VISUALIZATION OF VIBRATION OF IDEAL AND REALISTIC
STRINGS IN AN ACOUSTIC GUITAR BY USING
MATHEMATICA
by
AHMAD ALIF BIN KAMAL
Dissertation submitted in partial fulfilment
of the requirements for the degree
of Master of Science in Mathematics
August 2015
ACKNOWLEDGEMENT
Alhamdulillah, the dissertation for my study is finally completed. It is essential
that the guidance, help, and support given and provided by the people around me are
keys in completing this dissertation.
Firstly, I would like to express my sincere and precious gratitude to Dr. Yazariah
Mohd Yatim as the supervisor of this dissertation. Without her guidance and advice, this
thesis would have been impossible to be tackled and finished. Her support and patience
has been spearheading through the challenges as well as sincerity and encouragement
that provides the morale boosting, becomes an inspiration for me to topple the obstacles
in completion of this dissertation.
I would like to also thank my family and friends for the supports and
motivations. Without them, the strength to go on would not have been sufficed. They
were very encouraging, and always by my side, especially my beloved wife and son,
through thick and thin of the times spent for this dissertation.
Last but not least, thank you to University Malaysia Sarawak and Kementerian
Pendidikan Malaysia for providing me SLAB throughout my postgraduate study for the
whole year and semester of studies.
ii
TABLE OF CONTENTS
ACKNOWLEDGEMENT
ii
TABLE OF CONTENTS
iii
LIST OF TABLES
vi
LIST OF FIGURES
vii
ABSTRAK
ix
ABSTRACT
xi
CHAPTER 1: INTRODUCTION
1.1
The Acoustic Guitar String
3
1.2
Vibrations of Damped String
5
1.3
Vibrations of Stiff String
6
1.4
Vibrations of Damped Stiff String
7
1.5
Problem Statement
8
1.6
Objectives
8
1.7
Layout of Dissertation
9
CHAPTER 2: LITERATURE REVIEW AND METHODOLOGY
2.1
Literature Review
11
2.2
Methodology
16
2.3
Mathematica Software
21
iii
CHAPTER 3: VIBRATION OF IDEAL STRING
3.1
Mathematical Formulation
26
3.2
Algorithm for Modes of Vibration and Sound
31
3.3
Results and Discussion
33
CHAPTER 4: VIBRATION OF DAMPED STRING
4.1
Mathematical Formulation
43
4.2
Algorithm for Modes of Vibration and Sound
47
4.3
Results and Discussion
49
CHAPTER 5: VIBRATION OF STIFF STRING
5.1
Mathematical Formulation
59
5.2
Algorithm for Modes of Vibration and Sound
63
5.3
Results and Discussion
65
CHAPTER 6: VIBRATION OF DAMPED STIFF STRING
6.1
Mathematical Formulation
75
6.2
Algorithm for Modes of Vibration and Sound
78
6.3
Results and Discussion
80
CHAPTER 7: SOUND SYNTHESISING WITH DAMPED STIFF STRING
7.1
Guitar Notes and Their Frequencies
92
7.2
Guitar Tabs and Songs
95
iv
CHAPTER 8: CONCLUSION AND FUTURE WORK
8.1
Conclusion
100
8.2
Future Work
102
REFERENCES
103
APPENDICES
v
LIST OF TABLES
Table 3.1
Number of nodes and antinode for given harmonics
35
Table 7.1
The guitar notes and their frequencies
94
vi
LIST OF FIGURES
Figure 1.1
A classical guitar
1
Figure 1.2
The anatomy of a guitar
2
Figure 3.1
The initial plucking position of guitar string
27
Figure 3.2
Mode of vibrations for (a) n  1 , (b) n  2 , and (c) n  3
34
Figure 3.3
Displacement of yx, t  against x and t
35
Figure 3.4
Tone for (a) n  1 , (b) n  2 , and (c) n  3
37
Figure 3.5
Tone for (a) f1  220 , (b) f1  275 , and (c) f1  330
38
Figure 3.6
Amplitude distribution for (a) h  0.01 , (b) h  0.1 , and
(c) h  1
40
Figure 3.7
Timbre for (a) d  0.1 , (b) d  0.23 , and (c) d  0.5
41
Figure 4.1
Mode of vibrations for (a) n  1 , (b) n  2 , and (c) n  3
51
Figure 4.2
Displacement of yx, t  against x and t
52
Figure 4.3
Tone for (a) n  1 , (b) n  2 , and (c) n  3
53
Figure 4.4
Tone for (a) f1  220 , (b) f1  275 , and (c) f1  330
54
Figure 4.5
Amplitude distribution for (a) h  0.01 , (b) h  0.1 , and
(c) h  1
55
Figure 4.6
Timbre for (a) d  0.1 , (b) d  0.23 , and (c) d  0.5
56
Figure 4.7
Timbre for (a)   1 , (b)   0.01 , and (c)   0.0001
58
Figure 5.1
Mode of vibrations for (a) n  1 , (b) n  2 , and (c) n  3
66
Figure 5.2
Displacement of yx, t  against x and t
67
vii
Figure 5.3
Tone for (a) n  1 , (b) n  2 , and (c) n  3
68
Figure 5.4
Tone for (a) f1  220 , (b) f1  275 , and (c) f1  330
69
Figure 5.5
Amplitude distribution for (a) h  0.01 , (b) h  0.1 , and
(c) h  1
71
Figure 5.6
Timbre for (a) d  0.1 , (b) d  0.23 , and (c) d  0.5
72
Figure 5.7
Timbre for (a)   0.1 , (b)   0.01, and (c)   0.001
74
Figure 6.1
Mode of vibrations for (a) n  1 , (b) n  2 , and (c) n  3
81
Figure 6.2
Displacement of yx, t  against x and t
82
Figure 6.3
Tone for (a) n  1 , (b) n  2 , and (c) n  3
84
Figure 6.4
Tone for (a) f1  220 , (b) f1  275 , and (c) f1  330
85
Figure 6.5
Amplitude distribution for (a) h  0.01 , (b) h  0.1 , and
(c) h  1
86
Figure 6.6
Timbre for (a) d  0.1 , (b) d  0.23 , and (c) d  0.5
87
Figure 6.7
Timbre for (a)   1 , (b)   0.01 , and (c)   0.0001
89
Figure 6.8
Timbre for (a)   0.1 , (b)   0.01, and (c)   0.001
90
Figure 7.1
Fret numbers on a guitar’s fretboard
93
Figure 7.2
Guitar tab
95
Figure 7.3
Playing a guitar note
96
Figure 7.4
Playing a guitar chord
96
Figure 7.5
Playing a guitar arpeggio
97
Figure 7.6
Playing a guitar with pauses
97
Figure 7.7
Guitar tab for “Negaraku”
98
Figure 7.8
Guitar tab for intro part of song “Daybreak’s Bell” by
L’arc en Ciel
98
viii
PENGGAMBARAN GETARAN TALI GITAR AKUSTIK YANG IDEAL DAN
REALISTIK MENGGUNAKAN MATHEMATICA
ABSTRAK
Dalam disertasi ini, semangat mendalam terhadap matematik dan muzik, di
mana muzik merupakan salah satu subjek popular dalam fizik, telah menyuntik minat
untuk menjalankan penyelidikan atas getaran tali gitar akustik. Model untuk tali gitar
akustik yang ideal dan sebenar menggunakan permodelan matematik, yang diselesaikan
dengan
kaedah
analitikal
dan
mengaplikasikan
penyelesaiannya
dalam
perisian
Mathematica telah disiapkan. Persamaan gelombang satu dimensi digunakan dan
diubahsuai untuk pemodelan tali ideal dan sebenar dan diselesaikan dengan kaedah
pemisah pembolehubah bagi mendapatkan fungsi sesaran tali. Persamaan umum bagi
tali yang ideal, terlembap, kaku, dan kaku terlembap menjurus kepada pelbagai andaian
serta memerlukan aspek fizik lain untuk mendapatkan penyelesaiannya. Siri Fourier
turut digunapakai untuk mendapatkan nilai pemalar penting dalam penyelesaian. Satu
program telah dibuat bagi membolehkan pengguna untuk memasukkan tinggi petikan,
posisi petikan, frekuensi semula jadi tali, pekali lembapan, dan parameter kekakuan
untuk memerhatikan pelbagai mod getaran tali. Hasil daripada program ini ialah nada
asas dan nada lebihan atau harmonik tali, dan juga timbre bagi bunyi yang dihasilkan
oleh model tali tersebut. Tinggi petikan mempengaruhi amplitud sesaran tali. Dengan
memanipulasi posisi petikan, perubahan dapat diperhatikan pada kepelbagaian taburan
amplitud. Bagi frekuensi semula jadi, semakin tinggi semakin banyak ayunan yang
ix
berlaku dalam selang masa yang sama. Pekali lembapan yang tinggi menyebabkan
getaran tali mencapai posisi seimbang dengan lebih cepat. Nilai parameter kekakuan
yang besar mencetuskan lebih frekuensi dalam bentuk getaran. Bagi tali ideal, sesaran
berayun dalam keadaan malar tak terhingga, bertentangan dengan tali terlembap di
mana sesaran menghampiri posisi seimbang apabila masa menghampiri infiniti. Dalam
tali kaku, ayunan tak terhingga tetapi tempoh getaran dikurangkan dan kekakuan
merumitkan bentuk
gelombang.
Tali kaku terlembap
memiliki kedua-dua kualiti
padanya dan digunakan untuk menghasilkan semula bunyi gitar akustik.
x
ABSTRACT
In this dissertation, the passion for both mathematics and music, while the latter
being a very popular subject in physics, had instilled an interest to conduct a research on
vibration of acoustic guitar string. The model for ideal and realistic guitar strings using
mathematical modelling, solved with analytical method and applied the solution in
Mathematica software is done. The one-dimensional wave equation is used and
modified to model the ideal and realistic string and solved by the method of separation
of variables to obtain the displacement function of the string. The general equations that
govern the ideal, damped, stiff, and damped stiff string lead to different assumption and
require other physics aspects to obtain the solutions. Fourier series is applied to obtain
values for important constant in the solutions. A program is made to allow users to input
plucking height, plucking position, natural frequency of string, damping coefficient, and
stiffness parameter to observe the different modes of vibration on the string. The
outputs of the program are the fundamental tone and overtones or harmonics of the
string, as well as the timbre of the sound produced by the modelled string. Plucking
height affects the amplitude of the displacement of the string. With manipulation of
plucking position, the changes can be seen in the diversity of amplitude distribution. As
for natural frequencies, the higher it is the more the oscillation happened in equal time
interval. Large damping coefficient makes the vibration of string to reach the
equilibrium position faster.
Greater value of stiffness parameter induces more
frequencies in the shape of vibration. For an ideal string, the displacement oscillates in
xi
an infinitely constant manner, in contrast to the damped string where the displacement
approaches its equilibrium position as time approaches infinity. In stiff string, the
oscillation is infinite, but the period of vibrationis reduced and stiffness complicates the
waveform. The damped stiff string has both qualities in it and is used to reproduce the
sound of acoustic guitar.
xii
CHAPTER 1
INTRODUCTION
Music and mathematics, how well they blend together, is frequently questioned
by people of both fields. Music is definitely among the popular topics in physics and
engineering. As someone with passion for both music as well as mathematics, this
dissertation proves to be a good start for any further studies and projects that can be
done within this field. Any big dream almost always begins with small steps. Gottfried
Wilhelm von Leibniz, a German mathematician who co-discovered calculus, once said
that music is the pleasure the human soul experiences from counting without being
aware that it is counting (Hammond & Kelly, 2011). Figure 1.1 below shows the
classical guitar known throughout the world.
Figure 1.1: A classical guitar
Source: http://www.interstatemusic.com
1
Guitar is a popular musical instrument, and classified as stringed instruments. It
has anywhere from four to eighteen strings, but commonly having six. A guitar is an
instrument defined having a long neck with frets, flat wooden soundboard, ribs and back
as well as curvy sides. Strumming or plucking the strings while “fretting” them
(pressing the strings on the neck against the fret) is how people typically played this
instrument. Guitars can be divided into two broad categories which are acoustic and
electric (Kumar 2011). Figure 1.2 shows the anatomy of a guitar, where the fret, bridge,
and nut are among the most important part to incorporate for this dissertation.
Figure 1.2: The anatomy of a guitar
Source: http://www.guitarfriendly.net
For modern acoustic guitar, there are three main types, which are the classical
guitar (nylon-string), steel-string guitar, and archtop guitar. The sound and tone of an
acoustic guitar is emitted by the vibration of strings, amplified by the body of the guitar
that acts as a resonating chamber. The classical guitar is commonly played as a solo
instrument by using different fingerpicking techniques (Gove 1999). The guitar is used
in wide variety of musical genres include blues, jazz, metal, punk, rock, and pop.
2
1.1
The Acoustic Guitar String
For the guitar, plucking its strings is the basic of how a person produces the
sound from it. This action causes a single string to vibrate and has a small displacement.
In this dissertation, the main focus will be on modelling the vibration of an acoustic
guitar string.
Many literature
sources such as MacDonald (2013), Gulla and
Katedralskole (2011), Kreyszig (1999), and Pelc (2007), indicated that vibration of
string in stringed instruments, guitars included, can be modelled by a one-dimensional
wave equation.
Firstly, the wave equation must be defined and derived. Consider the forces
acting on a small portion of a string. Since the string does not offer resistance to
bending, the tension force is tangential to the curve of the string. Let T1 and T2 be the
tensions at the endpoints of this portion. The points of the string move in vertical
direction with no horizontal motion, and then the horizontal components of the tension
must be constant that is
T1 cos   T2 cos   T  const ,
where T means an equal, constant value of tension force. For the vertical components,
there are two forces, namely  T1 sin  and T2 sin  , where the negative sign means a
force directed downward (Kreyszig, 1999). From the Newton’s second law, the
resultant of these two forces is equal to the product of linear mass density of string (or
some called it the mass of undeflected string per unit length)  ( kgm 1 ), length of
portion of the undeflected string x and acceleration  2 y t 2 ( ms 2 ), evaluated at
some point between x and x  x . Therefore
T2 sin   T1 sin    x
2 y
t 2
.
When this is divided with T1 cos   T2 cos   T , the result will be
3
T2 sin   T1 sin  T2 sin  T1 sin 
 x  2 y


 tan   tan  
.
T
T2 cos  T1 cos 
T t 2
Both the tangents are the slopes of the string at x and x  x , which is
 y 
 y 
tan     and tan    
.
 x  x
 x  x  x
Substitute these back into the equation above and divide with x will yield
2
1   y 
 y     y
  
.
 
x   x  x  x  x  x  T t 2
By letting x approaches zero, the one-dimensional wave equation is given by
1 2 y
v 2 t 2

2 y
(1.1)
x 2
where v is the speed of the waves traversing along the string for 1 v 2   T , with
v  2 f1 l , f 1 is the natural frequency of the string (Hz or s 1 ) and l is the length of the
string (m). The yx, t  is the displacement of the guitar string at any point and time
(Kreyszig, 1999).
Equation (1.1) is used to model displacement of the string in an ideal
environment, where most real physical aspects are neglected. Guitars strings are fixed at
both end, thus, the vibrations of the string are called standing waves. The note that
particular string is tuned to depends on the fundamental mode of vibration, while the
sound produced is obtained from the temporal solution of the governing partial
differential equation. Gulla and Katedralskole (2011) studied in detailed, step-by-step
approach of the wave motion of a guitar string, and they stated the some potential
improvements for this model include string stiffness, damping factor, and others.
4
1.2
Vibrations of Damped String
From the general wave equation in (1.1), it can be modified to model the
vibration of guitar string in a much more realistic environment. A vibrating string has
energy associated with it when it is plucked. In a real situation, when a string is plucked,
it will slowly die out and cease to vibrate. The situation of it dying out is what portrayed
as dissipation of energy, also known as a form of damping of the vibrating string. As a
matter of fact, all the initial energy associated will wind up as heat in the viscous
medium such as the air. There are many physical mechanisms associated with
dissipative loss of energy, but in this dissertation damping is taken in a general manner.
Consider a stretched string of length l as before with the whole system immersed
completely in viscous medium that causes damping. The generic damping force, acting
on a small portion of the string at a certain time is given by
dFdamping x, t   
y
,
t
where  is the constant proportionality for the damping force, and the negative sign
indicates that damping force is always directed in oppose to the motion of the vibration
of the string (Errede, 2000). With this, the wave equation is modified to

2 y
t
2
T
2 y
x
2

y
.
t
Dividing both sides with T to obtain
 2 y
T t
2

2 y
x
2

 y
T t
.
The wave equation to model a damped string is then (Errede, 2000)
1 2 y
v 2 t 2
y  2 y


,
t x 2
(1.2)
where v 2  T  and the damping coefficient    T .
5
1.3
Vibrations of Stiff String
Other than adding the damping factor, stiffness is another point that can refine
the model in order to achieve a realistic mathematical description of the vibrations of
guitar string for this dissertation. According to Gunther (2012), the term stiffness is used
to characterize the force necessary to bend the string. A thorough study on the stiffness
of string had been made by Testa et al. (2003), as an important characteristic in a real
vibrating system. This study is carried out by comparing the vibration of stiff string and
stiff bar. To start off, the equation for the transversal vibration of rods is given by
2 y
t 2

EI  4 y
 x 4
,
featuring the Young modulus of the material E, the moment of inertia I with respect to
the transversal axis of the cross section of the rod (that is the y-axis if the rod length is
on the x-axis), and the linear mass density of the rod  (Testa et al., 2003). Note that
stiffness property made the governing equation to be a fourth order differential
equation. The difference between this equation and the standard form of wave equation
is actually the effects of tension force. In rods, the substance or matter to make up the
rod is not affected by any form of tension force, hence, it is negligible.
In reality, vibrating strings have a restoring force due to its tension T, saying that
the string is not flexible or perfectly elastic. Therefore, the modification done to the
original wave equation in (1.1) to include the stiffness property of the string, where T
must not be neglected, is

2 y
t 2
T
2 y
x 2
 EI
4 y
x 4
.
Dividing both sides with T to obtain
 2 y
T t 2

2 y
x 2

EI  4 y
.
T x 4
6
Finally, the wave equation to model a stiff string is
1 2 y
v 2 t 2

2 y
x 2

4 y
x 4
(1.3)
,
where v 2  T  , and stiffness parameter   EI T (Testa et al., 2003).
1.4
Vibrations of Damped Stiff String
The two types of string properties previously covered will bring some needed
insight to simulate and visualize the vibrations of realistic string. To even enhance the
model of vibrating guitar string, this study will propose to add up both the damping and
stiffness factors into the one-dimensional wave equation for this dissertation. The
addition can be done simply by adding the terms related to these aspects, one by one.
Moreover, the flexibility and strength of mathematical modelling enables the obtaining
good findings for validation, verification, and various extensive applications from the
modification applied. For this type of model, the damping coefficient  and stiffness
parameter  are added independently to (1.1), and the proposed governing
1 2 y
v 2 t 2
y  2 y
4 y


 4 .
t x 2
x
(1.4)
There is damped string piano string model presented by Chaigne and Askenfelt (1994a),
which contains a third order time dependant damping term, then later revised by Bensa
(2003) into a third order spatial-temporal dependent term. The later has been
implemented in study by Saitis (2008). The two models also include hammer
interaction, which does not exist in a guitar. Rather, both the terms to represent damping
and stiffness in the proposed differential equation here are based on Errede (2000) and
Pelc (2007) works as mentioned in the two previous sections.
7
1.5
Problem Statement
For an acoustic guitar, a mathematical modelling of an idealized string had been
done and became a popular example even in mathematics syllabus. Researchers carried
out many studies on mathematic modelling for a realistic string based on a piano and the
string also consists of stiffness and damping factor. In addition, modelling on the
vibration of string has been more popular using digital waveguides. The main interest
would be to observe the difference between stiffness and damping, and to add them into
a single model to make the model even more realistic and perform studies on the
solutions obtained with mathematical approach. The application of Mathematica to
perform simulation and visualization as well as the sound reproduction will be carried
out in this dissertation.
1.6
Objectives
This dissertation aims to study the mathematical model for vibration of ideal, damped,
stiff, and damped stiff guitar string. So, the objectives are:
(a) To better understand the phenomena and values involved in the vibration of a
guitar string in different environment.
(b) To design an interactive code that enables users to input values for plucking
height h, plucking position d, natural frequency of string f 1 , damping coefficient
γ, and stiffness parameter α by using Mathematica software.
(c) To simulate the modes of vibration of ideal and realistic string that consists of
damping and stiffness factor.
(d) To reproduce the sound of guitar with different model and create songs in
Mathematica.
8
It is hoped that this research may help in understanding the vibration of a guitar string
and enhance the sound synthesis field by mathematical approach in addition to the
common physics approach.
1.7
Layout of Dissertation
This dissertation is divided into six chapters. Chapter 1 provides the introduction
to the acoustic guitar, basic knowledge of wave equation for ideal string and various
characteristic for realistic strings includes damping and stiffness. This chapter also
provides the motivation behind, objectives and the layout of the dissertation.
In Chapter 2, some of the previous works relevant to this research are reviewed
and the methodology that will be applied in carrying out this dissertation is discussed.
Chapter 3 investigates the mathematical formulation for the vibration of acoustic
guitar string in ideal environment. The governing differential equation is presented; the
problem is solved by using the method of separation of variables, the boundary initial
conditions are imposed and finding the constant by Fourier series is applied. By using
Mathematica, the plots and simulation of the modes of vibration is made and sound is
produced.
In Chapter 4, the mathematical formulation for the vibration of acoustic guitar
string in damped environment is investigated. The governing differential equation is
presented. Then, the problem is solved by using the method of separation of variables,
the boundary initial conditions are imposed and finding the constant by Fourier series is
applied. By using Mathematica, the plots and simulation of the modes of vibration is
made and sound is produced.
For Chapter 5, the investigation is on the mathematical formulation for the
vibration of acoustic guitar string in stiff environment. The governing differential
9
equation is firstly presented, next, the problem is solved by using the method of
separation of variables, the boundary initial conditions are imposed and finding the
constant by Fourier series is applied. By using Mathematica, the plots and simulation of
the modes of vibration is made and sound is emitted.
Chapter 6 investigates the mathematical formulation for the vibration of acoustic
guitar string in both damped and stiff environment simultaneously. The governing
differential equation is presented first; then the problem is solved by using the method
of separation of variables, the boundary initial conditions are imposed and finding the
constant by Fourier series is applied. By using Mathematica, the plots and simulation of
the modes of vibration is made and sound is synthesized.
Next, Chapter 7 discusses the result from the three previous chapters. Discussion
is made on even some musical aspects to implore more understanding of the
observations made and obtained.
Lastly, Chapter 8 summarized the main result of the research problems
presented in all chapters. This chapter also suggests some possible extension of the
present work for future studies.
10
CHAPTER 2
LITERATURE REVIEW AND METHODOLOGY
2.1
Literature Review
Guitars, being among the most popular stringed instrument exist since the
beginning of history definitely attract many researchers, be it from mathematicians,
musicians, and physicians. There are a lot of researches have been done on many
aspects of guitars, in terms of the appearance, materials to build each part of it, sound
synthesis, energy distribution, and many more. To set a foothold in understanding the
behavior of the vibration of the guitar string, several investigations have been reported.
Hammond and Kelly (2011) mentioned it is true that mathematics and music are
deeply connected with each other since long ago. Mathematics is present in the natural
occurrence of the ratios and intervals found in music and modern tuning systems.
Therefore, a study of Mathematics and music background was carried out. Fourier
series representations to sound waves and how they relate to music in terms of
harmonics and tonal color of instruments were examined. The history of sound
synthesis and modelling is even further explored by Valimaki (2006) being very
specific on musical instruments. Specifically for guitars, Kentor and Michaels (2003),
Guy (2007) and Bloodworth (2013) had written an extensive history on evolution of
guitar, where Prinson (2008) touched on electric guitar, which is important for this
dissertation to know the borderline and main differences between acoustic and electric
guitar.
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Karjalainen (2004) stated physics-based modeling of sound sources, can be
carried out in many forms. An overview focusing on methods that are potential in realtime simulation and synthesis, in contrast to solving equations that characterize the
behavior of the sound source is well presented. Modeling paradigms discussed include
digital waveguides, finite difference time domain schemes, wave digital filters, modal
decomposition
techniques,
and
source-filter.
The
interrelations
between
these
approaches are discussed and cases of applying them for real-time sound synthesis are
presented. Smith (1992) mentioned music or sound synthesis based on a physical model
promises the highest quality when it comes to imitating natural instruments. Also, the
digital waveguide is said to follow a different path to the physical model where the
wave equation is solved to obtain traveling waves in the medium interior, then they are
simulated in the waveguide model, in contrast to computing physical variable. Laurson
et al. (2001) reported sound synthesis and physical modelling of stringed instruments
has been among the active field of research using digital waveguides. The study using
this method is very extensive that various assumptions, properties, characteristics, and
criterion have been tested and experimented. This is shown that even sophisticated
software had been developed in order to better enhance the control and manipulation of
data in the model to achieve better results. The advancement of modelling and sound
synthesis in their research is all based on the principle of commuted waveguide
synthesis to be implemented on string instrument model. Salo (2002) on the other hand,
researched on the finite difference method for vibrating string model sound synthesis,
with
discussing
stability
conditions
and
reviewing the needed
initial boundary
conditions for piano and guitar like strings. Seno (2002) underlined some of drawbacks
of digital waveguide which includes the simplicity, though inexpensive it deprived the
model to be introduced with more sophisticated excitations in a string, and second is
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