelength — that defines the acoustic length of the flute.  Next, calculate k1 which is simply D/3.  Finally, subtract the bore length and k1 from the acoustic length.  The answer is k2.  An example of how to do this is in the Appendix.

After all that, the last paragraph is probably all a flute maker really needs to know.  But for the curious like me, I hope this has been helpful!

Appendix:  Equivalent Tube Sections

To find the acoustic length of any tube section containing an air column at an opening in the flute, the tube must first be converted to an equivalent tube with the same cross-sectional area as the main bore of the flute.

An equivalent section will have the same resistance to air flow as the original section.  So if the cross-sectional area is increased, then the length must be increased as well.  In this way, the inertia provided by the longer column of air will balance the lowered resistance from the bigger opening.  To achieve this, the ratio of length to cross-sectional area in both tubes must be equal.  This is also what Lew Paxton Price describes as the “magic ratio.”

Magic ratio adjustmsnt

Appendix: Measuring k2

When measuring the frequency, you will ether measure the frequency directly in “Hertz” or “Hz,” or you will get a note and a number of “cents” that indicates how sharp or flat the note is.  If the latter is the case, you will need to convert that to a frequency.  The basic musical notes (F#, G, G#, A, etc.) are identified by a number used by MIDI systems.  For instance, a basic midrange F# flute is denoted F#4 (in the 4th octave on the piano) and is given the MIDI number 66.  You can look up F#4 to find that its frequency is f = 370 Hz, but you can also find that using the equation:

MIDI to frequency conversion

or

MIDI to frequency conversion

The next semitone is G4 and the MIDI number is 67, which yields f = 392 Hz.

Let’s say you measure a note on the tuner that reads F# + 18 cents.  The “18 cents” represents how sharp the F# is, and in this case it means the note is 18 percent of the way between F# and G.  If you know your F# is F#4, then you can add the MIDI number of F#4 to the fraction of the way from F# to G, where 18 % = 0.18.  So the new MIDI number is 66 + 0.18 = 66.18, so

MIDI to frequency example

If the F# was flat and the tuner showed F#- 18 cents, you would simply subtract 0.18 from 66 instead.  Once you have found the frequency, you can find the wavelength, and then divide that in half to get the acoustic length.  The frequency and wavelength are related to each other by the speed of sound so that the (wavelength) = (speed of sound) / (frequency).  The letter “c” is usually used to represent the speed of sound, which is 13,552 inches per second at 72°F, and the Greek letter lambda (λ) is usually used for wavelength.  So, in our example

Acoustic length calculation example

The acoustic length is half that, or 18.12 inches.  If you are shooting for a G, you can figure out that the acoustic length 17.29 inches, so the flute is 0.83 inches too long.  Note that if you wanted to find the speed of sound at a temperature other than 72°F, the equation is

Speed of sound at a given temperature

where T = temperature in °F.

So let’s say you have a G flute that has a measured bore length of L = 15.2 inches with a bore diameter of D = 0.75 inches.  You can find k1 by using D/3 to get 0.25 inches.  To find k2,

Calculation of K2

Calculation of K2, example

Calculation of K2, example

 

 
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