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Acoustical Impedances: Calculations
and Measurements on a Trumpet
by
Jonathan Kipp
A Bachelor’s Thesis submitted in
October 2015
to the
Faculty of Mathematics, Computer Science and Natural Science
Department of Physics
at
RWTH Aachen University
with
Prof. Dr. rer. nat. Jorg Pretz
Declaration of Authorship
Hiermit erklare ich, Jonathan Kipp, an Eides statt, dass ich die vorliegende Bachelorar-
beit selbstandig verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel
benutzt sowie Zitate kenntlich gemacht habe.
Signed:
Date:
ii
Symbols and Constants
Symbols
k Wave number m−1
S Cross section m2
p Pressure Nm−2
u Particle velocity ms−1
U = Su Volume flow m3s−1
Z = paUa
Impedance Nsm−5
z = paua
Acoustical Impedance Nsm−3
ω angular frequency rads−1
Constants
Speed of Sound c = 343 ms−1
Density of air ρ = 1.3 kgm−3
iii
Contents
Declaration of Authorship ii
Symbols and Constants iii
List of Figures vii
1 Introduction 1
2 Theoretical Background 3
2.1 Defining a Horn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Sound Propagation in Air . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2.1 The Infinite Cylindrical Tube . . . . . . . . . . . . . . . . . . . . . 4
2.3 Finite Tubes with Flares . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3.1 Impedance of the Cylindrical Tube . . . . . . . . . . . . . . . . . . 9
2.3.2 Impedance of the Exponential Tube . . . . . . . . . . . . . . . . . 10
2.3.3 Impedance of the Conical Tube . . . . . . . . . . . . . . . . . . . . 12
3 Simulation 15
3.1 Cylindrical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Conical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Exponential Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3.1 Influence of single Diameters on the Spectrum . . . . . . . . . . . 23
4 Measurement 25
4.1 Diameter Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2 Measuring Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.3 Measuring Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5 Comparison of Simulation and Measurement 33
6 Conclusion 37
A Appendix 39
v
Contents vi
Bibliography 43
List of Figures
2.1 Cross Section and Section of Infinite, Cylindrical Tube . . . . . . . . . . . 4
2.2 First higher Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Sound Propagation in Cylindrical Tube . . . . . . . . . . . . . . . . . . . 7
2.4 Finite Cylindrical Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.5 Finite Exponential Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.6 Finite Conical Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.1 Diameter along Horn Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Cylindrical Model for infinite ZL . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 Exponential and Cylindrical Model in Comparison . . . . . . . . . . . . . 18
3.4 Impedance, Conical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.5 Impedance, Exponential Model . . . . . . . . . . . . . . . . . . . . . . . . 21
3.6 Horn in Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.7 Influence of 24th Diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.8 Simulation with new Value, Exponential Model . . . . . . . . . . . . . . . 22
3.9 Forth Resonance vs 24th Radius . . . . . . . . . . . . . . . . . . . . . . . . 24
3.10 Forth Resonance vs 10th Radius . . . . . . . . . . . . . . . . . . . . . . . . 24
4.1 Shape of the Trumpet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2 Diameter Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.3 Measurement Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.4 Flange for Trumpet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.5 Impedance of Trumpet without Mouthpiece . . . . . . . . . . . . . . . . . 31
4.6 Impedance of Trumpet without Mouthpiece, second Measurement . . . . . 31
5.1 Impedance of Conical Model and Measurement . . . . . . . . . . . . . . . 35
5.2 Impedance of Exponential Model and Measurement . . . . . . . . . . . . 35
A.1 Influence of Friction, Exponential Model . . . . . . . . . . . . . . . . . . . 40
A.2 Influence of Friction, Conical Model . . . . . . . . . . . . . . . . . . . . . 40
A.3 Comparison of Conical and Exponential Model . . . . . . . . . . . . . . . 41
A.4 Influence of 24th Diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
All photographs are taken by the author. All figures are compiled by the author.
vii
Chapter 1
Introduction
Manufacturing brass instruments is a highly complicated matter. The manufacturers
main goal is to build an instrument with good tuning, which means that the instrument’s
natural resonances match the frequencies of the corresponding notes. These natural
resonances are influenced not only by the instrument’s flare and length, but also by e.g.
the exact position and diameter of valves and tuning slides. To cover all these effects
would require a great amount of work, which is not compatible with the scope of this
work. This thesis aims on predicting and measuring the natural resonances of brass
instruments within a simple model. The comparison of prediction and measurement is
another goal as well as giving an outlook on which parameters will need more careful
treatment in further going simulations of these natural resonances. The focus will be
on the instruments properties, not taking into account that the stimulation of sound
waves by the players lips is highly complicated: professional players are able to play
up to three tones out of center just using tongue and lips1. The term horn will be of
great importance, it will be necessary to define the term horn in the context of this
work as well as developing ideas on how this term must be extended to improve the
results of e.g. simulations. The natural resonances can be obtained by measuring the
acoustical impedance Zin = pa/ua, where pa is the acoustical pressure and ua is the
acoustical particle velocity. The natural resonances occur at those frequencies, where
|Zin,throat|, the impedance at the throat of the horn, reaches a local maximum. So the
task will be to simulate and measure the impedance at the throat of the instrument,
because the excitation takes place at this point. The impedance depends on the flare of
the instrument from throat to mouth. Modeling the shape of the instrument in different
ways will give different results for the simulation. One part of this work is to study the
weaknesses and advantages of the two models proposed, the exponential and the conical
1Playing out of center: the musician bends the note and excites acoustic waves not with the initialfrequency, but with another frequency. Hence the musician forces excitation at a frequency which is notintended to be played with the valve combination that is chosen.
1
Theoretical Background 2
flare. The measurement of the impedance will be helpful to interpret the simulation
with respect to these weaknesses or advantages. This work also offers insight into the
dependence of the natural resonances on the instruments radius at one particular point.
This is done by varying single radii and computing the impedance for an instrument
shape with these variated radii. The aspiration of this work is to do a prediction with
a simple understanding of the term horn, which is within a halftone range of the actual
measurement. If this aspiration is met, all effects neglected will be just minor corrections
to the simple understanding. A next step, which is beyond the scope of this work, would
be not just to study the natural resonances, but the radiation from the bell and the
spectrum of harmonics, too. These aspects are of great interest when studying, if the
sound of the instrument fulfills a certain ideal.
Chapter 2
Theoretical Background
2.1 Defining a Horn
In the context of this work, a horn is a straight tube, which has a defined length L, ideally
rigid walls and a flare which can be described analytically. It has one open, driven end,
where the acoustical excitation takes place. The other end would be ideally closed on
the first look, but we will introduce a description to the behaviour of a radiating pipe.
In this understanding of the term horn, neither the brass instruments curvature nor the
material from which the horn is built is taken into account. This simple understanding
will be tested with measurements on an ordinary trumpet to learn if the model respects
the important characteristics for the natural tone scale and which neglected facts may
be worth considering. The most important characteristics, which are not respected here,
are the curvature of wave fronts, vibration of the walls, radiation effects at the bell and
the complicated stimulation at the throat.
2.2 Sound Propagation in Air
The phenomenon called sound is the modulation of pressure pabs = patm + pa and local
particle velocity vabs = vatm + va in a medium dependent on time and position. A
differential equation describing the particle displacement ξ in a tube with a flare from a
simple, one dimensional model, will be derived. However, propagation of sound in three
dimensions will first be described in a very academical case, to learn why this simple
one dimensional model is valid. If we want to describe the propagation of sound waves
with a simple time dependence eiωt (which is obviously an assumption) in any medium,
3
Theoretical Background 4
we have to solve Helmholtzs Equation for the acoustical pressure:
∆pa + k2pa = 0, (2.1)
[1], where k = ωc . Note that
ρ∂u
∂t= −∇p. (2.2)
The focus is set on the natural resonances of our system, which occur at those fre-
quencies maximizing the reflexion coefficient R (the systems border reflects most of the
wave energy back into the system). While the reflection coefficient is also measurable
and maybe the plausible property of a sound system, it is common to work with the
impedance z. It is defined as z(~x) = pa/ua or Z(~x) = pa/Ua, where ua is the local
particle velocity and Ua = S · ua is the volume flow (S is the general cross section here).
2.2.1 The Infinite Cylindrical Tube
Now a solution to Helmholtz’s equation in an infinite, cylindrical pipe is proposed. Tak-
ing advantage of the system’s translational symmetry along the pipe axis, the equation
will be solved in cylindrical coordinates. Choosing the coordinate system with the z-axis
being aligned to the pipes axis, the boundary conditions for this case are as follows:
∂p
∂ρ
∣∣∣∣ρ=a,φ,z
= 0 (2.3)
Φ(φ) = Φ(φ+ 2πn) (2.4)
Figure 2.1: Cross section of infinite, cylindrical tube of radius a and infinite, cylin-drical tube of radius a in section
Theoretical Background 5
Using the Laplace Operator in cylindrical coordinates1 and assuming the solution p to
be separable, p = R(ρ) · Φ(φ) · Z(z), yields
ρ2R′′ + ρR′ + (µ2ρ2 − α2)R = 0, (2.5)
where:
• Z(z) ∼ eiγz
• Φ(φ) ∼ eiαφ
• µ =√k2 − γ2
Note, that Φ fulfills condition 2.4. In the following, r = µρ is the new, dimensionless
coordinate, which gives:
r2R(r)′′ + ρR(r)′ + (r2 − α2)R(r) = 0. (2.6)
This is Bessels differential equation, which is solved by the Bessel and Neumann func-
tions. The latter are not interesting because we do not expect the solution to have poles
on the z-axis. Here α is an integer and indicates the order of the Bessel function. Hence
our general solutions are of the form
pα(r, φ, z, t) = const · Jα(r) · ei(γz+αφ+ωt), (2.7)
Jα being the Bessel function of order α. The solution for the pressure in an infinite
cylindrical pipe will be of no practical use within the context of this work, since the focus
is set on finite pipes with arbitrary flare. However, important facts can be determined
from this result, if boundary condition 2.3 is respected, which yields
Jα(µa) = 0⇒ µ =qm,αa∨ µ = 0, (2.8)
where qm,α is the mth root of the Bessel function of order α. Solving for γm,α yields
γ2m,α = k2 −
q2m,α
a2. (2.9)
The first mode (0,0) with γ2m,α = k2 ( µ = 0, pa is constant over the cross section) will
always propagate, but higher modes (m,α) will only propagate if the condition
1∆ = 1ρ∂∂ρ
(ρ ∂∂ρ
)+ 1
ρ2∂2
∂φ2 + ∂2
∂z2
Theoretical Background 6
Figure 2.2: First higher mode(0,1), appearing at approximately6300 Hz in a tube of radius 21 mm.The figure shows the pressure am-plitude over the pipes cross section,markers are parts of the tubes radius.
k >qm,αa
(2.10)
is fulfilled. Only then k is real and the wave is not damped along the axis. This
result is central for this work, because it legitimates a crucial simplification to the three
dimensional problem of sound propagation in pipes. The pipes used for trumpets have
radii in the cm range, while the excited wavelengths are at least two orders larger, about
1 m. In example, the minimum frequency for propagation of the first higher mode (0,1),
appearing at q0,1 = 2.40482, is approximately
f0,1 =2.4048 · 343.3
0.021 · 2π1
s= 6256.82 Hz, (2.11)
for a tube of radius 21 mm (typical radius in the conical section of a trumpet, see fig-
ure 3.1, while the played frequencies range from approximately 50 Hz up to 2000 Hz.
This constellation allows to neglect all higher modes of (0,0) and therefore a one dimen-
sional model along the z-axis can be used to describe the influence of the flare on the
propagation. This model and its results are discussed in section 2.3.1, 2.3.2 and 2.3.3.
2.3 Finite Tubes with Flares
In the frequency range of interest (50Hz up to 2000Hz) no higher modes will propagate,
so pressure and particle velocity will be in good approximation constant over the area
of the propagating wavefront (at given time and position along the tube’s axis). A one
dimensional model will be introduced now, which describes the influence of the cross
sectional flare of a tube on the acoustical impedance neglecting the influence of higher
2Value taken from http://mathworld.wolfram.com/BesselFunctionZeros.html, from 17th of June, 2015
Theoretical Background 7
Figure 2.3: A tube with cross section S. A wave travels along the tube and displacesthe volume V = S · dz by dξ. New borders of V are dashed red.
modes. For the one dimensional model the starting point is the following idea: When a
plane wave travels through a cylindrical pipe, a volume element of thickness dx moves
from ABCD to abcd and will be displaced by dξ (take a look at figure 2.3). So the
change in volume is given by
V + dV = Sdz
(1 +
∂ξ
∂z
), (2.12)
because the cross section does not change with z. Respecting that the total pressure is
ptot = patm + pa and using the definition of the bulk module dptot = −K dVV , yields:
pa = −K∂ξ
∂z(2.13)
Respecting that the elements motion must obey Newton’s equations (pressure gradient
in z direction must be equal to mass times acceleration) and substituting 2.13 we get:
− S∂pa∂z
dz = ρSdz∂2ξ
∂t2⇔ ∂2ξ
∂t2=
K
ρ
∂2ξ
∂z2(2.14)
This differential equation describes the propagation of a plane wave in a cylindrical pipe
in one dimension.
Theoretical Background 8
Now for the case with a flare: the change in volume is
V + dV = Sdz
(1 +
1
S
∂(Sξ)
∂z
), (2.15)
which gives us
pa = −KS
∂(Sξ)
∂z. (2.16)
Again using Newton’s equation yields
− S∂pa∂z
dz = ρSdz∂2ξ
∂t2. (2.17)
Differentiating again with respect to z and swapping the differential operators ∂∂z and
∂2
∂t2gives:
− ∂
∂zS∂pa∂z
= ρS∂2
∂t2∂ξ
∂z(2.18)
Substituting 2.16 yields Webbsters Equation:
1
S
∂
∂zS∂pa∂z
=ρ
K
∂2pa∂t2
(2.19)
The solution to this differential equation is of great value. It describes the propagation
of plane sound waves in tubes with arbitrary cross sectional flare, which will allow
the computation of the acoustical impedance Z = pa/Ua and prediction of the natural
resonances for such a tube. Now solutions to this equation will be presented for a
cylindrical tube as well as an exponentially and a conically flaring tube. For convenience,
2.19 will be brought into a more convenient form. Substituting P = S12ψ and writing
S = πa2 (a being the radius of the tube at position z) as well as ρ/K = c yields:
∂2ψ
∂z2+
(k2 − 1
a
∂2a
∂z2
)ψ = 0 (2.20)
Note, that the wave ψ and hence the real pressure pa is non propagating if k2 < F ,
where F is the horn function F = 1a∂2a∂z2
. This fact will be discussed for the exponential
tube later on. Please note that the discussion following is made exclusively for plane
waves. This is not necessarily correct, since a flare of the pipe forces the wavefronts to
be curved, but the diameter of the instruments described here are quite small. Hence
the difference between area of the wavefront and the pipes cross section is small enough
to neglect the curvature for the goals of this work.
Theoretical Background 9
Figure 2.4: A finite, cylindrical tube with length L and radius a
2.3.1 Impedance of the Cylindrical Tube
The cylindrical pipe of radius a has the easiest of shapes one could think of. The horn
function F is zero in this case, so no cutoff is obtained. Equation 2.20 takes the form:
∂2ψ
∂z2+ k2ψ = 0 (2.21)
The solution to this equation is
ψ(z) = Aeikz +Be−ikz. (2.22)
This yields the pressure:
pa =ψ
S12
=1
a·(p0e
ikz + p1e−ikz
)eiωt (2.23)
Using 2.2 gives the volume flow Ua:
Ua = −Sρ·∫ t
0
∂pa∂z
dt′ = − Sρc
(p0e
ikz − p1e−ikz
)eiωt + const (2.24)
Theoretical Background 10
If the computation of the input impedance Zin of the cylindrical tube of length L at
z = 0 is needed, two equations to eliminate p0 and p1 are required. Simple application
of the boundary condition determining the impedance at z = L to be equal to ZL gives
the two equations:pa(L)
Ua(L)= −ρc
S· p0e
ikL + p1e−ikL
p0eikL − p1e−ikL= ZL (2.25)
pa(0)
Ua(0)= −ρc
S· p0 + p1
p0 − p1= Zin (2.26)
Solving 2.25 for p0/p1 and substituting into 2.26 results in:
Zin,cyl =ρc
S·ZL cos(kL) + iρcS sin(kL)
iZL sin(kL) + ρcS cos(kL)
(2.27)
2.3.2 Impedance of the Exponential Tube
One of the many models for the flare of a tube is the exponential one, a(z) = a0 · emz,where z is measured from the narrow part of the pipe and a0 is the radius at the narrow
Figure 2.5: A finite, exponential tube with length L and radii a and a2
Theoretical Background 11
end of the pipe. For this kind of slope, the equation 2.20 takes the form:
∂2ψ
∂z2+(k2 −m2
)ψ = 0 (2.28)
This kind of differential equation has the solution (with λ2 = k2 −m2)
ψ(z) = Aeiλz +Be−iλz (2.29)
This gives for the pressure (with the assumed time dependence eiωt and a = a0 · emz):
pa =ψ
S12
=1√πa
(Aeiλz +Be−iλz
)· eiωt =
(p0e
i(λ+im)z + p1e−i(λ−im)z
)· eiωt (2.30)
Note that a cutoff is obtained here (as mentioned in the last section). For k < m waves
are not propagating. This happens especially for small frequencies, if the tube is quite
short and the flare then is to rapid, which results in a big flaring parameter m. This is a
central problem for the simulation, because small pipe segments are used, the impedance
being computed for these segments separately. This method requires very short pipe
segments, because we want to describe the flare as exact as possible. Choosing the
segments to be satisfyingly infinitesimal will produce difficulties measuring the segments
length and the radii, but will solve the problem of imaginary λ at frequencies above
50Hz. Using 2.2 gives the volume flow Ua:
Ua = −Sρ·∫ t
0
∂pa∂z
dt′
= − S
ωρa0
(p0(λ+ im)ei(λ+im)z + p1(−λ+ im)e−i(λ−im)z
)· eiωt + const
Again, two equations are needed to eliminate p0 and p1. Simple application of the
boundary condition determining the impedance at z = L to be equal to ZL (as in
section 2.3.1) gives the two equations:
pa(L)
Ua(L)= − ρω
S(L)· p0e
iλL + p1e−iλL
p0(λ+ im)eiλL + p1(−λ+ im)e−iλL= ZL (2.31)
pa(0)
Ua(0)= − ρω
S(0)· p0 + p1
p0(λ+ im) + p1(−λ+ im)= Zin (2.32)
Now solving 2.32 for p0p1
p0
p1= e−2iλL ·
(λ− im)ZL − ρωS(L)
(λ+ im)ZL + ρωS(L)
(2.33)
Theoretical Background 12
and substituting into 2.31 (note that λ2 +m2 = k2, see 2.22):
Zin =ρω
S(0)·
e−2iλL(
(λ− im)ZL − ρωS(L)
)+ (λ+ im)ZL + ρω
S(L)
e−2iλL(
(λ− im)ZL − ρωS(L)
)(λ+ im) + (λ+ im)ZL + ρω
S(L)(−λ+ im)
=e−iλL
((λ− im)ZL − ρω
S(L)
)+ ((λ+ im)ZL + ρω
S(L))(k2ZL − ρω
S(L)(λ+ im))e−iλL −
(k2ZL − ρω
S(L)(λ− im))eiλL
Writing λ+ im and λ− im in their polar forms, which are(Θ = arctan(mλ )):
λ+ im = k · eiΘ (2.34)
λ− im = k · e−iΘ (2.35)
will simplify this equation. Substituting these forms into Zin gives (using k = ωc )
Zin,exp =ρc
S1·ZL cos(bL+ Θ) + i ρcS2
sin bL
iZL sin bL+ ρcS2
cos(bL−Θ). (2.36)
2.3.3 Impedance of the Conical Tube
A more simple flare is the conical flare, a(z) = a0 · z, where the distance z is measured
from the conical apex, see 2.6. 2.20 now takes this form:
∂2ψ
∂z2+ k2ψ = 0. (2.37)
Figure 2.6: A finite, conical tube with length L and radii a and a2
Simulation 13
The solution to this differential equation is well known to be
ψ(z) = Aeikz +Be−ikz. (2.38)
Using equation 2.2, the volume flow Ua is given by:
Ua = −Sρ·∫ t
0
∂pa∂z
dt′ = − S
ρca
(p0e
ikz(1− 1
ikz)− p1e
−ikz(1 +1
ikz)
)eiωt (2.39)
Analogous to the section before, the boundary conditions
pa(z1)
Ua(z1)= − ρc
S(z1)· p0e
ikz1 + p1e−ikz1
p0(1− 1ikz1
)eikz1 + p1(1 + 1ikz1
)e−ikz1= Zin (2.40)
pa(z2)
Ua(z2)= − ρc
S(z2)· p0e
ikz2 + p1e−ikz2
p0(1− 1ikz2
)eikz2 + p1(1 + 1ikz2
)e−ikz2= ZL (2.41)
are used. Again solving 2.41 for p0p1
and substituting into 2.40 yields:
Zin = − ρc
S(z1)
eik(z1−z2)(
(1 + 1ikz2
)ZL − ρcS(z2)
)+ e−ik(z1−z2)
((1− 1
ikz2)ZL + ρc
S(z2)
)eik(z1−z2)
((1 + 1
ikz2)ZL − ρc
S(z2)
)(1− 1
ikz1
)− e−ik(z1−z2)
((1− 1
ikz2) + ρc
S(z2)
)(1 + 1
ikz1
)(2.42)
We now introduce 1± 1ikzj
in their polar form:
1± 1
ikzj=
1
i
(i± 1
kzj
)=
1
i√
1 + 1k2z2j
e±iΘj , (2.43)
where Θj = arctan(kzj). Accordingly:
1√1 + 1
tan2(Θj)
e±iΘj =tan(Θj)√
1 + tan2(Θj)e±iΘj = sin(Θj)e
±iΘj (2.44)
Substituting and using L = z2 − z1 yields
Zin = − ρc
S(z1)
e−ikL(−ZL i
sin(Θ2)eiΘ2 − ρc
S(z2)
)+ eikL
(ZL
isin(Θ2)e
−iΘ2 + ρcS(z2)
)i
sin(Θ1)e−i(kL+Θ1)
(−ZL i
sin(Θ2)eiΘ2 − ρc
S(z2)
)− i
sin(Θ1)ei(kL+Θ1)
(ZL
isin(Θ2)e
−iΘ2 + ρcS(z2)
) .(2.45)
Now all terms with the same exponent are collected and Euler’s formula is used, yielding:
Zin,con = − ρc
S(z1)
ZLi
sin(Θ2) sin(kL−Θ2) + ρcS(x2) sin(kL)
ZLsin(kL+Θ1−Θ2)sin(Θ1) sin(Θ2) − i
ρcS(z2)
sin(kL+Θ1)sin(Θ1)
(2.46)
This formula describes the impedance at one end of a conical horn, while the other end
Simulation 14
is determined by ZL. It is, like the other formulas for the cylindrical and exponential
tube, starting point for the simulation of the impedance (see chapter 3).
Chapter 3
Simulation
The impedance at one end of a pipe of length L is determined by the pipe’s shape and
length, but also by its impedance at the opposite end, ZL. The simulation of the input
impedance at the throat of a horn will be done by separating the horn into satisfyingly
short segments, so that the actual flare is described in good approximation by the model
we apply on that segment. The previous segments input impedance is used as ZL. At
the bell, were a small part of the vibrational energy is radiated to the surrounding air,
the impedance takes a more complicated form. The impedance in general has the form
Z = R+ iX. (3.1)
Here, R is the reflection coefficient and X is the transmission coefficient (determining,
which amount of vibrational energy is transmitted through the border plane). In this
case, the mouth of the horn is not ideally closed, where R = ∞ and X = 0, but it can
be understood as a pipe being flanged to an infinite plane. The impedance then takes
the form
Z = ρcS
(1− J1(2ka)
ka+ i
H1(2ka)
ka
). (3.2)
Here H1 is the Struve Function of first order and J1 is the Bessel Function of first order.
This expression will determine the impedance at the bell, which we assume to be a
’baffle in an infinite plane’. This model is accurate in our context, because the space in
front of the bell is quite large compared to the bell diameter of 139 mm. The infinite
plane represents nothing more than the boundary condition of the acoustic pressure
being fixed to atmospheric pressure. From another point of view the volume flow is
fixed to the atmospheric value, which is zero or at least very low in comparison to the
acoustic volume flow, because of the enormous cross sectional jump from the bell to the
outside area. This model is based on the assumption, that the pressure and volume flow
are constant over the (bell’s) cross sectional area, which is a problematic simplification
15
Simulation 16
especially at the rapid flaring bell. Since this work is entirely based on ideas neglecting
higher modes, they are not considered here either. One more interesting point is friction.
Since the walls will never be ideally rigid and smooth, this effect is never to be eradicated.
Because dealing with vibrating walls in the context of this work is not possible, there
will be at least a simple attempt on dealing with friction. The frictional losses at the
walls are described with a simple correction to the phase velocity and the wavenumber
k, which is extended by an imaginary part. So for the case with friction:
vph = c
(1− 1.65 · 10−3
√fa
)(3.3)
α =3 · 10−5
√f
a(3.4)
k =ω
c+ iα (3.5)
Here a is the tubes radius and f is the frequency. Please note that the information
about friction and radiation which is given here is minimal. For further reading take
a look at [1], from where the method and numerical values are taken. Now the input
impedance for the horn is computed with two different models (conical, exponential),
using the diameters from figure 3.1. For the corresponding measurement technique see
section 4.1. Frictional losses as well as radiation effects are respected as described above.
Simulation 17
139115
88.9102.5
78.268.2
62.756.4
51.848.145.8
41.738.635.833.8
32.631.129.9
28.1
25.7
23.25
20.45
16.65
14.95
14.1
13.65
12.65
12.4
12.15
11.9
5
10
20
30
40
50
60
20
53
4053
36
25
90
VON EINEM AUTODESK-SCHULUNGSPRODUKT ERSTELLT
VON
EINEM
AU
TOD
ESK-SC
HU
LUN
GSPR
OD
UK
T ERSTELLT
VON EINEM AUTODESK-SCHULUNGSPRODUKT ERSTELLT
VON
EIN
EM A
UTO
DES
K-S
CH
ULU
NG
SPR
OD
UK
T ER
STEL
LT
17.45
70
Figure 3.1: Diameter along the horn axis in mm. The horn is shown in section.
Simulation 18
3.1 Cylindrical Model
There are two boundary cases which can be considered to test, weather the simulation
works correctly. The simulation for the cylindrical tube will return evenly spaced res-
onances, which are uneven multiples of the fundamental frequency f0 = c/ (4L), if the
impedance at the bell is ZL =∞. Figure 3.2 shows the expected behavior, reproducing
the required boundary case. The second case is the exponential prediction for equal
diameters. If the simulation with the exponential model is done for segments which all
Figure 3.2: Cylindrical model for infinite ZL. All resonances are uneven multiples off0 = 50 Hz. c = 200 in this example for simple values.
Figure 3.3: 30 segments with same radius after exponential model and cylindrical tubeof length equal to sum of segment’s lengths vs frequency. Exponential prediction wasmultiplied by 10 to optimize overview. Both predictions are equal over all frequencies.
Simulation 19
have the same diameter, the result must be the same as the one of one cylindrical seg-
ment with same radius and the length equal to the sum of the segment’s lengths. Figure
3.3 displays exactly this behavior, which allows to accept the simulation and compute
the impedance with the measured diameters.
3.2 Conical Model
The impedance at the throat of a conical segment of length L is given by (see chapter
2.3.3):
Zin,con =ρc
S1·
iZL sin(bL−Θ2)sin Θ2
+ ρcS2
sin bL
ZLsin(bL+Θ1−Θ2)
sin Θ1 sin Θ2− i ρcS2
sin(bL+Θ1)sin Θ1
(3.6)
Where
• Sj is the pipes cross section at zj
• ZL is the input impedance at the mouth of the horn
• Θj = arg( cω + izj)
Because the horn function equals zero for the conical case, we obtain no cutoff frequency
in this model. Below cutoff, the wavenumber k is imaginary, the wave does not prop-
agate (see Webbsters equation, Chapter 2.3) and the impedance is zero. Since waves
are propagating for each wavenumber k in a conical segment, the expression for Zfl
(developed above) can be used without modification. Figure 3.4 now shows the absolute
value of Zin,con at the throat of the horn plotted against frequency f on a log-scale. The
resonances are marked with black dashed lines, the tolerances are quarter tones. The
decrease in amplitude is caused by the real part of Zfl, which is anti proportional to ω
and therefore decreases with growing frequency. Recall, that the resonances are uneven
multiples of a ground frequency f0 = c/(4L) = 64 Hz for a cylindrical segment of length
L = 1.34m (typical length of a trumpet) with ideally closed end. The first resonance
of the conical model however is located at a much higher frequency. Whereas all higher
resonances are approximately multiples of the second resonance frequency f2 divided by
two (3/2f2, 2f2, 5/2f2, 3f2 . . . ), the first resonance breaks this pattern. One would
expect it to be at 1/2f2, which equals one octave.1 If compared to the second resonance,
it can be seen that the first resonance is not one octave, but approximately one octave
plus one quart (17 half tones) lower than the second. This surprising fact needs careful
comparison with actual measurements because this behavior is not expected.
1One octave is divided into halftones of 100ct,where the tonal difference I in cent between twofrequencies f1 and f2 is I = log2(f1/f2). So one octave (1200ct) equals a frequency ratio of 2/1.
Simulation 20
Figure 3.4: Absolute value of input impedance after conical model vs. frequency.Resonances are marked with black, dashed lines. Tolerances are quarter tones
3.3 Exponential Model
The impedance at the throat of a exponential segment of length L is given by:
Zin,exp =ρc
S1·ZL cos(λL+ Φ) + i ρcS2
sinλL
iZL sinλL+ ρcS2
cos(λL− Φ)(3.7)
• Sj is the pipes cross section at zj (where (0,1) is (0,L))
• λ =√k2 −m2
• Φ = arg(b+ im)
• m is the flare constant given by√F (see Webbster equation, Chapter 2.3)
In the exponential model the horn function F is not zero. For an exponential flare, the
solution to Webbster’s equation is proportional to eibz with b =√k2 −m2. So a wave will
not propagate unless ω/c > m, which means that the acoustical impedance Zin,exp = 0
for ω < mc. In the simulation, ZL = 0 if this condition is true. It is indeed true for a
quite large frequency range if the difference between the radii of the considered segment
is very large, so the flare is very rapid, which it is especially at the bell. This results
in Zin,exp to be not continuous (note the edges). Figure 3.5 shows the absolute value of
Zin,exp at the throat of the horn plotted against frequency f on a log-scale. The second
resonance immediately catches attention. It is positioned very much out of the pattern.
This mismatched position has to be a result of an error in the diameter measurement,
because the influence of the segments length and radii on the resonances dominate all
Simulation 21
Figure 3.5: Absolute value of input impedance after exponential model vs. frequency.Resonances are marked with black, dashed lines. Tolerances are quarter tones
Figure 3.6: Horn is shown in section. The 24th radius of 16.65mm is quite large incomparison with its two neighbors.
other effects. Since the lengths are all measured with high precision (take a look at
section 4.1), the diameters are more likely responsible for this mismatched position. By
looking at the shape displayed against the horn axis in figure 3.6, the diameter most likely
being measured wrong can be identified: the 24th diameter of 16.65 mm seems to be
too large. Figure 3.7 now shows the exponential prediction with the old value compared
to the exponential prediction done with the 24th diameter substituted with the mean of
the two neighbors, which is (17.45 + 14.95) /2 mm = 16.2 mm. The simulation with the
corrected value is much more consistent with the equally spaced pattern we obtained for
the conical prediction, so the obviously wrong diameter is substituted by the new value
Measurement 22
Figure 3.7: Influence of 24th diameter. Exponential prediction with old and newvalue vs frequency. The new value almost only influences the second resonance, the twocurves are nearly identical over the whole frequency range. Measured resonances are in
black, dashed lines.
Figure 3.8: Simulation after exponential model with new value for d24 vs frequency.Resonances after this model are marked with black, dashed lines.
of 16.2 mm. The new simulation is shown in figure 3.8 and will be used for comparison
with the measurement in chapter 5. As for the conical model the number of resonances
is correct, whereas the positions are different especially at high frequencies (see ticks or
comparison in Appendix A.4). The exponential model also predicts the first harmonic
to be approximately 17 halftones below the second resonance2.
2One octave is divided into halftones of 100ct,where the tonal difference I in cent between twofrequencies f1 and f2 is I = log2(f1/f2). So one octave (1200ct) equals a frequency ratio of 2/1.
Measurement 23
3.3.1 Influence of single Diameters on the Spectrum
Simulating the input impedance of a horn, allows to compute the natural resonances,
but denies insight on the effect of one particular part of the instrument on this spectrum.
It cannot be determined, how the variation of one radius procreates over the following
steps of the simulation. To obtain insight on this topic, selected radii ai will be variated
in steps of 0.5 percent of the actual value. The corresponding values for the resonances
will then be displayed against the radius. From the figures it is clear, that the resonances
are strongly influenced by the radii at the almost conical part of the instrument, whereas
the bell has not a great influence on the resonances. Figure 3.9 shows, that the forth
resonance, which should be highly stable because of its importance for the instrument
(note in the mostly played frequency area), decreases in frequency when the 24th radius
is variated. The horizontal scale is in steps of 0.005 · r24. As can be seen in figure
3.1, that equals steps of 0.005 · 20.45 mm ≈ 0.1 mm. The frequency steps are in the
order of 5 Hz for a change of about 15 percent, which would be 3 mm. In section
4.1 the measuring uncertainty for the diameters is approximated as 0.54 mm. If 3 mm
equal an uncertainty of 5 Hz in the frequency range, then the measuring uncertainty
of 0.54 mm equals an uncertainty of roughly 1Hz. This, on the one hand results in a
very good spectral resolution, on the other hand the ideas presented here can be seen as
not more than rough estimations. One more point can be made in this section. If the
correlation of the resonance and one of the first diameters (see figure 3.10) is compared
to the correlation shown in figure 3.9, it is clear that the shape of the bell has only little
influence on the resonances. Hence measuring the shape at the bell with higher precision
than the mostly conical part of the instrument is not recommended. Accordingly the
characteristics mostly influencing the resonance spectrum are the total length of the
instrument as well as the almost conical part from radius 20-25 (see figure3.1). This
however contrasts the aspiration of obtaining an almost continuous impedance for the
exponential model by choosing almost infinite segments, as explained above.
Measurement 24
Figure 3.9: Forth resonance vs 24th radius ± units of 0.5 percent. Left scale in Hz,right scale in ct. The correlation of radius increase/decrease and resonance position is
one issue which could be covered in following works.
Figure 3.10: Forth resonance vs 10th radius ± units of 0.5 percent. Left scale in Hz,right scale in ct. The correlation of radius increase/decrease and resonance position is
not as dramatic as in figure 3.9.
Chapter 4
Measurement
4.1 Diameter Measurement
Figure 4.1 displays the diameter of the instrument along the instruments axis. The mea-
surement is done with a caliper. To carry out the measurements as precise as possible,
the trumpet is lying on a sheet of paper. The positions for measurements are first drawn
onto this sheet of paper and then transferred onto the trumpet. Figure 4.2 shows the
setup for this technique in detail. The straight segments length is determined with very
good precision (only the systematic uncertainty of the caliper used, see below), whereas
the diameters are, especially at the bell, tainted with a great measuring uncertainty.
This uncertainty is caused by the fact, that the caliper may not always be perfectly
perpendicular to the pipe axis. In the curvature of the instrument the straight distance
between measuring points were chosen as L because in our model we assume the seg-
ments to be straight. The uncertainty on the measured diameters must be combined
from multiple uncertainties. Obviously, there is the raw measuring uncertainty of the
caliper, which is approximately the smallest unit as an upper limit (0.1 mm). While this
uncertainty is quite small, the dominating effect is the caliper not being perpendicular
to the instruments axis. If the caliper is off center by the angle α, then the measured
diameter m would differ from the real diameter r by
d = m− r = m (1− cosα) . (4.1)
If the experimentalist is able to keep α under 5◦, then the maximal measuring uncertainty
on diameters is the one for measuring the bell of 139 mm:
d = 139 · (1− cos 5◦)mm ≈ 0.53 mm (4.2)
25
Measurement 26
139115
88.9102.5
78.268.2
62.756.4
51.848.145.8
41.738.635.833.8
32.631.129.9
28.1
25.7
23.25
20.45
16.65
14.95
14.1
13.65
12.65
12.4
12.15
11.9
5
10
20
30
40
50
60
20
53
4053
36
25
90
VON EINEM AUTODESK-SCHULUNGSPRODUKT ERSTELLT
VON
EINEM
AU
TOD
ESK-SC
HU
LUN
GSPR
OD
UK
T ERSTELLT
VON EINEM AUTODESK-SCHULUNGSPRODUKT ERSTELLT
VON
EIN
EM A
UTO
DES
K-S
CH
ULU
NG
SPR
OD
UK
T ER
STEL
LT
17.45
70
Figure 4.1: Trumpet, diameter in mm measured with caliper. Lengths of the segmentsare also given in mm.
Measurement 27
Figure 4.2: Measurement of trumpet diameters on the sheet
Quadratic addition gives the total uncertainty σtotal:
σtotal =√
0.532 + 0.12 mm ≈ 0.54 mm. (4.3)
So the measuring uncertainty for the diameters, which are at least about 12 mm, can be
approximated as maximally 0.54 mm. This is an acceptable uncertainty for using these
diameters.
4.2 Measuring Technique
Consider a cylindrical tube expanding from z = 0 to z = L. The impedance Zin at z = 0
is given by (for pa and ua have a look at section 2.3.1):
Zin =pa(0)
ua(0)= ρc
A+B
A−B= ρc
1 +R
1−R, (4.4)
where R = B/A is the reflection coefficient. R can be obtained by measuring the
pressure amplitude at different positions zj of the tube and computing the transfer
Measurement 28
functions, defined as
H1,2 =pa(z2)
pa(z1)=
1 +Re−2ikz2
1−Rei2ikz1· eik(z2−z1). (4.5)
Solving for R yields
R =H1,2 − eiks
e−iks −H1,2, (4.6)
where s = z2 − z1 is the distance between the two measuring positions. Zin accordingly
is
Zin =e−iks −H1,2 +
(H1,2 − eiks
)e2ikz1
e−iks −H1,2 − (H1,2 − eiks) e2ikz1. (4.7)
This calculation is automatically done by the ITA1 software, the output used for this
work already is the complex number Zin. The impedance at a point is anti proportional
to the cross section at this particular point. So if the impedance is measured at the
point z = 0 of the Kundts tube with radius b and the impedance of the horn of radius
a at this point is of interest, the measured impedance has to be multiplied by the ratio
r = SbSa
:
Zin,horn = r · Zin,measured (4.8)
4.3 Measuring Setup
We use a Kundt’s tube for our measurements. At one end of this tube the source
generates the signal (exponential sweep, sinus signal), the other end holds the test object.
The tube with diameter 2 inches, which equals 5.08cm2, is build from aluminium and
has a satisfying wall thickness to prevent vibrations of the walls. It is closed, hence
no vibrational energy is lost to the air surrounding the tube. The tube has four holes
for microphones, which are closed when the microphone position is not used. For the
measurements at the four different positions the same microphone is used, so there are
no systematic errors caused by different microphone sensitivities or other inequalities
between microphones. A large distance between the microphones is desirable, because
the distance between two microphones must exceed 0.05λmax, where λmax is the maximal
wavelength for which results are acceptable[2].Therefore a minimal frequency is obtained
for each microphone pair, for which measurements are acceptable. The distances of
the second, third and forth position are given in the table, with the estimated lower
frequency limit for measurements. Each pair of microphone positions (1,2),(1,3),(1,4)
allows measurements in different frequency ranges, based on their distance. With these
1’Institut fuer technische Akustik’ at RWTH21 inch equals 2.54 cm, value taken from http://www.din-formate.de/kalkulator-berechnung-laenge-
masse-groesse-einheiten-umrechnung-inch-zoll-in.html from 10.7.2015
Measurement 29
Table 4.1: Microphone position distances and corresponding lower frequency limit(roughly).
Microphone pair Distance in mm Lower frequency limit in Hz (roughly)
(1,2) 17 1009
(1,3) 110 156
(1,4) 514.05 33
frequency ranges combined, we are able to measure the impedance from approximately
40Hz up to 9000Hz. The software handling the raw data output of the microphones was
developed at ITA3. As can be seen in figure 4.3, the instrument is connected to the tube
by one of two special flange (shown in figure 4.4). The flanges assure that no energy losses
occur at the junction from tube to instrument. These flanges, one for measurements with
mouthpiece and one for measurements without mouthpiece, were crafted in the workshop
of the physics department. When measuring without mouthpiece, the throat must be
extended by a distance d = 2 cm, otherwise the flange cannot be connected to the
instrument. The measurement is done as follows: The source generates a sinus signal, in
Figure 4.3: Trumpet, flanged to the Kundt’s tube. The instrument is supported by arock wool cuboid to prevent strains. The signal generator is on the right, red parts areplugs for the microphone position. Note that the tubes diameter is constant, regardless
of the outer shape.
3’ITA Toolbox’, providing a GUI for measurements and data handling with Matlab.
Comparison 30
Figure 4.4: Special flange used to connect trumpet and Kundts tube.
this case with a sampling rate of 44100, from 0−22050 Hz. The sweep consists of roughly
65000 entries, so the step length is approximately 0.3 Hz. First a test measurement is
performed, where the nonlinear parts of the signal are controlled. If the amplitude of
these nonlinear parts (mainly caused by the source) is too high, then the amplitude of
the signal must be reduced. Generally, the intensity of the nonlinear parts should not
be greater than one tenth of the linear signal. On the other hand, one must be able to
distinct the signal from noise, which requires the amplitude of the signal to be about
ten times greater than the noise. For all of the measurements done, the quality of the
setup is controlled and rated as being acceptable. Now the sound pressure is measured
at the different microphone positions while scanning trough the sweep interval each
time. The software then computes the transfer functions H1,2, H1,3, H1,4 and returns
the impedance.
4.4 Results
This section shows the results of the measurements. The focus is set on the absolute
value of the impedance, so in the graphics |Zin,measured| is shown. These measured
Comparison 31
Figure 4.5: |Zin,measured| of the trumpet without mouthpiece vs. frequency. Reso-nances are marked with black, dashed lines.
Figure 4.6: |Zin,measured| of the trumpet without mouthpiece vs. frequency, secondmeasurement. Resonances are marked with black, dashed lines.
impedances will be compared to the simulation in the following chapter. Figures 4.5
and 4.6 show the absolute value of Zin,measured for a trumpet of length L = 1.34 m
without mouthpiece. Mark the pattern below the first resonance and the double peak
at the second resonance, which are two obvious differences to the simulation. The
pattern may be due to the fact that the values are taken very near or even below the
measuring border at approximately 40 Hz. The double peak however is not explained
so easily, because it is very surprising to have a resonance so consequently out of any
pattern. The amplitude of the impedance decreases towards higher frequencies. The
most evident characteristic is the amplitude of the forth resonance. It is approximately
Comparison 32
one order greater than all other resonances and quite sharp. This resonance turns out
to be the keynote in the middle playing octave and it is very much useful, that this
resonance is the clearest one (this note may be the most important note on a brass
instrument).
Chapter 5
Comparison of Simulation and
Measurement
Now the measured impedance is compared with the results of our simulation. The figures
5.1 and 5.2 comparing measurement and simulation show, that the impedances shape is
not exact, but the resonances are within a half tone range of the prediction. It is espe-
cially evident, that the measurement confirms the position of the first resonance, which
is not, as already mentioned in chapter 3, at the half of the second resonance frequency,
but nearer to a frequency which is approximately 17 halftones lower than the second res-
onance. However these values have the greatest difference from measurement to model,
the measured resonance being at an even smaller frequency than the model’s prediction.
For higher frequencies, measurement as well as prediction decrease in absolute value
and the simulation tends to predict the resonances at higher frequencies as well. Also
mark, that the measurement confirms the second resonance of the exponential model
Table 5.1: Positions of the resonances in Hz, conical as well as exponential predictionand measurements
Prediction in Hz Measurements in Hz
Resonance Conical Exponential 1st 2nd
1 82.44 94.06 75.37 76.37
2 226.48 233.32 243.25 241.91
3 345.21 352.21 358.66 354.96
4 473.17 482.24 483.48 484.15
5 598.26 604.63 605.61 605.28
6 727.66 735.93 729.43 728.75
7 856.25 864.69 856.60 854.59
8 986.44 994.88 977.05 974.70
9 1115.04 1127.61 1099.52 1097.50
10 1237.11 1245.86 1224.68 1220.98
33
Comparison 34
after correcting the 24th radius. Table 5.1 shows the measured and predicted values for
the resonances, contrasting all values. Comparing the measurements with both models,
it can be said that both models are equally successful (within the aspirations of this
work). Both conical and the exponential model are close to the actual measurement.
They also have weaknesses though: the exponential model brings discontinuities with
it on the one hand, this is due to the cutoff frequencies for the short segments being
not small enough to fall out of the frequency range of interest. On the other hand,
the exponential model is more sensitive to variations of the radii along the instruments
axis (see appendix figure A.4). It can be seen in this figure, that the exponential model
reacts more extremely to changes of the shape than the conical model. In contrast with
the exponential model, the conical one has no cutoff and therefore is continuous over
all frequencies. It also causes fewer problems with the simulation, but is not as sensi-
tive to changes of radius as the exponential model. The differences between simulation
and measurement are acceptable within the frame of this work. Considering additional
effects like vibration of the walls and curvature of the wavefronts as well as providing
better understanding of the effects considered will much likely improve the result. The
understanding of friction and radiation is quite poor in this model used and improving
this understanding would surely contribute to further insights.
Conclusion 35
Figure 5.1: |Zin,measured| of trumpet without mouthpiece and conical prediction vs.frequency. Resonances are marked with black, dashed lines
Figure 5.2: |Zin,measured| of trumpet without mouthpiece and exponential predictionvs. frequency. Resonances are marked with black, dashed lines
Chapter 6
Conclusion
The goal of this work is to obtain a prediction of the natural resonances of a trumpet,
which is within a half tone range of the actual resonance. These actual resonances are
extracted from the measured acoustical impedance. Chapter 5, presenting the compar-
ison of prediction and measurement, shows that this aspiration is met. Especially the
position of the first resonance is quite dissimilar in the prediction than it was measured
though, but definitely not at the half frequency of the second resonance. The shape
of |Zmeasured| is in some regions not reproduced with great success also. However, the
quite simple model already allows predictions of the resonances with, for the scope of
this work, satisfying accuracy. It has been made clear that the shape of the instruments
bell (see section 3.3.1) has no great influence on the resonances, whereas the mostly
conical section after the bell is of great importance for the spectrum. Measuring the
diameters in this region with higher precision would certainly improve the simulations
results, but a higher precision is not realistic with the setup chosen for this work (con-
sider measuring uncertainties presented in section 4.1). Using an optical measurement
technique, especially to minimize the experimentalists influence on the measured values,
would be necessary here. The technique for the impedance measurements however needs
no immediate improvement, the results are satisfyingly exact. The only weakness of this
setup is that the lower frequency limit for acceptable measurements is in the same or-
der as our first resonance. An additional microphone with greater distance to the first
microphone would solve this problem. Overall the aspirations of this work are met and
the simple, one dimensional model for sound propagation in tubes turns out to be very
successful in predicting the instruments resonances.
37
Appendix A
Appendix
39
Conclusion 40
Figure A.1: Influence of friction, exponential model with and without friction vs fre-quency. Influence increases towards greater frequencies but has almost no influence onthe resonances positions. The influence on the amplitude however is obvious. Reso-
nances after exponential model with friction are in black, dashed lines.
Figure A.2: Influence of friction, conical model with and without friction vs frequency.Influence increases towards greater frequencies but has almost no influence on the res-onances positions. The influence on the amplitude however is obvious, the frictionallosses result in smaller amplitude. Resonances after conical model with friction are in
black, dashed lines.
Conclusion 41
Figure A.3: Comparison of conical and exponential model. Both models give thesame number of resonances, but differ in position and shape. Especially the edges in
the exponential prediciton are an obvious difference to the conical prediction.
Figure A.4: Influence of 24th diameter. Exponential prediction with old and newvalue vs frequency. The new value almost only influences the second resonance, the twocurves are nearly identical over the whole frequency range. Measured resonances are in
black, dashed lines.
Bibliography
[1] Neville H. Fletcher and Thomas D. Rossing. The Physics of Musical Instruments.
Springer.
[2] Komitee ISO/TC 43/SC 2 Bauakustik. EN ISO 10534-2. Springer.
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