TThe effects of the mouthpiece and bell on the frequencies of the vibrating air column in a trombone can be demonstrated quite readily by first calculating the expected resonant frequencies of a piece of PVC pipe that is the same length as a trombone, then replacing portions of the PVC pipe of the same length with first a cup-shaped mouthpiece and then a flaring-bell section of a trombone. The trombone components used were from a new pBone plastic trombone1 that is not only much less expensive than a regular instrument, which is made mostly of brass, but also features components that are much lighter than those of a conventional brass trombone and is therefore easier to work with.

2.
J.
Backus
,
The Acoustical Foundations of Music
, 2nd ed. (
Norton
,
New York
,
1977
), pp.
260
268
.
3.
R. W.
Pyle
,
“Effective length of horns,”
J. Acoust. Soc. Am.
57
,
1313
1314
(
1975
).
4.
A. H.
Benade
,
Fundamentals of Musical Acoustics
(
Dover
,
New York
,
1990
), pp.
414
416
.
5.
http://www.phys.unsw.edu.au/jw/brassacoustics. html#mouthpiece.
6.
See http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html, under Sound and Hearing; Musical Instruments; Brass Instruments; Brass Concepts; Mouthpiece Effect.
7.
Ref. 4, pp. 407–409.
8.
N. H.
Fletcher
and
T. D.
Rossing
,
The Physics of Musical Instruments
, 2nd ed. (
Springer
,
New York
,
1998
), pp.
431
433
.
10.
See http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html, under Sound and Hearing; Musical Instruments; Brass Instruments; Brass Concepts; Bell Effect.
11.
Figure from http://hyperphysics.phy-astr.gsu.edu/hbase/ hframe.html, under Sound and Hearing; Musical Instruments; Brass Instruments; Brass Concepts; Forced Harmonic. Sequence used with permission of webmaster RodNave@gsu.edu.
12.
http://flexpvc.com/PVCPipeSize.shtml.
13.
Data taken with Logger Pro, Vernier Software & Technologies, http://www.vernier.com/. Data could be taken with any data collection hardware and software able to capture sound waves, display their form, and calculate frequencies.
14.
T. D.
Rossing
,
F. R.
Moore
, and
P. A.
Wheeler
,
The Science of Sound
, 3rd ed. (
Addison-Wesley
,
San Francisco
,
2002
), p.
232
.
16.
See http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html, under Sound and Hearing; Musical Instruments; Brass Instruments; Brass Concepts; Pedal Tone.
17.
From Ref. 14. Used with permission.
18.
Ref. 14, p. 135.
19.
An online appendix is available at TPT Online, http://dx.doi. org/10.1119/1.4849149, for teachers interested in a more mathematically advanced calculation.

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