**sitar**meaning on the internet then this is the right place where you came. In this article we have briefly described about

**what is sitar**?,

**sitar instrument**and

**sitar history**.

**Sitar music**and

**sitar sounds**is very beautiful to listen when you play

**sitar instrument.**I will guarantee you, In this article you will get awesome information about

**sitar instrument.**Infact, we've read many books on the same topic and then, we've made this content. There are two famous historical instruments of India,

**Veena and Sitar**, and what is the difference between

**Veena vs Sitar**, you can read about this topic on the Internet.

**Sitar price ** is very costly but if you have that type of budget then you can easily afford it. In this article we will also cover the topic of **sitar strings ****. **Some peoples also search about **sitar indian **on the internet. I am an Indian and I have an experience about **sitar instrument **for 5 years. And I know how to make a sitar and what improvements have been made in it from time to time. If you are not believing me, then read this article at once. In the table of contents click on a particular topic and, you can directly access that topic.

**Contents**

**1. Introduction - Sitar**

The sitar is an ancient Indian string instrument that features heavily in Indian classical music. This is usually accompanied by a Tambura, a similar drone-type instrument that is used to set the tonic of the piece being performed.

The origin of the sitar can be dated back as far as the Middle Ages and is usually found in the northern part of India. It does not feature at all in southern Indian classical music.

The sitar became popular in the western world through the music of Pandit Ravi Shankar during the 1950s and George Harrison of the Beatles in the 1960s (Park 2008).

It is known for its unique timbral quality, which is attributed to its sympathetic strings, the construction of its bridge, long hollow neck, and its resonating chamber.

It is usually played by balancing the instrument between the player's left foot and right knee. This position then allows the player's hands to move freely around the instrument's neck without supporting its weight.

**2. Mechanics of Sitar**

The sitar has a very unique and distinguishable body. On the neck of the instrument, all the frets are moveable, allowing for fine-tuning and the use of microtones.

There are generally around 14 of those depending on the type of sitar. They are also suspended off the neck, allowing the sympathetic strings to run underneath and resonate freely.

Usually, the sitar has about 21 strings, most of these being sympathetic. These sympathetic strings are also known as tarb. These strings are never really ever touched as they are just meant to vibrate sympathetically.

Although, sometimes, you may hear a player strum all of these at once for effect. Along with the sympathetic strings, you have the main six or seven strings. Three of these, called chikaari, provide the drone while the rest are used to play the melody (Courtney 2010).

The most important parts of the sitar are the two bridges. There is the large bridge called the badaa goraa for holding the drone and melody strings in place, and then there is the smaller bridge for the sympathetic strings called the chota goraa.

These bridges are collectively known as jawari and are generally made of camel bone. The shape of the jawari are like slopes, and it is the way the string interacts with these slopes when plucked that give the sitar its particular timbre.

Initially, when the sitar string is plucked is, there are a shortening and lengthening of the string relative that is relative to the slope, which leads to the string generating overtones.

This particular process is explained in better detail further on in this dissertation.

The resonator of the sitar is called the Kadu. These are very delicate and usually are just made of a gourd.

On some sitars, there are two resonators, the other one being at the top of the neck. The gourds may also sometimes have strings inside of them that are there to resonate sympathetically.

**3. Bridge Structure**

As mentioned before, it is the sitars sloped bridge construction and its relationship to the strings that give it its specific buzzing sound. What happens is explicitly a type of nonlinear distortion occurs when the string is plucked due to the interaction between the camel bone and the string.

This nonlinear distortion gives rise to additional overtones, somewhat similar to what happens when amplitude clipping occurs. Due to the square wave-like properties forced upon by the clipping, it creates odd harmonics that are not present in the original signal (Park 2008).

The previous section mentioned that a shortening and lengthening of the string relative to the jawari occurs. This also lends to the buzzing timbre and affects the string's pitch ever so slightly since its length is changing.

Concerning the actual model, this requires that there be a dynamically evolving delay line.

How this was implemented is explained further in the documentation.

Further to how the nonlinear distortion occurs due to the friction between the sting and jawari, the transverse waves traveling along the string interact with the jawari just before they reach the point of termination.

Generally, in a stringed instrument with a typical style bridge such as that of a western classical guitar, this termination point is where the waves usually flip over and travel in the opposite direction. However, in the sitar, before this happens, the larger amplitude transverse waves in the string interact with the jawari earlier than the smaller ones, altering the shape of the strings and causing it to bulge.

The transverse waves are not terminated but interact with the jawari, unlike the smaller ones, which mainly reflect. This dramatically increases the higher partial content at large wave amplitudes but not at smaller wave amplitudes.

This interaction between the string and jawari reduces gain substantially, as the energy is transferred to the louder, higher partials. The imprecise termination point of the sitar is akin to the fretless electric bass.

**4. Sitar Physical Model**

The wave equation is a fundamental second-order partial differential equation that describes the propagation of waves with speed v. Initially, the idea was to use the Karplus-Strong algorithm for the main and sympathetic string synthesis because of its low computational costs.

Still, on further research, investigation, and testing, it was decided to use the bi-directional digital waveguide approach instead. The Karplus-Strong algorithm was reserved for another modeling approach.

The bi-directional digital waveguide approach is a much more realistic model of how a one-dimensional string vibrates as it takes into account two acoustic waves are traveling in opposite directions. It is known that the vibration of an ideal string can be described as the sum of two traveling waves going in the opposite

directions (D’Alembert 1747).

Another modeling approach that is unique to this particular model concerns the non-linear bridge structure or jawari as it is officially called. The jawari, because of its design, requires that there be a dynamically changing delay line. The amount of delay length modulation that occurs in this delay line is all relative to how much energy is in the plucked string.

The length of the string changes more rapidly at the attack portion of the signal, gradually becoming less random and settling into a more periodic pattern as the energy dissipates through the termini. This particular problem of non-linearity was solved using the Karplus-Strong algorithm and a feedback loop from the main sitar string itself.

This technique is explained with more clarity further on in this document.

This model also makes use of fractional delay filtering. Fractional delay filtering is a modeling technique that allows for the accurate cancellation and dampening of musical tones (Lehtonen et al. 2008).

Usually, delay lines in these particular types of models could only be of an integer sample length, causing the physically modeled instrument to be slightly out of tune, but by using fractional delay filtering, this can be avoided.

The other modeling techniques used in this particular model are all techniques that have been used for the modeling of the western classical guitar, but they have been adapted to the sitar. These techniques include the sympathetic resonance of strings, comb filtering, all-pass filtering, and body resonance filtering.

**5. A Brief History About Sitar**

The first use of physically-based models to synthesize sound was by John Kelly and Carol Lochbaum (Kelly and Lochbaum 1962). They implemented a simplified model of the human vocal tract as a one-dimensional acoustic tube of varying cross-sections.

The most widely heard example of physical modeling for many years is Stanley Kubrick's 2001: A Space Odyssey. Most of the early work on physical modeling of musical instruments was focused on vibrating strings.

This was due to them being computationally efficient to calculate. It was Pierre Ruiz in 1970 that was the first person to synthesize a musical instrument using a physical model.

It was then Ruiz, and Lejaren Hiller discovered that the quality of a vibrating string sound was mainly defined by how the string loses energy (Hiller and Ruiz 1971a, 1971b). Similar to Ruiz and Lejaren Hiller's publication by McIntyre and Woodhouse, some approaches would describe theoretical results to a realistically lossy vibrating string equation (McIntyre and Woodhouse 1979).

These techniques were then to be followed by the Karplus-Strong algorithm (Karplus and Strong 1983). The Karplus-Strong algorithm was discovered as an effortless computational technique that arose from work being conducted on wavetable synthesis.

It works by feeding a burst of white noise into a feedback loop of length L samples. On each loop, the white noise is filtered over and over again by a simple averaging filter. The frequency-dependent decay of the white noise created for the first time on a computer sounded very string-like.

What made this algorithm so successful was that the realistic string timbres produced with great ease were very computationally efficient. This was very relevant at the time as modern-day standards would have limited processing power.

Seemingly this technique had nothing to do with physics, and it wasn't until David Jaffe and Julius O. Smith did further work with it and showed a clearer understanding of it about the physics of a plucked string (Smith 1983; Jaffe and Smith 1983).

After this, Julius Smith introduced the theory of digital waveguides and generalized the underlying ideas of the Karplus-Strong algorithm (Smith 1987). Karjalainen says that digital waveguides are physically relevant abstractions yet computationally efficient

models, not only for plucked strings but also for various one-, two-, and three-dimensional acoustic systems (Karjalainen et al. 1998).

These digital waveguides proved to be an efficient model of many linear and physical systems such as strings and acoustic tubes. One of the advantages of these waveguides over analytical methods was introducing non-linearity into models, just like those that would have to be considered when modeling the sitar (Smith 1987).

This enabled researchers to produce a variety of different realistic instrumental sounds. To this day, digital waveguides are still a crucial modern research topic concerning the field of physical modeling.

They are still used extensively in many commercial synthesis systems, whether it is hardware or software.

The first commercially available systems to include digital waveguides were at the beginning of the 1990s. These were Bontempi-Farfisas MARS

in 1992 and then this was followed by Yamahas VL1 in 1993

**6. A Taxonomy**

Below the image is a taxonomy of the different types of physical modeling synthesis techniques used. Only DWGs (Digital Waveguides) will be covered in this

dissertation. This figure has been given so the reader can see where this technique is derived from and how it relates to other methods.

**7. Simple Physical Model of Sitar**

### 7.1. What is a Model ?

Model-Building is a fundamental human activity. For our purposes, the model can be defined as all forms of calculations that predict the behavior of physical objects or phenomena based on the initial state and the power of "input."

Our first successful model occurred in our head. It is effectively constructing a simplified abstract view of what usually may be a very complex system.

Gaining an understanding of a complex natural system such as a musical instrument is generally accomplished by combining or building upon more straightforward and more basic models.

I say we were to look at a guitar. The guitar is comprised of many mechanical parts such as the strings, resonator, and bridge. Each of these parts is the building block to the overall complex model.

For virtual musical instruments and audio effects, the model replaces the real thing, allowing us to understand how it works (Smith 2010).

### 7.2. Basic Vibrating String Model

The basic model of a vibrating string is based on Newtonian principles. The vibrations in the string are transverse waves or, in this case, transverse acoustic waves.

To derive the equations governing small transverse vibrations of an elastic string, which is stretched to length L, you have to make simplifying assumptions so that the resulting equation does not become too complex.

First of all, place the string along the x-axis, stretch it to length L, and fix it at the ends x = 0 and x = L. The string is then distorted instantly, say t = 0; it is then released and allowed to vibrate.

The problem is to determine the vibrations of the string, that is, to find its deflection at u(x; t) at any point x and where t > 0. To do this, the following can be assumed.

- The mass of the string per unit length is a constant. The string is perfectly elastic and doesn't offer any resistance to bending.
- The tension caused by stretching the string before fixing it at the endpoints is so significant that the action of the gravitational force on the string can be neglected.
- The motion of the string is a slight transverse vibration in a vertical plane; that is, each particle of the string moves strictly vertically.

T is the string tension in the string and p is the string's linear mass density.

The derivation of this equation is beyond the scope of this dissertation but can be found in any applied mathematics textbook (Kreyzieg 1999). This PDE is the starting point for both digital waveguide models and finite difference schemes.

### 7.3. Mass-Spring System

This section discusses the principles behind a basic mechanical system's motions, a mass on an elastic spring. The string of a musical instrument is a mass-spring system. If you are to take an ordinary spring and suspend it vertically from support and then at the other end attach a body of mass m.

Here you are to assume that m is so large you can disregard the mass of the spring. Then pull the body down a certain distance and

then release it. You will notice that it undergoes a motion. This motion is governed by Newton's second law:

Mass x Acceleration = my^{n}= Force

Where *Force* is the resultant of all the forces acting on the body. Here*, **y*^{n}*= d ^{2}y / dt^{2}*

**where**

*,*

*y*

*(*

*t***is the displacement of the body and**

*)*

*t**is time. At first the string is unstretched, but then when the body is attached, the body stretches the string by an amount*

*s***.**

*0*This causes and upward force F0 in the spring. It has been experimentally shown that this restoring force F0 is relative to stretch, say,

*F**0 **= **-**ks**0*

This is known as Hookes law.* *** k** is called the spring constant. Where the larger the value for

**, the more stiff the spring is, hence giving a smaller**

*k*

*s***.**

*0*

*s***being the amount of displacement. The extension of**

*0*

*s***is such that**

*0*

*F***balances the weight**

*0*

*W*

*=*

*mg***Consequently**

*.*

*F*

*0*

*+*

*W*

*=*

*-*

*ks*

*0*

*+*

*mg*

*= 0*

*.*** These forces do not affect the motion. The entire system is at rest, this is what is called the static equilibrium of the system. The position of the body at the static equilibrium position is ***y *** = 0**. We measure the displacement of the body from the static equilibrium position as

*y*

*(*

*t*

*)*

*.***The main point is that ***F*** 0** is the restoring force. It has the tendency to restore the system back to its static equilibrium position

*y*

*= 0***With this understanding of a how a mass-spring system works, it brings us on to damped and undamped mass-spring systems.**

*.*Every system has damping otherwise it would just keep moving forever. It would be like if a string was plucked and it kept vibrating forever. Although, to explain the next point we are going to look at an undamped system first.

Let’s take for an example an iron weight on the end of a spring. In this situation F1 is

the only force *Mass x Acceleration = my***^{n}= Force** causing the motion. Hence, making

*my*^{n }*= **-*** ky** from

*Mass x Acceleration = my***. This means that the model for the mass-spring system without damping becomes:**

^{n}= Force*my*^{''}* **+ **ky **= 0*

By finding the complex roots of this equation we get the general solution

The corresponding motion to this equation is called a harmonic oscillation. These harmonic oscillations are similar to the waves that occur when a string is plucked. When

the string is plucked or in this case when the iron weight is displaced the spring makes these harmonic oscillations. By applying the addition formula for cos, this equation can be written as

*y**(**t**) = **C **cos(**w*_{0}*t **- **g**)*

And, since the period of the trigonometric function *y**(**t**) = **C **cos(**w*_{0}*t **- **g*** ) ** is

**the body executes at**

*2 x 22/7 (PI) /W*_{0 }**cycles per second. This quantity is called the frequency of the oscillation and is**

*W*_{0 }/2 x 22/7 (PI)**measured in Hertz.**

In the case where the system has been damped which is more likely to be the situation. We connect the mass to a dashpot to demonstrate its properties. By looking at the equation governing the system we can derive three different cases. The damped system equation being

*m''*^{}* **+ **cy*^{’}* **+ **ky **= 0*

* *

Where *-**cy*^{’ }*= **F*** 2**, this being the force imposed by the dashpot. The three different cases are, overdamping, critical damping and underdamping. It is the roots of equation

*my*

^{''}

*+*

*cy*

^{’}

*+*

*ky***that determine this.**

*= 0** *

**Case 1:** In the overdamping case the body does not oscillate since the damping takes the energy from the the system and there is no external force that keeps the motion going. The equation *my*^{n}* **+ **cy*^{’}* **+ **ky *** = 0 **has distinct real roots _1, _2 in this case

**Case 2:** The critical case marks the border between the non-oscillatory motions and oscillations; this explains its name ”critical case”. It has to do with the fact equation *my*^{''}* **+ **cy*^{’}* **+ **ky *** = 0 **has a real double root.

** **

**Case 3:** Underdamping is the most interesting case. Underdamping occurs when the roots of the equation are complex conjugate roots.

Underdamping would be similar to the case in most strings on an instrument. When the string is initially plucked it settles into a periodic behaviour corresponding to a harmonic oscillation. These three cases are illustrated in

There is a particular modeling technique based solely on this mass-spring paradigm, as mentioned before (Hiller and Ruiz 1979). As can be seen, it requires a detailed description of all the physical characteristics of the vibrating objects.

Furthermore, it requires that you stipulate the boundary conditions for the PDE of the one-dimensional wave equation. It also requires the physical description of the excitation mechanism. The difference equations presented earlier are the equations that are then used to compute the resulting sound output (Bianchini and Cipriani 2008).

### 7.4. D’Alemberts Solution of the Wave Equation

With D’Alemberts travelling wave solution it can be shown that the vibration of an ideal string can be described as the sum of two travelling waves going in opposite directions using the wave equation. We will start with the wave equation

If we are to denote the right travelling waves and the left travelling waves by the following equations:

*v = x + ct, z = x - ct*

Then u becomes a function of v and z. The derivates of the wave equation in

can now be expressed in terms of the derivatives with respect to v and z by the use of the chain rule. This becomes

*u*_{x }*= **u*_{v}*v*_{x}* **+ **u*_{z}*z*_{x }*= **u*_{v }*+ **u*_{z}

We now apply the chain rule to the right side of the equation giving us

*u _{xx} = (u_{v} + u_{z})_{x} = (u_{v }+ u_{z})_{v}v_{x} + (u_{v} + u_{z})_{z}z_{x} = u_{vv }+ 2u_{vz }+ u_{zz}*

Now we transform the other derivative in

giving

*u _{tt }= c^{2}(u_{vv}2u_{vz} + u_{zz})*

By inserting the two results into

we get

This resulting equation can now be solved by two successive integrations with respect to *z.*

where* h(v)* is an arbitrary function of v. Integrating this with respect to* v* gives

where u(z) is an arbitrary function of z. Since the integral is a function of v, say, 0(v), the solution u is of the form u = 0(v) + u(z). Then because of *my*^{''}* **+ **ky **= 0 *We get

This is known as D’Alemberts solution of the wave equation. The traveling-wave solution of the wave equation was first published by d’Alembert in 1747 (D’Alembert 1747)(Kreyszieg 1999). The bi-directional digital waveguide is based on this very principle and will be discussed further on the dissertation.

### 7.5. Sampled Traveling-Wave Solution

In order to use the traveling wave solution in the ”digital domain” it is neccesary that you sample the traveling-wave amplitudes at intervals of T seconds. The continuous traveling-wave solution to the wave equation given in (3.16) can be sampled to give

where *x **= *** cT **denotes the spatial sampling interval in meters,

*T***denotes the time sampling interval in seconds, and**

*y***and**

^{+}

*y***are defined for notational convenience (Smith 2010) .**

^{-}## 8. Digital Waveguides

Here the Karplus-Strong algorithm and the extended version of it will be explained in detail. This chapter will also introduce the bi-directional digital waveguide; this is the modeling technique that is central to the modeling of the sitar strings.

### 8.1 Karplus-Strong algorithm

The Karplus-Strong algorithm was discovered by two men around 1980. Their names being Alan Karplus and Kevin Strong. The paper on this algorithm was published in1983.

It was Alex Strong in December of 1978 that conceived its most straightforward modification and called it the Plucked-String algorithm. How it works is by merely averaging two successive samples (Karplus and Strong 1983). This can be written mathematically as

It was discovered that this averaging process produced a slow decay of whatever waveform was being computed by it. This algorithm made a pitch of period p+12 samples and sounded similar to the decline of a plucked string.

What was so remarkable about this algorithm was that there was no multiplication required. Making it exceptionally computationally efficient.

Back then, they did not have anywhere near the same microprocessing power that we have nowadays, so this would have been fast and easy to implement considering the limitations (Karplus and Strong 1983).

Strong says the naturalness of the sound derives largely from differing decay rates for the various harmonics. No matter what initial spectrum a tone has, it decays to an almost pure sine wave, eventually decaying to a constant value (silence) (Karplus and Strong 1983).

The actual excitation of the algorithm requires that a noise burst be fed into the system. How Strong initially did this was by feeding the algorithm with a wavetable filled with random values.

Every time, the use of a different random wavetable had the advantage of giving each repetition of the same pitch a slightly different harmonic structure.

This gave each note its character, sort of like a real instrument. Usually, what would be used to excite the system would be a burst of pink or white noise (Karplus and Strong 1983).

Once the noise burst is fed into the system, it is immediately output and then fed back into a delay line of L samples long. The result of this delay line is then fed into the averaging filter as described already.

This is usually a first-order low pass filter. Also, the filter gain must always be less than one, or else the signal will never decay and could make the system unstable.

The output of the averaging filter is then output and at the same time sent back into the delay line. This process keeps repeating until the signal is averaged out to silence (Karplus and Strong 1983).

The length L in samples of the delay line determines the fundamental pitch of the note being played. L is determined by the equation L = Fs=F1, where Fs is the sampling frequency. The overall effect of the algorithm is quite realistic and very similar to a plucked string sound considering it is such a simplistic procedure.

It may not have a natural-sounding guitar string tone, but various extensions can be applied to help this, which will be discussed next. Alan Karplus conceived a simple variation of the algorithm for drum timbres. Since we are only interested in strings, this will not be discussed.

### 8.2. Karplus-Strong Extended

Around the same time that the paper about the original Karplus-Strong algorithm was published, David A. Jaffe and Julius O. Smith published an article about various extensions to the original algorithm.

The need to implement these extensions came from the musical needs that arose out of the composition of May All Your Children Be Acrobats (1981) and Silicon Valley Breakdown (1982), both by David Jaffe (Jaffe and

Smith, 1983).

One of the first modifications made was about the tuning. The fact that the delay line length L had to be an integer caused tuning problems. The tuning problems occurred at high frequencies.

The fundamental frequency f1 = fs / (N+1/2 ) meant that the pitches were rounded off. This was barely noticeable for low pitches (large N), but as the pitch increased, it becomes more and more off-sounding (Jaffe and Smith, 1983).

The solution to this problem was fractional delay filtering. It can be shown experimentally that by using a fractional delay filter, there is a more accurate cancellation and dampening of musical tone partials (Lehtonen et al. 2008).

What was needed was introducing a filter into the feedback loop, which would delay the signal slightly without altering the loop gain. The filter that was raised was an all-pass filter.

It ensured there was no change to the growth of the signal. The equation for this filter and its transfer function is as follows.

The only thing that the all-pass filter affected was the phase of the signal (Jaffe and

Smith, 1983).

Another problem with the algorithm was decay-time; the difference between the decay times for a low pitch and a high pitch was drastically different. The ability to control decay time is very important if you want to have a realistic realization of a plucked string.

Consequently, Jaffe and Smith found methods that could be used to control decay time. One of the ways was to introduce a loss factor p. Where equation

Where |p| <= 1 if the string is to be stable. Essentially what decay shortening does is

produce a damped version of the Karplus-Strong algorithm. Where low-pitched notes are comparable to low notes on real strings.

Another technique that was employed was decay stretching. This was done by changing the feedback average ** H_{a }**to a two-point weighted average. This reduces the amount of energy loss at high frequencies.

For the greatest control it is said both the uniform loss method and two-point-averaging method should be used together (Jaffe and Smith, 1983).

Dynamics was another issue that was dealt with.

Where the output of the system was directly related to the noise burst being input into the system. What enabled this to work was, since the strings that were plucked hard had more energy in the higher partials than the strings plucked lightly, a one-pole low pass filter could be used to attenuate these higher partials before they were fed into the system.

This allowed the user to be able to set if the string was to sound muted when it was plucked or alternatively sound like an open string. All that the user had to do was adjust the cut-off point of the one-pole low pass filter and you could get varying excitation timbres (Jaffe and Smith, 1983).

Some of the other extensions had to do with pick position and pick a direction. Pick position involved implementing a comb filter just after the noise burst.

Depending on the comb filter's delay length, you can choose the string at various positions, allowing you to suppress specific harmonics.

Pick direction can then also be controlled by lowpass filtering the noise burst before it is fed into the delay line or using a rich harmonic spectrum instead of a noise burst. Another way to affect the noise burst is to change the duration of the noise burst.

To model sympathetic string vibration, Jaffe and Smith sent a small percentage of the string output from a plucked string to another string that had been tuned to a different pitch.

Since the sympathetic string was tuned to a different pitch, all the partials of the plucked string that did not coincide with the sympathetic string would have been attenuated (Jaffe and Smith, 1983).

There will be a further discussion about sympathetic strings in this dissertation, as it is central to the sitar model. It can be seen here that these extensions it can make the fundamental algorithm much more expressive and realistic sounding.

Usually, although very similar to a plucked string, the Karplus-Strong algorithm does have a very artificial sound.

### 8.3. Bi-directional Digital Waveguides

A bi-directional digital waveguide is essentially a bi-directional delay line at some wave impedance. This is also considered a lossless digital waveguide. Wave impedance is the ratio between the force of a wave to the velocity of a wave.

For linear time-invariant systems, impedance may vary with angular frequency (** w**). The bi-directional waveguide works because each delay line contains a sampled acoustic traveling wave (Smith 2010).

Since it is a bi-directional waveguide, this means that there is a sampled acoustic wave traveling from left to right and right to left in each of the delay lines.

In this model, d’Alemberts are traveling wave solutions whereby it can be shown that the vibration of an ideal string can be described as the sum of two traveling waves going in the opposite direction (d’Alembert 1747).

The type of bi-directional digital waveguide we will be dealing with in this dissertation is rigid terminations. If we terminate a length L ideal string at x = 0 and x = L, we then have the boundary conditions

*y**(**t, **0) = 0 **y**(**t, L**) = 0*

How this system works is, the excitation is fed into the system at an arbitrary point given in the image below. The acoustic traveling waves proceed to travel around the bi-directional waveguide being delayed by ** N/2** samples by the delay lines. It can be seen in the diagram