Acoustical Impedances: Calculations and Measurements on a ...

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


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


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


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


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.


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.


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)


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


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)


and substituting into 2.31 (note that λ2 +m2 = k2, see 2.22):

Zin =ρω

S(0)·

e−2iλL(


)+ (λ+ im)ZL + ρω

S(L)

e−2iλL(


)(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

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

AU

TOD

ESK-SC

HU

LUN

GSPR

OD

UK

T 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


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


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.

43

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