Building an Instrument from Formulas

Lenore Horner

The Seven Hills Schools

April 18, 2015

Goal

Starting Point

End Correction

Finger Holes

Outline

1 Goal

2 Starting Point

3 End Correction

4 Finger Holes

Lenore Horner (7 Hills) Instrument April 18, 2015 2 / 8

Goal

Starting Point

End Correction

Finger Holes

Goal

Have my physics students build PVC flutes based on simpleextensions of the formulas for standing waves.

The pipes are cylindrical and open at both ends. (Quick check:overblow a transverse flute and the frequency doubles)

Lenore Horner (7 Hills) Instrument April 18, 2015 3 / 8

Goal

Starting Point

End Correction

Finger Holes

Basic Formulas

string; open pipe

f =nv

2L(1)

n = all integers (2)

stopped pipe

f =nv

4L(3)

n = odd integers (4)For this project, only worry about the fundamentals so n = 1 fromnow on.

Lenore Horner (7 Hills) Instrument April 18, 2015 4 / 8

Goal

Starting Point

End Correction

Finger Holes

End Correction Approximation

It takes time for the information that the pipe has ended topropagate to the center of the column of air in the pipe.

stopped pipe: standard correction

f =v

4(Lcorr + 0.3D)(5)

D is the interior diameter of a cylinder

This can be approximately motivated by the idea that we have toadd a cylinder with volume equal to the hemisphere at the end ofthe pipe.

1

2

4

3πr3 = πr2Lextra (6)

Lextra =2r

3=

D

3(7)

Lenore Horner (7 Hills) Instrument April 18, 2015 5 / 8

Goal

Starting Point

End Correction

Finger Holes

Testing End Correction

pipe dimensionslength: 28.0 cminterior diameter: 2.0 cm

results

f =v

4L= 306 Hz (8)

f =v

4(L+ 0.3D)= 300 Hz (9)

fplayed = 298 Hz (10)

Lenore Horner (7 Hills) Instrument April 18, 2015 6 / 8

Goal

Starting Point

End Correction

Finger Holes

Hall

Formulas for ONE Hole

te = wall thickness pipe + 0.8Dh − 0.5D2

h

Dp(11)

y = Lcorr, lowest − Lcorr, current (12)

x =te

r2 + tey

(13)

Ldrill hole at = Lcorr, current − x (14)

Lenore Horner (7 Hills) Instrument April 18, 2015 7 / 8

Goal

Starting Point

End Correction

Finger Holes

Reality Check: 1”

starting from empirical end correction for each end of the flute

Lenore Horner (7 Hills) Instrument April 18, 2015 8 / 8

Goal

Starting Point

End Correction

Finger Holes

Reality Check: 1-1/4”

starting from empirical end correction for each end of the flute

Lenore Horner (7 Hills) Instrument April 18, 2015 9 / 8