Appendix: Comparing Measured and
Calculated Frequencies
Table I. Comparison of the measured and calculated frequencies
of the cylinder with the mouthpiece section and a cylinder with
both the mouthpiece and bell section.
for “Mouthpiece and Bell Effects on Trombone
Resonance,” by Michael C. LoPresto, Henry Ford
Community College
Harmonic
This portion is likely beyond the scope of a non-mathematical Science of Sound or Musical Acoustics course, but could
possibly be of interest as an independent study for a more
mathematically advanced student.
The playing frequencies of a trombone can be calculated
with the use of an expression for the frequencies of what is
known as a Bessel horn.1,2 The frequencies depend on the parameter g that defines the rate of flare of the bell in the equation a = b(x+x0)g, where a is the radius of the bell a distance
x from the large open end and x0 is an end correction.1,2 For
trombones, usually g = 0.7 and x0 = 0.1 cm,3 and β = 0.0.639
is another parameter.
fn =
c
(2n − 1) + β[g ( g + 1)]1/ 2 

4( L + x0 ) 
(3)
The effective length that a mouthpiece adds to the air column to which it is attached can be calculated with the expression 2,3
 (2π f / c ) L0 
c
.
tan−1 
1 − (4 fL 1 / c ) 2 
2π f


L=
(4)
This expression depends on frequency and L0, which is the
length of the tubing to which the mouthpiece is attached that
has the same volume of the mouthpiece and L1 that is equal
to the length of cylindrical tubing that has the same resonant
frequency as the mouthpiece.4 A mouthpiece can be considered a Helmholtz resonator with its own resonant frequency.5,6 For a trombone mouthpiece the resonant frequency is
about f = 550 Hz.4 Both of the lengths are easily calculated.
L0=V/πr2 = 6.5 cm, where V is the volume of the mouthpiece
(about 10 cm3 for a pBone mouthpiece, measured by filling
the mouthpiece with water and then pouring the water into a
graduated cylinder) and r is the inner radius, about 0.75 cm
for the ½-in PVC pipe used. L1 = c/4f = 15.6 cm, where c is the
speed of sound and f = 550Hz, the above mentioned mouthpiece resonant frequency.
Table I shows the measured frequencies of the cylinder
with a mouthpiece section and the cylinder with both a
mouthpiece and bell section. Note the close agreement with
the calculated values. The frequencies for the cylinder with a
mouthpiece section were calculated by subtracting the actual
8-cm length of the mouthpiece from the 274-cm length and
replacing it with the effective mouthpiece length calculated
with Eq. (4). The frequencies for the cylinder with a mouthpiece and bell were calculated with Eq. (3), using a length L of
274 cm –8 cm = 266 cm plus the effective mouthpiece lengths
calculated with Eq. (4).
Cylinder
with
mouthpiece
(measured)
Cylinder
with
mouthpiece
(measured)
Cylinder
with
mouthpiece
& bell
(measured)
Cylinder
with
mouthpiece
& bell
(calculated)
n
f (Hz)
f (Hz)
f (Hz)
f (Hz)
2
99
94
117
113
3
158
156
176
174
4
219
218
234
235
5
275
280
294
295
6
336
341
350
355
7
389
401
411
413
8
458
460
466
470
References
1. 2. 3. 4. 5. 6. http://www.phys.unsw.edu.au/jw/brassacoustics.
html#mouthpiece.
See http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html,
under Sound and Hearing; Musical Instruments; Brass Instruments; Brass Concepts; Mouthpiece Effect.
F. J. Young, “The natural frequencies of musical horns,” Acustica 10, 91–97 (1960).
A. H. Benade, Fundamentals of Musical Acoustics (Dover, New
York, 1990), pp. 414–416.
M. Morse, Vibration and Sound, 4th ed. (Acoustical Society of
America, Melville, NY, 1991), pp 234–235.
N. H. Fletcher and T. D. Rossing, The Physics of Musical Instruments, 2nd. ed. (Springer, New York, NY, 1998), pp. 433–437.
Michael C. LoPresto, Henry Ford Community College, Dearborn, MI
48128 ; lopresto@hfcc.edu
The Physics Teacher ◆ Vol. 52, January 2014