ギターのシミュレーションをやってみた

1. simulation object

The acoustic guitar is the more classical version of the guitar, a stringed instrument that transmits sound directly from the instrument. It has six standard strings and is played by fingerpicking or plucking with a pick. The vibrations of the strings and sound waves reach the ears of the player and audience through the resonance of the body of the instrument and the cavities inside the body [1].

Structure of a guitar Adapted from Reference [2]
<img src=“https://nature-architects.com/wp-content/uploads/2025/05/image-12-805x1024.png” alt=“” width=“402” height=“612” class="size- large wp-image-2455” />
Internal Structure of a Guitar Adapted from Reference [3]

For the 3D model used in the simulation, we used the following simple model from the Ansys tutorial page [4]. (The reinforcing beams that are in the backside of the board in the real one are not modeled.)

2. Organizing vibration transmission paths

In complex vibration phenomena, vibration often follows complicated transmission paths from the excitation source to the point where the vibration is observed, and it is very difficult to treat them all together as a single problem. Therefore, it is important to organize the vibration transmission path and divide the problem into units for each element on the path. In the case of an acoustic guitar, as shown below, there are two possible paths: the path where the plucked string is the excitation source and its vibration propagates to the air around the string and the path where it propagates to the body through the junction between the string and the body (bridge).

3. string vibration

First, we consider the properties of the excitation source, the string by itself.

3.1. Background knowledge (string theory in continuum mechanics)

According to the theory of strings in continuum mechanics, strings have the following eigenfrequency modes, and the vibration of a string during string repulsion is represented by the superposition of these eigenfrequency modes. (For a detailed derivation, see reference [5])
!
Here, the frequency of the nth-order mode (nth-order natural frequency) is expressed by the following equation.

\(f_n=\frac{n}{2L}\sqrt{\frac{T}{\rho}}\)

(L : length of string, T : tension, ρ : mass per unit length)

The important properties that can be found from this equation are as follows.

• All natural frequencies are integer multiples of the first-order natural frequency. The first natural frequency is called the fundamental frequency and determines the pitch of the sound. The sounds produced by the second and subsequent eigenfrequency modes are called overtones, and the timbre of the sound is determined by the ratio of the volume of each overtone.
• The natural frequency is proportional to the square root of tension. Therefore, increasing the tension raises the pitch of the note.
• The natural frequency is inversely proportional to the length of the string. Therefore, shortening the length of the string will raise the pitch.

3.2. Simulation

Let us now check the detailed behavior of the string by simulation. First, the eigenfrequency analysis is used to obtain the eigenfrequency modes. It is important to note that when we treat strings using the finite element method, we need to take into account the nonlinear phenomenon that the tension acting on the string causes a large deformation of the string, which in turn changes the stiffness of the string. Therefore, we first perform a nonlinear analysis with large deformation where the tension is applied to the string as a static load, and then perform a natural frequency analysis at the obtained stationary point. The analysis setup is as follows (The amount of forced displacement is manually tuned so that the first natural frequency matches the pitch of the first string (329.6 Hz, “mi” note))

The analysis results are shown below.

• List of natural frequencies (up to 9th order)

• Natural Vibration Modes (up to 3rd order)

<img src=“https://nature-architects.com/wp-content/uploads/2025/05/新規-2.gif” alt=“” width=“379” height=“159” class="size-full wp-image- 2459“ /><img src=”https://nature-architects.com/wp-content/uploads/2025/05/新規-3.gif“ alt=”“ width=”379“ height=”159“ class=”size-full wp- image-2459“ /><img src=”https://nature-architects.com/wp-content/uploads/2025/05/新規-4.gif“ alt=”“ width=”379“ height=”159“ class=”size- full wp-image-2459” />

The first-order natural frequency is close to the theoretical value of 329.5 Hz calculated from the obtained tension, and the second-order and subsequent natural frequencies are almost an integer multiple of the fundamental frequency, confirming that the mode shape is sinusoidal. (The natural frequency is slightly higher than the theoretical value because string theory does not take into account the shear stiffness of the string.)

Next, we will check the behavior of the string when it is repelled by a time history response analysis. The analysis setup is as follows. The repelling string is modeled by releasing the string from a state of forced displacement at the repelling position.

The analysis results are shown below.

• Deformation Shape (The vibration is occurring at a high speed, so the frame drops and the vibration looks like it is occurring slowly.)
  

• Time history waveform of force in Y direction applied to the right fixed end (bridge)
  

• Time waveform of string velocity at soundhole position
  

• Frequency amplitude characteristics obtained by Fourier transforming the above velocity waveforms
  

• The above velocity waveform written out as audio data

The amplitude characteristics confirm that the peak is at an integer multiple of the primary eigenfrequency, and a single pitch sound is produced aurally. Theoretically, no components other than integer multiples of the primary natural frequency are produced, but in practice, the response is produced over a frequency range with some width centered on integer multiples of the primary natural frequency due to transient changes caused by damping.

4 Body and air vibration

Next, we consider vibrations propagating through the body and air.

4.1. Background knowledge (transfer function)

Assuming a linear and time-invariant system that generates an output waveform from an input waveform, the output waveform when a short pulse is input to the system is called the impulse response. (In this example, the input waveform corresponds to the velocity of the string at the soundhole position or the force applied to the body through the bridge, and the output waveform corresponds to the pressure variation at the observation point.)

Input pulse and impulse response Adapted from reference [6].

However, linear here means that when multiple input waveforms enter the system, the effects of each can be considered separately, and that if the input is multiplied by a constant, the output is also multiplied by a constant. Then, for any input waveform, the following operation is possible: “decompose the input waveform into an addition of pulses ⇒ add the impulse responses to each pulse to obtain the output waveform.

Adapted from reference [7]

In other words, it is the same as knowing all the characteristics of the system if only the impulse response is known. This operation can be expressed in a mathematical expression as follows, which is a form of so-called convolution integral.

\(y(t)=\int^{\infty}_{-\infty} x(\tau)h(\tau-t)d\tau\)

(x : input waveform, y : output waveform, h : impulse response)

This convolution integral has an important property: by Fourier transforming both sides of the equality, we obtain the following equality.

\(Y(\omega)=X(\omega)H(\omega)\)

(X : frequency response of input waveform, Y : frequency response of output waveform, H : frequency response of impulse response = transfer function)

From this formula, it can be seen that the frequency response of the output waveform is expressed as the product of the frequency response of the input waveform and the transfer function, which is the frequency response of the system. It also means that the output waveform can be obtained by Fourier transforming the input waveform, calculating the product of the input waveform and the transfer function, and then inverting the result by Fourier transform. Therefore, it can be said that the transfer function, like the impulse response, represents all of the system's characteristics. The transfer function can be easily obtained by H=Y/X given an example input and its output. (X must not be zero.)

4.2. Simulation

Let us now check the vibration transfer characteristics of the body and air by simulation. Since the governing equation for the structure (body) is the equation of motion and the governing equation for the acoustics (air) is the Helmholtz equation, this is a multi-physics problem with structure-acoustics coupling. The first step is to analyze the vibration from the string to the body.

First, we will simulate the path of vibration propagation from the strings to the body. The analysis setup is as follows Since the finite element method cannot represent the infinite open space outside the body, only the area around the observation point is modeled out of the open space, and an absorbing layer is placed at the edge of the open space to prevent non-physical sound wave reflection. From the response at each excitation frequency, the transfer function is obtained.

The results of the analysis are shown below.

• Transfer function (amplitude)
  

• Impulse response
  

• Vibration at major response peak frequencies (body displacement, air pressure fluctuations)

264Hz
<img src=“https://nature-architects.com/wp-content/uploads/2025/05/新規-6.gif” alt=“” width=“506” height=“226” class="size-full wp-image- 2479“ /><img src=”https://nature-architects.com/wp-content/uploads/2025/05/新規-7.gif“ alt=”“ width=”506“ height=”226“ class=”size-full wp- image-2479” />

784Hz
<img src=“https://nature-architects.com/wp-content/uploads/2025/05/新規-8.gif” alt=“” width=“506” height=“226” class="size-full wp-image- 2479“ /><img src=”https://nature-architects.com/wp-content/uploads/2025/05/新規-9.gif“ alt=”“ width=”506“ height=”226“ class=”size-full wp- image-2479” />

It can be seen that both the body and the air vibrate with different vibration modes depending on the frequency, which causes the transfer function to form multiple resonance peaks. In addition, the main response is in the low frequency range where the body has a highly responsive vibration mode, and the response is smaller in the high frequency range because the body does not have a highly responsive mode.

Next, we simulate the path of vibration propagation from the strings to the air. Only the method of excitation is varied in the analysis setup, and the velocity is applied to the surface of an elongated cavity that mimics a string as follows.

The results of the analysis are as follows

• Transfer function (amplitude)
  

• Impulse response
  

• Vibration at major response peak frequencies (body displacement, air pressure fluctuations)

433Hz
<img src=“https://nature-architects.com/wp-content/uploads/2025/05/新規-10.gif” alt=“” width=“506” height=“226” class="size-full wp-image- 2479“ /><img src=”https://nature-architects.com/wp-content/uploads/2025/05/新規-11.gif“ alt=”“ width=”506“ height=”226“ class=”size-full wp -image-2479” />

The peaks due to resonance are smaller than with body excitation, and the overall tendency of the response to increase gently toward the high frequency range is more influential.

Finally, let us calculate the speech waveform at the observation point from these two transfer functions. We take as input the force waveform at the fixed end and the velocity waveform at the soundhole position, both obtained from the string simulation, and product each with the transfer function in the frequency domain, then back in the time domain, and finally sum the two pressure fluctuation waveforms. (In reality, the string motion is also affected by the air and the body, but we ignore that effect here.)

The results obtained were as follows

• Sound at the observation point
  

• Sound propagating from the strings to the body
  

• Sound propagating from the strings to the air
  

• Comparison of amplitude characteristics (body propagation vs. air propagation, string velocity waveform vs. pressure waveform at observation point)
  

Compared to the pure velocity waveform in section 3, the frequency characteristics are changed by resonance and colored as a guitar. In addition to the change in the amplitude ratio of the overtone, the characteristic feature is that the non-pitched components (components other than integer multiples of the primary natural frequency) generated slightly by transient changes due to decay resonate with the transmission path, creating a peak at a frequency other than integer multiples of the primary natural frequency, which adds a non-pitched strike-like nuance immediately after the plucked string. This gives the strings a non-pitched percussive nuance immediately after the plucked strings. The thickness of the sound in the low to middle register comes from the sound propagated from the strings to the body, while the sharpness in the upper register, which is particularly noticeable when the strings are plucked, comes from the sound propagated from the strings to the air. Of course, this transfer function varies greatly depending on the shape and material properties of the guitar, and it is the designer's job to create a transfer function for the guitar that is pleasing to the ear through shape design and material selection.

• Scalloped bracing (Yamaha, etc.)[8].

By controlling the gouge shape of the reinforcement on the back of the top material, the sound in the bass range is improved while maintaining the necessary strength. (Of course, the placement of the reinforcement is also designed to improve strength and acoustic characteristics.)

• Acoustic Resonance Enhancement Technology (Yamaha)[9].

This is a technology to change the physical properties of wood to the desired ones (“good sound” and strength as a musical instrument) by placing the wood in a cylindrical metal pressure vessel, controlling the internal temperature and humidity, and further processing the wood by changing the pressure in stages. Specifically, the wood is hardened in the grain direction, which is the direction of string tension, to ensure strength against string tension, and softened in the thickness direction, which is the direction of vibration when the strings are plucked, to improve the sound in the bass register.

• Ovation Guitar[10].

The back material is made of reinforced plastic with a curved surface structure, so that all the sound reverberated in the body is concentrated in the sound hole. The sound hole is also unique and impactful in terms of design, achieving uniform and balanced acoustic characteristics. The aggressive design creates a unique sound that is very interesting.

5. reproduction of playing technique (extra)

Although we have been able to output guitar sounds, we can still only play a single pitch, so let's try to reproduce the guitar's playing technique as well.

5.1. push strings

By pressing the string against the fingerboard with the fingers, we can constrain and change the effective length of the string, and by changing the natural frequency, we can make notes sound at various pitches.
Here, let's change the “mi” note on the first string to a “si” note by pressing the string. First, here is the definition of the most common musical scale, called the mean scale

• The frequency doubles as you go up an octave.
• There are 12 different notes within an octave (do, do#, re, re#, mi, fa, fa#, so, so#, la, la#, si)
• The frequency ratio between adjacent notes on the scale is always constant
• The frequency of the middle “la” note is defined as 440 Hz.

That is, the frequency of the note n above the middle “la” note is expressed as 440*2^(n/12)[Hz]. (same if n is negative)

Here, the frequency ratio is 2^(7/12)=1.498 times higher than the mi note because the si note is 7 notes higher. Therefore, from the string theory formula, it is sufficient to push the string at the point where the string length is 1/1.498=0.667 times the length of the string. In practice, the string length should be slightly longer to account for the increase in tension during pressing. According to reference [8], the string length should be 1/0.995=1.005 times the theoretical value.

In the analysis, XY displacement constraints are given to the point where the string is to be pushed. (The displacement in the X direction is 0, and in the Y direction the displacement is the distance from the string to the fingerboard)

The analysis results (deformed shape of the string and sound data) are shown below. The first-order natural frequency was 492.6 Hz, which is close to the pitch definition value of 493.9 Hz.

5.2. chords

By adjusting the natural frequencies of several strings by pressing them, and then simultaneously plucking them by swinging the hand down, chords can be played. This will allow you to play at a minimum.

Let's play a C major chord (do, mi, so) as an example. A major chord is a chord in which the frequency ratio of the component notes can be expressed in a neat integer ratio of approximately 4:5:6, and is perceived as a harmonic sound. (Conversely, the definition of the scale was determined so that chords made by regularly selecting notes from the scale like this would sound in a clean integer ratio.)

The rules for sounding this chord on the guitar are as follows. (Octave differences are treated as the same note.)

• Play any of the chord's component notes on all strings.
• Every note of the chord is played on at least one string.
• The bass note (Do for a C major chord) is sounded on the lowest string.

Also, the most common tunings for strings 1 through 6 are as follows (The numbers in the note names distinguish octave differences.)

Considering these factors, an example of how to push the strings would look like this. The pitches are, starting from the 1st string, mi4, do4, so3, mi3, do3. (the 6th string is not played)

In this model, there is no interaction between the strings, so if we calculate them one by one and add them together, they will be the same, but in reality, resonance will occur between the strings. (For example, the third-order eigenmode of Do 3 and the second-order eigenmode of So 3 resonate because their eigenfrequencies are almost the same.) Therefore, we approximate the resonance component by convolving the impulse response created by exponentially damping the waveforms of the other strings with the original waveform, and add it thinly to the original waveform.

<img src=“https://nature-architects.com/wp-content/uploads/2025/05/image-44-846x1024.png” alt=“” width=“423” height=“512” class="size- large wp-image-2509” />

In addition, the timing of the forced displacement is shifted slightly as the strings are plucked in order from the bass strings by swinging the hand down.

The results of the analysis (deformed shape of the string and sound at the observation point) are as follows.

This is too long to go into here, but the guitar still has many other varied techniques, many of which are interesting from a physical standpoint as well!

6. Summary

In this article, we have explained the concept of vibration/acoustic problems and the pronunciation mechanism of the guitar through guitar simulations. When listening to music, I think you will enjoy music more if you think about the physics behind it.

References.

[1]https://ja.wikipedia.org/wiki/%E3%82%A2%E3%82%B3%E3%83%BC%E3%82%B9%E3%83%86%E3%82%A3%E3%83%83%E3%82%AF%E3%83%BB%E3%82%AE%E3%82% BF%E3%83%BC

[2]https://acousticguitarmagazine.jp/beginners/2024-0510-parts-name/

[3][https://ja.namu.wiki/w/클래식 기타/재료 및 구조](https://ja.namu.wiki/w/%ED%81%B4%EB%9E%98%EC%8B%9D%20%EA%B8%B0%ED%83%80/%EC%9E%AC%EB%A3%8C %20%EB%B0%8F%20%EA%B5%AC%EC%A1%B0)

[4]https://innovationspace.ansys.com/courses/courses/large-deformation/lessons/homework-quizzes-simulation-examples-large- deformation-in-structural-mechanics-lesson-5/

[5]https://ds.cc.yamaguchi-u.ac.jp/~tsaito/D02/DynamicsII11.html

[6]https://ja.wikipedia.org/wiki/%E3%82%A4%E3%83%B3%E3%83%91%E3%83%AB%E3%82%B9%E5%BF%9C%E7%AD%94

[7]https://www.noe.co.jp/technology/18/18inv1.html

[8]https://jp.yamaha.com/sp/myujin/11120.html

[9]https://www.yamaha.com/ja/tech-design/research/technologies/are/

[10]https://nihon-meisho.com/144/

[11]http://www.crane.gr.jp/MakingGuitarZero/CRANE_027_01flet_formalize.html