Fundamentals of Guitar Modes

Basic Modal Concepts

Modal Definition: Modal Analysis (Modal Analysis) is a theory and method for studying the inherent vibration characteristics of structures. It mainly includes parameters such as modal frequency, modal shape, damping ratio, modal mass, modal stiffness, and modal participation factor. For linear systems, each mode can be understood as a spatial vibration form that is most likely to occur near a certain natural frequency of the structure.

When the structure is excited near a certain modal frequency, if the excitation position and direction match the mode shape, the mode will be effectively excited. The position in the mode shape where the displacement is close to zero or the phase is reversed is called a nodal line or a node region; the region with a large displacement is called an anti-node region. The vibration of the guitar top, back, cavity air and bridge system is not a simple movement of a single frequency, but the result of the coupling of multiple structural modes and air modes. The modal frequency is mainly determined by the mass distribution, stiffness distribution, boundary conditions and coupling relationships of the system.

Equations of motion and eigenvalue issues: In a linear elastic system, undamped free vibration can be described by the following differential equation of motion (for linear, lightly damped systems, any response can be expanded into a linear superposition of modes).

Let the generalized displacement of the system be q(r,t), then

q(\mathbf{r},t)=\sum_k \phi_k(\mathbf{r})\,\eta_k(t),\qquad \eta_k(t)=A_k\cos(\omega_k t+\varphi_k)e^{-\zeta_k\omega_k t}

Differential equation of motion of a linear elastic system (undamped free vibration):

[M]\{\ddot{x}\}+[K]\{x\}=\{0\}

\begin{bmatrix} m_{11}&m_{12}&\cdots&m_{1n}\\ m_{21}&m_{22}&\cdots&m_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ m_{n1}&m_{n2}&\cdots&m_{nn} \end{bmatrix} \begin{Bmatrix}\ddot{x}_1\\\ddot{x}_2\\\vdots\\\ddot{x}_n\end{Bmatrix} + \begin{bmatrix} k_{11}&k_{12}&\cdots&k_{1n}\\ k_{21}&k_{22}&\cdots&k_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ k_{n1}&k_{n2}&\cdots&k_{nn} \end{bmatrix} \begin{Bmatrix}x_1\\x_2\\\vdots\\x_n\end{Bmatrix} = \begin{Bmatrix}0\\0\\\vdots\\0\end{Bmatrix}

Modal participation factor and resonance: When the frequency of external excitation coincides with a certain modal frequency, the structure will undergo resonance in this mode, resulting in a significant vibration response. However, the strength of the resonance response not only depends on the degree of frequency agreement, but also depends on the ” coupling efficiency” of the excitation to the mode. This is measured by the Modal Participation Factor: For an excitation force or displacement input in a given direction, the Modal Participation Factor is a scalar quantity indicating the degree of participation of that mode in the overall response of the structure. If the participation factor of a certain mode in this direction is large, when the excitation frequency is close to its modal frequency, the mode will be strongly excited to produce a significant response; the modal participation factor is used to describe the contribution of a certain mode to the response under a specific excitation position, excitation direction and measured quantity. Even if the excitation frequency is close to a modal frequency, the mode may be weak in the measured response if the excitation point is located near the mode node or if the excitation direction does not match the main direction of motion of the mode. On the contrary, if the excitation position and direction are highly matched to the mode shape, an obvious response may occur even if the input energy is not large. Therefore, the actual dynamic response not only depends on whether the frequency is close to the natural frequency, but also depends on the spatial matching relationship between the excitation and the mode shape.

Acoustic guitar structure and main resonance components

Acoustic guitar (including classical guitar and steel-string acoustic guitar) is a complex acoustic structure, which is composed of strings , bridge , soundboard (panel) , back plate , side panels (side walls of the body) , braces , soundhole and the air in the cavity. Each part plays a different role in the modal vibration of the guitar:

Soundboard (panel) : The soundboard is the main structural sound radiation component of the guitar. The strings input force and speed into the soundboard through the bridge, causing the soundboard to vibrate, and then the sound energy is radiated outward by the air movement on the surface of the soundboard and near the sound hole. The low-order main mode of the soundboard usually manifests as large-area in-phase agitation, which has certain “approximate piston” characteristics, so the radiation efficiency is high; but the real soundboard is not a rigid piston, but a complex anisotropic thin plate with bending, torsion, local vibration and bracing constraints. Soundboard thickness, camber, longitudinal and transverse elastic modulus of wood, density, damping, bracing layout and bridge mass will significantly affect its modal frequency and vibration shape.
Back : The back is the wooden board on the back of the guitar, generally made of hardwood (such as rosewood, etc.), and is usually slightly thicker than the soundboard. The back also has its own bending modes and can be coupled to the soundboard through the structural paths of the side panels and cavity air pressure. When discussing the phase of the panel and the back panel, it is necessary to first define the direction: If the outward normal direction of each is positive, then the panel and the back panel will “bulge outward at the same time” and the volume of the guitar cavity will increase, and “retract inward at the same time” will reduce the volume of the guitar cavity. This movement will strongly couple the cavity air; if the direction of the same spatial coordinate is positive, the phase description will be opposite. Therefore, when analyzing the coupling between the backplane and the panel, you cannot just say “in phase” or “anti-phase”, you must also specify the reference direction. Backplate resonance is not usually solely responsible for the sound of a guitar, but it can affect the peaking, width, and decay of the low- and mid-bass response. Moderate back panel participation can increase the fullness and airiness of the sound; if a certain back panel mode is excessively concentrated with the panel and air modes, it may cause some pitches to be abnormally prominent or attenuated.
Side Panel : The side panel is the side wall that connects the soundboard and back panel to form the body cavity. It is generally made of hardwood and bent into the shape of the body. The side plate itself is relatively narrow and has high stiffness. It is usually regarded as an approximate rigid boundary in low-order modes, limiting and guiding the vibration boundary conditions of the soundboard and back plate. The main function of the side plates is to form the volume and shape of the resonance box, which affects the air mode frequency (for example, the volume of the guitar cavity affects the Helmholtz resonance frequency). The side panels themselves also have high-frequency bending modes, but due to their arc-shaped structure and close connection with the soundboard/back panel, these modes have a relatively small impact on the overall sound radiation. They generally appear as local high-frequency vibrations and are not the main source of sound radiation.
Braces : Braces are slender wooden strips adhered to the inside of the soundboard and back plate to strengthen the structure and adjust vibration patterns.Different types of guitars use different bracing layouts (for example, classical guitars mostly use fan-shaped bracing, steel-string guitars mostly use X bracing, etc.). Bracing significantly changes the soundboard’s local stiffness, mass distribution, and vibration energy flow paths, thereby affecting modal frequencies, nodal line locations, and mode zoning. In some modes, the node line may be close to the bracing or guided by the direction of the bracing, but the position of the bracing is not necessarily the node line. The more precise function of the bracing is to change the equivalent bending stiffness, degree of anisotropy and local coupling relationship of the panel, thereby indirectly regulating the modal frequency, mode shape and radiation efficiency. For example, X bracing causes the soundboard to vibrate along an Overall, the bracing layout is an important means of controlling modes: you can fine-tune the frequency of a specific mode by trimming the thickness and shape of the bracing, so that the guitar can obtain the desired response on the bass or treble.
Sound hole and cavity air : The sound hole is a hole opened in the soundboard (usually in the middle of the soundboard). The sound hole connects the internal air of the guitar body with the external air, making the guitar cavity constitute a typical Helmholtz resonator. The vibration of the air in the cavity forms the air cavity mode : the lowest order air mode (called the A0 mode) is similar to a mass-spring system, in which the air in the cavity acts as a “spring” and the air column at the sound hole acts as a “mass”, oscillating in and out. This mode corresponds to one of the lowest resonant frequencies of the guitar, generally in the 80~120Hz range (equivalent to around the E2 to A2 pitches), and is often called “air cavity resonance” or “air mode”. In this mode, the air in the guitar cavity moves in and out like a piston through the sound hole, while the soundboard and back plate mainly act as elastic boundaries. The air cavity mode enhances the radiation of the guitar’s lowest range: when the string frequency approaches this frequency, the cavity air resonates with the soundboard, enhancing the volume and sustain of the bass. In addition to the lowest A0 Helmholtz mode, there are also higher-order air standing wave modes (A1, A2, etc.) in the guitar cavity, corresponding to the high-order standing wave forms of air vibration in the cavity (such as longitudinal, transverse 1/2 wavelength, etc.). These higher-order air modes are not as strongly coupled to the soundboard vibration as A0, are higher in frequency and disharmonic with each other (not integer multiples of A0), and also contribute to the resonance of the guitar’s mid-band.
Strings and Bridge : Strings are the direct sound source of the guitar, and the strings are plucked by the player to stimulate vibration. The vibration frequency of an open string is determined by the string length, tension and linear density, and produces a series of harmonics.The vibration of the string itself hardly radiates sound directly to the air (due to the thin string diameter and poor impedance matching), but the vibration of the string transmits the force to the soundboard through the bridge, thereby driving the entire resonance box to vibrate. The bridge is bonded to the surface of the soundboard and is the connection interface between the strings and the soundboard. Its mass and stiffness also affect the soundboard mode: the bridge increases the local mass of the soundboard, reduces the frequency of some modes of the soundboard, and affects the local distribution of vibration shapes. The spectrum of string-bridge excitation is very wide, but the response of the guitar resonance box to different frequencies is not uniform – it acts as a filter and radiation converter, redistributing the energy input by the strings between various modes and outputting it as sound. Among them, the vibration components that match the guitar’s modal frequency are significantly enhanced and effectively radiated into audible sound, while the components that do not fall near the resonance frequency are relatively suppressed. Therefore, the overall timbre of the guitar is closely related to the modal structure: if the fundamental frequency or overtone of the string corresponding to a certain pitch happens to be close to the modal frequency, the sound will be particularly loud, with a fast and clear attack transient; conversely, if there is no resonance support near a certain sound, it will sound weak or have a short sustain. A well-designed guitar’s modal frequency distribution should cover the string pitch range as evenly as possible, with modal support from the lowest to the highest notes, to ensure balanced volume and uniform timbre in each sound zone.

Ideal flexible string (no bending stiffness), tensile string with span L (tension T, line density μ=ρA), small deflection transverse vibration satisfies the wave equation:

\frac{\partial^2u}{\partial t^2}=c^2\frac{\partial^2u}{\partial x^2},\qquad c=\sqrt{\frac{T}{\mu}}

Fixed end boundary u(0,t)=u(L,t) =0. The eigensolution is:

u_n(x,t)=\sin\!\left(\frac{n\pi x}{L}\right) \cos(\omega_nt+\varphi_n),\qquad \omega_n=\frac{n\pi c}{L}

The corresponding frequency is:

f_n=\frac{\omega_n}{2\pi}=\frac{n}{2L}\sqrt{\frac{T}{\mu}},\qquad n=1,2,\dots

Real metal strings have bending stiffness using the Euler–Bernoulli extended one-dimensional equation:

\mu\frac{\partial^2u}{\partial t^2}+EI\frac{\partial^4u}{\partial x^4} =T\frac{\partial^2u}{\partial x^2}

where E is Young’s modulus and I is the second moment of area. The eigenfrequency is approximately:

\omega_n\approx\frac{n\pi}{L}\sqrt{\frac{T}{\mu}} \sqrt{1+\frac{\pi^2EI}{TL^2}n^2}

f_n\approx\frac{n}{2L}\sqrt{\frac{T}{\mu}}\sqrt{1+Bn^2},\qquad B=\frac{\pi^2EI}{TL^2}

The B is the dissonance coefficient , causing the overtones to deviate by integer multiples.

In general, the structural components of the acoustic guitar form a complex modal vibration system through mutual coupling: the strings provide excitation, and the bridge couples the excitation to the soundboard; the soundboard, back and side panels form a wooden cavity to provide structural elasticity; the sound hole and the internal air provide cavity acoustic resonance. The vibrations of the entire system couple with the surrounding air and are ultimately radiated as sound. The quality of a guitar depends largely on the distribution of these modal frequencies and vibration shapes, and how well they match the sounding frequencies of the strings. During the design and tuning process, luthiers will pay attention to the material and structural parameters of the main resonant components (such as soundboard thickness, bracing thinning, sound hole size, etc.) to adjust the modal frequency and vibration shape to shape the sound characteristics of the guitar.

Mathematical modeling of modal analysis

Finite Element Modeling: Modal analysis of guitar usually requires the establishment of a mathematical model with the help of the finite element method (FEM). Since the guitar structure includes multiple structures/media such as thin plates (soundboard, back plate), shells (side plates), beams (bracing beams, necks, etc.) and closed air cavities, it is more complicated to accurately simulate. In the finite element model, the soundboard and back plate are often discretized into shell elements or plate elements (considering anisotropic wood elastic constants, because the stiffness of wood along the grain direction and perpendicular to the grain direction is greatly different), the side plates can be defined as frame boundaries with shell elements, and the bracing beams as stiffeners can be modeled with beam elements or equivalent shell elements. Local additional mass stiffness such as the bridge should also be included in the model. For the analysis of acoustic modes, it is necessary to model the air inside the guitar cavity, for example, using sound field finite elements (fluid elements) to fill the cavity, and setting open boundary conditions at the sound hole to connect to the outside world. This fluid-structure coupling model can simultaneously calculate the natural mode of the wooden structure box and the natural mode of the air cavity, as well as the coupled mode of the two. However, the entire model is computationally intensive, especially when it includes the radiation sound field in the external infinite space, which can be further solved with the help of the boundary element method (BEM). Therefore, simplification is often used in research: for example, first assuming a rigid box to calculate the air mode A0 frequency, and then simulating aerodynamic flexibility through concentrated mass-springs in the structural model, or only considering the first few coupled modes, etc., to reduce computational complexity.

Modal calculation and solution: After establishing a finite element model and applying corresponding boundary conditions (usually the guitar is regarded as hanging in free space with a free boundary to approximate the actual suspension test situation), the modal parameters can be obtained by solving the eigenvalue problem. Specifically, the system motion equation after finite element discretization can be written in a matrix form similar to Equation (1). The actual calculation can be completed using the eigenfrequency analysis module of commercial finite element software, which directly outputs a series of modal frequencies and corresponding mode shapes. It should be noted that there are uncertainties and anisotropy in the material properties of wood in practice, so the calculation results need to be corrected and verified based on experiments. For example, if the Young’s modulus of the material is changed in the same model, the modal frequency will change significantly; changes in wood parameters caused by humidity changes will also cause modal frequency shifts. Therefore, experimental modal analysis (such as tapping method to measure natural frequency) is often used to calibrate the simulation model to make the calculated modal consistent with the actual measurement. Furthermore, in order to focus on the main dynamic characteristics of the guitar, it is generally only necessary to extract modes within a certain frequency range (e.g. the 0–1000 Hz range covers the main radiation modes). Although there are many high-order modes, they have little impact on the overall radiation and can be appropriately truncated in the analysis.

Application of modal participation factors: After establishing the model to obtain the modes, the participation factors of each mode can be further calculated to predict which modes will dominate the response under specific excitations. For example, when simulating playing a certain string, the excitation direction at the bridge is vertical to the soundboard, then calculating the modal participation coefficient of the vertical force at the bridge can screen out the mode that contributes the most to the bridge drive response (usually the mode in which the soundboard vibrates in a “piston” manner). These modes will mainly affect the acoustic radiation of the guitar. Through the modal superposition method, any external force response can be expressed as a linear combination of the modal responses. The actual sound of a guitar is the superposition of many modes that are excited by the strings and participate in varying degrees. Among them, low-order modes often dominate, while high-order modes provide details and timbre modifications. Due to modal orthogonality, a certain mode can be adjusted independently during analysis or debugging (such as changing part of the structure to modulate frequency) without relatively affecting other modes. This has become an important theoretical basis for guitar tuning and optimal design.

Here’s a look at damping, quality factor, and attenuation. Quality factor:

f_n\approx\frac{n}{2L}\sqrt{\frac{T}{\mu}}\sqrt{1+Bn^2},\qquad B=\frac{\pi^2EI}{TL^2}

The Q of the string is determined by a combination of material internal friction, mechanical impedance matching to the bridge, and radiative coupling to the air/panel.

Typical mode shapes and their impact on timbre

Acoustic guitar has numerous modes distributed throughout the audible frequency range. The following selects several typical modes to illustrate their vibration shape characteristics and corresponding frequencies, as well as their contribution to the guitar’s tone.

(a) Air cavity mode (Helmholtz resonance , A0 mode): The lowest order mode is usually Helmholtz resonance of the air cavity, which is mainly manifested by the oscillation of the air inside the guitar body in and out of the sound hole. Therefore, it is also called “air cavity mode” or “A0 mode”. Strictly speaking, if the guitar body is assumed to be rigid, the A0 mode shape is a simple reciprocating motion between the air column at the sound hole and the air volume in the cavity. However, in actual guitars, the soundboard and back plate are not completely rigid. They will deform as the air pressure in the cavity changes, which is equivalent to an elastic membrane coupled with an air spring. This coupling makes the actual lowest mode often appear as the reverse vibration of the soundboard and the back plate squeezing the air in the cavity, which is the so-called “breathing mode” (as shown in Figure 1). In the breathing mode, when the soundboard and back plate bulge outward at the same time, the volume of the cavity increases and the internal air pressure decreases, forcing outside air to be sucked in through the soundhole; when the soundboard and back plate retract inward, the volume inside the cavity decreases and the air is squeezed out of the soundhole. The whole process is like the box “breathing”, hence the name. The frequency of this mode is typically around 100 Hz. The air cavity mode is the main source of guitar bass resonance: when the fundamental frequency or the first few harmonics of the string are close to this frequency (for example, the six-string open E2 tone is about 82 Hz, which is adjacent to the A0 frequency of many guitars), the speaker will have strong resonance, enhancing the volume and persistence of the tone, making the bass sound richer and fuller. On the contrary, if the A0 mode frequency is designed too high, the lowest bass range may appear thin; if it is too low, excessive “rumbling” low frequencies may be produced. Guitar makers often control the Helmholtz frequency by adjusting parameters such as body cavity volume and sound hole diameter to balance the guitar’s low-frequency response. It should be noted that the A0 mode is mainly dominated by air movement. Although the midrange board/backplate participates in the mode, the amplitude is relatively small; therefore, most of the sound radiated in this mode is radiated from the sound hole. It is an omnidirectional low-frequency radiation that plays a decisive role in the low-frequency directivity and resonance of the guitar.

(b) Main soundboard mode (soundboard-air coupled resonance): Above the air cavity mode, the soundboard’s first bending mode is often one of the most important structural modes of the guitar. This mode is sometimes called the “(0,0) mode”, which means that the soundboard as a whole moves in and out in the same phase (the center is the belly and the edges are the knots), similar to the movement of a piston. For a stand-alone soundboard without a sound hole, this modal frequency might be in the 200–300 Hz range; but in an actual guitar, the soundboard, air, and backplate are coupled to form a set of related modes. A typical situation is that two modes with similar frequencies appear: one is dominated by strong coupling between the soundboard and the air in the cavity, and the other is affected by the movement of the backplate. Taking an acoustic guitar as an example, its experimental modal analysis shows that a pair of modes appear at about 188 Hz and 203 Hz: in both cases, the soundboard and the sound hole air vibrate in the same phase. The difference is that the back plate vibrates in different phases – in the lower frequency 188 Hz mode, the back plate moves in the opposite direction with a small amplitude, while in the 203 Hz mode, the back plate and the soundboard move in the same direction with a larger amplitude. This set of modes can be seen as the result of the interaction between the soundboard and air coupling resonance (the main mode of the soundboard) and the backplate resonance. For classical guitars, the common main soundboard mode frequency is about 180~200Hz, which together with the air cavity mode constitutes the two pillar resonance of the guitar’s low-frequency band. The vibration shape of the main soundboard mode usually shows that the central area of the soundboard bulges and sinks, and the border is a node line; the local stiffness decreases due to openings near the sound hole, and deformation concentrated areas often appear on the vibration shape, but in general, most parts of the soundboard move in the same direction. In this mode, the air in the sound hole often moves together with the sound board and enters and exits the sound hole in the same phase. Therefore, sound waves can be radiated through the sound board surface and sound waves through the sound hole. It is a mode with high radiation efficiency. The main soundboard mode has a decisive influence on the mid-low frequency tone of the guitar: it gives the guitar a rich mid-frequency resonance, making the open mid-bass (such as the A~D range) loud and “resonant box” deep. If this mode frequency is too high (the soundboard is too hard or too small), the guitar will sound bright and lacks bass foundation; if it is too low (the soundboard is too soft), the sound may appear dull. Many of the top luthiers of classical guitars adjust the main soundboard resonance close to G3 (~196 Hz), which is considered an important factor in the “loose” tone of a typical classical guitar.Similarly, they will adjust the modal frequency of the back panel to be slightly higher than the soundboard (such as A3~220 Hz) to provide a certain coordinated resonance without completely coinciding with the soundboard mode. The soundboard main mode and the air cavity mode jointly shape the overall balance of the guitar’s bass and mid-range: the former provides thickness and body volume, and the latter provides depth and low-frequency extension. The two work well together to support the guitar from the lowest bass to the midrange. If the frequencies of the two modes are too close, a certain tone may be too prominent to form a wolf sound; if the difference is appropriate (usually one to two whole tones), they can complement each other to enhance the frequency bandwidth and make the guitar’s low-frequency response smoother.

(c) High-frequency local modes: In addition to the above-mentioned low-order global modes, guitars have a large number of higher-order modes in higher frequency bands. These modes usually correspond to more complex vibrational deformations of the soundboard and back plate, including the emergence of multiple node lines, local vibration areas, etc. For example, at mid-to-high frequencies (such as the 200–500 Hz range), the soundboard will exhibit first-order and second-order bending modes: common forms include a transverse node line along the width of the body (dividing the soundboard into upper and lower parts that vibrate in opposite directions, similar to the (0,1) mode); and a longitudinal node line along the length of the body (dividing the soundboard into left and right parts that vibrate in opposite directions, similar to the (1,0) mode). Experimental observations show that at about 262 Hz, a guitar appears in a mode of reverse vibration of the upper and lower sections of the soundboard and back plate (horizontal 1-section line). The air in the cavity “sways back and forth” and generates a certain net flow through the sound hole. At about 315 Hz, a mode of reverse vibration in the left and right sections (longitudinal 1-section line) appears. At this time, because the upper and lower parts of the soundboard are almost in opposite directions, the net movement of the air in the cavity is very small, and the air in the sound hole is almost motionless. Moving to higher frequencies, modes with more than two nodal lines will appear (such as (1,1) mode, (0,2) mode, etc., more complex vibration shapes). The characteristics of high-frequency modes are complex vibration shapes, the effective vibration area involved in each mode is small, and adjacent parts are often in anti-phase, resulting in low radiation efficiency. In other words, although there are many high-frequency modes, their contribution to far-field sound radiation is relatively limited, and they more affect the delicacy and extension of the guitar’s tone. When playing high registers or harmonics, these higher-order modes are excited, adding brightness and richness to the tone without significantly increasing the overall loudness. In fact, the human ear’s perception of bright/dark guitar timbre is related to the distribution and attenuation of high-frequency modes: there are many high-frequency modes and high Q values, which will bring “shining” treble details; if the high-frequency modes attenuate quickly due to material damping, etc., the guitar timbre will appear soft and warm but lack brightness. In reality, it is impossible to control each high-frequency mode as precisely as adjusting low-frequency resonance, but material selection (such as panel density, finishing process) and structure (such as fingerboard and bracing coupling) will affect the distribution and attenuation of high-frequency modes. Therefore, an excellent guitar will still maintain a certain response in the high frequency part, so that the sound has both rich low and mid-range frequencies and clear high-frequency details.

Generally speaking, modes of different orders contribute to the sound of the guitar: low-order modes (air cavity, main soundboard, etc.) determine the basic response curve and resonance characteristics of the sound, mid-order modes (mode shapes in the range of several hundred Hertz) enrich the timbre and projection of the mid-frequency band, and high-order modes give the sound unique overtones and texture. A well-designed guitar should have these modes well-distributed and staggered along the frequency axis to avoid having some pitches unsupported by resonance or having resonances that are too stacked at certain frequencies. Experienced luthiers will “tune” the soundboard and back plate during production so that their fundamental mode frequencies avoid the relationship of integer multiples of the musical sound, and at the same time disperse the higher-order mode frequencies, so that the guitar has a moderate resonance response to the notes all over the fretboard, and obtains a balanced and varied timbre.

Below we provide some mathematical explanations of the panel backplane and cavity.

The veneer is approximated as an isotropic thin plate (Kirchhoff–Love assumption), with thickness h, density ρ, Young’s modulus E, and Poisson’s ratio ν. Deflection w(x,y,t) satisfies:

D\nabla^4w+\rho_ph\frac{\partial^2w}{\partial t^2} +\eta D\nabla^4\!\left(\frac{\partial w}{\partial t}\right)=p(x,y,t)

in:

D=\frac{Eh^3}{12(1-\nu^2)}

D is the bending stiffness of the plate, eta is the structural loss factor, and p is the pressure difference on both sides of the plate. The boundary conditions are determined by the connections between the edging, the beams and the side panels; the actual panel with the bracing is a non-uniform anisotropic panel .

Modal superposition writing:

w(x,y,t)=\sum_k\phi_k(x,y)\eta_k(t),\qquad M_k\ddot{\eta}_k+C_k\dot{\eta}_k+K_k\eta_k=F_k(t)

Among them:

M_k=\iint\rho_ph\phi_k^2\,\mathrm{d}S,\qquad K_k=\iint D(\nabla^2\phi_k)^2\,\mathrm{d}S,\qquad F_k=\iint p\phi_k\,\mathrm{d}S

The Chladni pattern (sand method) actually shows the node lines (reflecting the constraint/beam layout).

For the cavity mode, the air in the cavity satisfies the following equation:

\nabla^2p-\frac{1}{c_0^2}\frac{\partial^2p}{\partial t^2}=0

The sound hole of a guitar is similar to a short neck cavity, and the low-frequency main resonance (Helmholtz) frequency satisfies the following equation:

f_H=\frac{c_0}{2\pi}\sqrt{\frac{A}{VL_{\mathrm{eff}}}}

Changes in volume and sound hole position and depth will affect this frequency.

Then we consider the coupling. Taking the place where the bridge and the top are in contact as an example, the string end transverse velocity V a is continuous with the bridge/top speed V b. The force balance equation is:

F_b(\omega)=Z_b(\omega)v_b(\omega)=-F_s(\omega)

in:

Y_b(\omega)=\frac{v_b}{F_b}

is the bridge point admittance (or mechanical sensitivity), the linear mode sum:

Y_b(\omega)=\sum_k \frac{i\omega\phi_k(\mathbf b)^2} {M_k\left(\omega_k^2-\omega^2+i2\zeta_k\omega_k\omega\right)}

The equivalent damping at the string end is significantly affected by Y b: when ω approaches the panel mode, the energy is coupled out faster, that is, “some sounds are louder and shorter.”

Then we look at the coupling of the panel to the cavity air. Let the normal velocity field of the panel be V n(x,y,ω) , the cavity volume velocity U cav and the acoustic hole volume velocity U hole satisfy:

U_{\mathrm{cav}}(\omega)=\iint v_n\,\mathrm{d}S,\qquad U_{\mathrm{hole}}(\omega)=\frac{p_{\mathrm{cav}}(\omega)}{Z_{\mathrm{neck}}(\omega)}

Z_{\mathrm{neck}}(\omega)\approx i\omega\rho_0\frac{L_{\mathrm{eff}}}{A}

p_{\mathrm{cav}}(\omega)= \frac{U_{\mathrm{cav}}-U_{\mathrm{hole}}}{i\omega C_{\mathrm{air}}}

By combining the above equations, the classic two (or three) degrees of freedom coupling model can be obtained, and the low-frequency “mainboard mode”, “cavity mode”, “anti-phase/in-phase” synthetic mode frequency and radiation efficiency can be obtained.

Then we look at sound radiation and loudness. The perceived loudness of a guitar is related to the far-field sound pressure p∞. In the frequency domain, the normal velocity radiated sound power of the panel is approximately:

\Pi(\omega)\approx\frac{\rho_0c_0}{2}\iint \sigma(\mathbf r,\omega)\lvert v_n(\mathbf r,\omega)\rvert^2\,\mathrm{d}S

Among them, σ is the radiation efficiency (related to the wave number k=ω/c 0, plate wave phase velocity, and the matching degree of structural waves and air acoustic waves). The low frequency is mainly dominated by the piston component (overall volume velocity) and the acoustic hole jet; the medium and high frequencies are contributed by the high-order modes of the plate.

Experimental measurement and visualization of modal shapes

In order to further study and verify the above-mentioned guitar mode shapes, scientists and luthiers used a variety of experimental methods to visualize the vibration modes.

kradny board

Chladny Method : The most classic in history is the Chladny Figure experiment. Sprinkle fine sand or tea leaves on the surface of the soundboard, and use a loudspeaker to sweep the frequency to excite the whole guitar or a separate soundboard. At certain frequencies (resonance frequencies), the vibration causes the sand grains to be shaken to the node lines, forming a visible vibration pattern. This method is simple to operate. As early as the 19th century, the physicist Chladni used it to observe the vibration shape of the plate, hence its name. For guitars, the Kladni method can measure the modal frequency and vibration shape of the soundboard and back plate before assembly. Luthiers use these patterns to judge the effect of soundboard thinning or bracing adjustments: a symmetrical soundboard produces a symmetrical pattern, and unusually asymmetrical node lines may indicate uneven material or structural asymmetry. It should be noted that the modal frequencies of the independent panel do not exactly correspond to the modal frequencies of the instrument assembled, but this method can provide useful feedback during the production process. The Chladny figure mainly shows the node topology of the modes, which is limited for quantitative research, but it has been proven to be very helpful in adjusting resonance in actual lutherie.
Force hammer method : Record modal information through the acceleration sensor patch and excitation hammer
Holographic Interferometry and ESPI: Laser holographic interferometry is another technology for high-precision visualization of vibrational shapes. By interferometrically measuring the subtle displacements of the vibrating surface, the hologram can display the equal displacement profile of the vibration in the form of a fringe pattern. Therefore, the spatial resolution is extremely high and complex vibration shapes can be clearly depicted. Electronic speckle interferometry (ESPI) is a digital implementation of holographic interferometry that also allows for real-time observation of vibrational modes. Researchers have used holographic interference to conduct modal measurements on musical instruments such as guitars, and the obtained vibration shapes correspond well to the finite element calculation results. For example, Chatziioannou et al. verified the accuracy of simulated modalities via ESPI on instruments such as double bass and viol. For guitars, similar methods can be used to observe the vibration patterns of the soundboard in various modes, including subtle local warping and phase differences, which are difficult to see directly with the naked eye. The advantage of holographic interference is that it can obtain continuous full-field information without placing points on the structure. It is currently one of the modal visualization methods with the strongest spatial resolution.
Experimental modal analysis and acoustic-radiation measurement: In addition to observing mode shapes directly, researchers often combine impedance measurements with sound-field measurements. Mechanical admittance or impedance at the bridge can be measured by applying a unit force and recording vibration velocity, which reveals modal frequencies and damping. Microphone arrays can simultaneously map the radiated sound field and identify modes with high radiation efficiency. These measurements help determine which modes contribute most strongly to a guitar’s sound and guide design improvements.

Modal testing

Modal testing

In summary, with the help of modern experimental techniques, we are able to “see” abstract modal shapes and connect theoretical models to real instrument behavior. These tools not only verify the existence and shape of each mode of the guitar mentioned above (for example, the measured breathing mode and soundboard mode of the guitar are very consistent with the calculated predictions), but also provide scientific basis for guitar manufacturing and improvement. Nowadays, in addition to empirical judgment, people are increasingly referring to modal test results to fine-tune musical instruments to make their frequency response more ideal. The secret of acoustic guitar timbre is contained in the rich and diverse mode shapes : Through theoretical modeling analysis and experimental visualization, we have deepened our understanding of how these modes vibrate in various frequency bands and couple with each other, and are closer to explaining the intertwined science and art problem of “why a good guitar sounds charming”.

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Fundamentals of Guitar Modal Concepts

Fundamentals of Guitar Modes

Basic Modal Concepts

Acoustic guitar structure and main resonance components

Mathematical modeling of modal analysis

Typical mode shapes and their impact on timbre

Experimental measurement and visualization of modal shapes

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