© The Author(s), Published by EDP Sciences, 2024

1 Introduction

The violin’s f-holes play a critical role in the instrument’s sound [1]. These stylized cuts into the violin top plate enable an air resonance that functions similarly to a Helmholtz resonator and provides substantial acoustic support for the lowest tones of the violin [2–4]. As the f-holes modify top-plate geometry, they also influence the violin’s structural modes [5, 6]. Consequently, understanding the f-hole’s acoustic and structural function is of principal concern for luthiers and acousticians.

Researchers have undertaken numerous studies on the acoustics of the violin’s f-holes. A series of experiments by Saunders [7] in 1953 showed that increasing the f-hole length increased the radiated pressure levels not only for the A0 (compliant-walled Helmholtz resonator) mode, but also at higher frequencies. He inferred that these increased levels at higher frequencies were partly due to stronger vibrations of the bridge-island (or bridge-hill) region, the top-plate area between the two f-holes. Because plugging the f-holes with cotton only strongly impacted radiated levels near the A0 mode, he suggested that sound radiation through the f-holes may be limited at higher frequencies. These early results highlight the complex relationship between the f-hole shape and the violin’s modal response.

Although structural-acoustic coupling is an essential feature of violin acoustics [4, 8, 9], studying either the isolated acoustic or structural system has provided important insights. For example, models treating the lowest air mode A0 as a rigid-body Helmholtz resonator using lumped-element representations are common in the literature. Itokawa and Kumagi [2], Cremer [10], and Schelleng [3] all used different variations of equivalent circular and elliptical apertures to estimate the acoustic inertance of the f-hole shape. Once calculated, the f-hole inertance and cavity compliance yielded estimates of the Helmholtz resonance of this mode under a rigid-body assumption [2], often within very reasonable agreement with experimentally derived values. Consequently, these early studies were successful in explaining the existence of the A0 mode by applying resonator theory.

A later work by Shaw [11] further developed the classical resonator theory to a two-degrees of freedom (2DOF) model to predict the behavior of both the Helmholtz mode A0 and the first longitudinal air resonance of the violin cavity A1. His 2DOF model also incorporated a more complicated equivalent aperture representation of the f-hole by representing its shape as an elongated ellipse with two circles to account for the f-hole eyes. Experiments on a violin called “Le Gruyère” (the swiss-cheese violin), a violin with 65 plug-able small holes cut into the violin ribs, validated the lumped-element model. Shaw’s fitted 2DOF model gave excellent predictions of A0 and A1 frequency changes for at least three reported variations of open holes. His results also suggested coupling between these two modes, a prominent feature of violin acoustics [4].

In another study on the air modes of the violin, Bissinger [12] used an aluminum violin-shaped cavity dubbed “La Empierre” to remove structural-acoustic coupling. A removable top-plate insert enabled variations to the f-hole’s shape, including circular, slot, and traditional f-holes. The results showed that A0 frequency rose with increasing f-hole aperture size, experimentally validating theoretical assumptions based on resonator theory. He also found that f-hole size and position did not strongly impact the other cavity modes. Bissinger [13] later used La Empierre to further investigate Shaw’s 2DOF model by filling the metal cavity with water to create measurable changes in its volume. Theoretical and experimental results, including earlier measurements by Hutchins [14], gave a different volume dependence than the V^−0.5 typical of a rigid-walled Helmholtz resonator. This result led Bissinger to suggest coupling between A0 and A1 as an obstacle in predicting violin A0 frequency from volume alone, such as in the case of the Schelleng-Hutchins scaled violin octet.

Application of numerical methods such as the boundary element method (BEM) have allowed more precise investigations of the violin’s air modes compared to simpler lumped-element models. For example, Bissinger [15] used the BEM to predict the acoustic modes of a closed violin cavity with a particular focus on developing semi-empircal models to predict the A1 resonance frequency. More recently, Nia et al. [16] applied a BEM technique to numerically evaluate the conductance (inversely proportional to the inertance [17]) of the f-hole shape. This improved calculation method gave better estimates of A0 frequency using Helmholtz resonator theory, obtaining less than 1% RMSD errors between measured and predicted resonance frequencies for a rigid cavity with f-hole shapes. Application of this model suggested that the violin f-hole shape slowly evolved during the classical Cremonese period to a larger aperture size, leading to an increased A0 frequency and a higher radiated sound power at this air resonance. However, similar to previous lumped-element models employing rigid-body resonator theory, their work could not directly account for the compliance of the violin’s wooden body and thus required elastic correction factors to compensate for the discrepancies between measured and predicted results.

Consequently, a large body of existing theoretical and experimental research has evaluated the impact of the violin’s f-holes on the A0 mode. The results have shown that increasing the f-hole size increases both the A0 mode’s resonance frequency and radiated sound power. However, the difficulty in modeling the violin’s structural acoustics, including the effect of the compliant violin cavity, have limited the scope of the analysis. Studies applying rigid-body Helmholtz resonator theory have required post-measurement elastic correction factors or other model adjustments to compensate for prediction discrepancies [11, 16]. Models based only on rigid-body behavior cannot completely describe the essential acoustic-structural interactions characteristic of the instrument.

Investigations on the impact of the f-holes on the structural modes of the instrument are few. Instead, researchers have tended to apply numerical methods to study the in-vacuo structural modes of the violin with the primary aim of tuning plate resonances. Bretos et al. [18], developed a finite element method (FEM) model of the instrument including the top and back plates, ribs, lining, and corner blocks. Their analysis first considered the modal frequencies of the unassembled top and back plates and found excellent agreement in both mode shapes and frequencies with previously reported experimental results. They then compared mode shapes and frequencies after assembling the violin body with further comparisons between previous works. Additional discussion included the impact of the carving process on tuning violin-plate resonance frequencies.

Although their work did not consider how the f-hole’s shape modifies top-plate resonance frequencies, their results underscore the utility of numerical methods to model musical instruments. Numerical approaches such as the FEM can incorporate the wood’s elasticity, which is challenging to represent using lumped-element models alone. The method’s accuracy is beneficial in virtual prototyping of musical instruments or their components, e.g., for the acoustic guitar or violin [19–21].

Gough [5] likewise used the FEM to study violin top and back plates, including the impact of arching, shape, plate thickness, and material anisotropy. His results showed that slowly introducing the f-holes into the top plate lowers the resonance frequencies of its vibrational modes. In particular, the two modes most commonly believed to determine the assembled violin’s acoustics saw modal frequency decreases of over 10%. In a later study, a similar FEM model provided the response of the bridge-island region and showed the importance of the f-holes in permitting vibrations in this area. However, it did not directly consider how controlled f-hole shape changes would impact bridge-island structural vibrations [6].

While studying the violin’s rigid-body acoustic cavity modes or in-vacuo structural modes has provided significant insight into the f-hole’s function, some of the instrument’s most interesting behavior occurs due to the instrument’s complex structural-acoustic coupling. These interactions can in part explain why works modeling the A0 mode as a rigid-walled resonator have consistently overestimated the A0 resonance frequency. In addition, even the best FEM model of the wooden body will fail to predict the A0 mode without accounting for acoustic loading and radiation from the vibrating body. Consequently, coupled structural-acoustic models and measurements are best suited to completely understand the violin’s complexities, just as they are used for analysis of other stringed instruments [20, 22, 23].

In a later work employing Le Gruyère, Weinreich et al. [9] studied air-wood coupling. The specialized violin allowed this analysis since modifying the total aperture area with plugs could control the air mode frequencies without significantly altering the wood modes. The results revealed “modal veering,” suggesting a strong coupling between the A0 and the lowest bending modes of the violin. Bissinger et al. [1] used patch near-field acoustical holography (pNAH) and scanning laser Doppler vibrometer (SLDV) measurements to identify the relative contribution of sound radiated through the f-holes and from structural vibrations. Significant sound radiation from the f-holes, even for the wood modes of the violin, suggested corpus-induced flow. More recently, Yokoyama used a coupled FEM-BEM simulation to study the acoustics of a violin made by the Guarneri family and digitally reconstructed with a CT scanner [23]. A simulation with around two million degrees of freedom predicted important signature modes including A0, B1⁻ and B1⁺. The results also provided information about the effect of wood density on the violin’s eigenfrequencies and insight into the violin’s sound radiation. Thus, the most complete and accurate descriptions and predictions of violin acoustics require careful application of fluid-structure interactions.

Consequently, research has verified that the f-holes influence the A0 mode and strongly impact the violin’s other modes. The works have also validated the utility of numerical methods such as the FEM and BEM to predict the structural and acoustic characteristics of the instrument. Nonetheless, outside of the A0 mode, which may be simply modeled using resonator theory, there remains limited information and understanding on how changes in f-hole shape modify the violin’s acoustic response. This work studies how f-hole shape changes influence the violin’s lowest strongly radiating modes. A coupled structural-acoustic model provides a straightforward method to study the impact of f-hole shape variations on the radiated sound power of the instrument. The model shows that enlarging the shape of the f-hole strongly impacts the B1⁻ mode by increasing radiation from the f-holes and the violin body. Similarly, enlarging the f-holes increases the radiated response in the bridge-island region and from other higher-frequency modes. Experimental measurements on a fractional-size violin validate these trends, confirming strengthened wood-mode radiation with increased f-hole aperture area.

2 Coupled simulation

2.1 Simulation method

Coupled FEM-BEM simulations are an effective method to study violin behavior as they incorporate the critical structural-acoustic coupling of stringed instruments [20, 22, 23]. As illustrated by Figure 1, a Yamaha 3/4 size violin served as the reference geometry for the coupled simulations. The violin CAD model includes the top and back plate, ribs, lining, block, neck, fingerboard, nut, soundpost, and bass bar. The model does not have the bridge, tailpiece, pegs, and strings for more straightforward implementation into simulation software.

Figure 1 CAD model of the 3/4 size violin used in the coupled FEM-BEM simulations.

Estimating the wood’s material properties is essential for reliable numerical simulations [19, 20, 22]. The properties of wood have long been known to impact the violin’s modal frequencies and thus its tonal characteristics. Nonetheless, the primary scope of the present work is to study how changes in geometry, rather than material properties, influence the violin’s bending modes. Consequently, rather than attempting to exactly reproduce the material properties of a single violin, which requires extensive fitting and tuning procedures [19, 22], the estimated wood material properties followed from tonewood samples used in violin construction.

Material property values of the samples followed from a fitting procedure similar to that used in [20]. First, SLDV measurements and analysis identified the lowest structural mode shapes and frequencies of tonewood sample plates. The modal analysis included at least one longitudinal mode, one transverse mode, and four torsional modes. Next, FEM models of the tonewood sample plates served as the basis for numerically estimating plate modal frequencies and allowed density estimation from the measured plate mass. Optimized least-squares fits of the tonewood’s orthotropic material properties, including its Young’s modulus, shear modulus, and Poisson’s ratio, ensured consistency between measured and simulated modal frequencies. Table 1 plots these fitted material properties for the top and back plate and compares them with previously published results. In general, the measured parameters agree very well with results from other works. However, the radial-tangential component of the shear stress G_RT for the spruce top plate appears higher than values available in [22, 24]. This inconsistency could be due to invalid assumptions of homogeneous material properties or to the tuning method which attempts to match simulation results with the measured modal frequencies.

The simulations used the commercial software ANSYS [25] to estimate the violin’s modal frequencies using the FEM without coupling to the acoustic medium. The high-order tetrahedral mesh used for the violin had roughly three million degrees of freedom, comparable to that used by Yokoyama [23]. Free boundary conditions enabled equitable comparisons between later measurements. The analysis identified about 110 structural modes between 0 to 4 kHz.

The BEM model allows computation of the air resonances of the violin cavity, which are later coupled to the structural modes produced by the FEM model. Figure 2 shows the violin cavity’s low-order quadrilateral BEM mesh of around 43,000 elements. Additionally, because the violin’s top plate, bottom plate, and ribs are very thin (∼3 mm) relative to the desired wavelengths (>340 mm), this surface mesh provides a reasonable estimate of the radiated acoustic pressure field, provided that radiation from other parts of the violin body, e.g., the neck and fingerboard, are negligible.

Figure 2 Violin cavity BEM mesh used for the coupled simulations.

An in-house software calculated the coupled FEM-BEM response of the instrument by loading the in-vacuo structural modes determined by the FEM analysis with the acoustic medium described by the BEM model using a strong coupling technique [26, 27]. The technique used the surface velocity for the structural to acoustic coupling and the surface pressure for the acoustic to structural coupling. A1 N force applied in the normal direction to a 2 mm square region of the top plate near the f-holes served as the excitation force. Six different f-hole scalings based on the same shape having a combined surface area of ∼1010 mm² allowed comparisons of how the f-hole size impacts the radiated response. The smallest pair of f-holes had a total aperture area of 494 mm² (70% scaling of the original size), and the largest pair of f-holes had a total aperture area of 1222 mm² (110% of the original size). Figure 3 plots these two extremal f-hole pairs on the violin model. Although some size variations are likely too radical for adoption by luthiers, the wide range of f-hole sizes provides essential insight into modal variations due to changes in the overall f-hole aperture.

Figure 3 Smallest (70% of original size) and largest (110% of original size) pair of f-holes on the fractionalsize violin model.

The present study applied three evaluation surfaces for analysis. The first surface was a 1.5 m radius sphere with 326 nodes centered about the violin. Calculated pressure values on this surface provided estimates of the radiated sound power, directivity, and other radiation properties (see Sect. 2.2). The second surface was a rectangular 697-node planar surface placed 1 mm above the apex of the violin top plate. This surface covered the top plate (excluding the fingerboard) to provide a means to study near-field radiation through the f-holes and from the top plate (see Sect. 2.3). The third surface was a plane set 15 cm away from the violin top plate and served as a validation between later measurements made at this same observation distance (see Sect. 3.1). Further details on the simulation and its results appear in [28].

2.2 Sound power analysis

A first investigation considers the influence of the f-hole size on the signature, or lowest modes, of the violin. Rather than analyzing the pressure at a single measurement position for these modes, the sound power followed by numerically integrating the mean-squared pressure over the r = 1.5 m spherical evaluation surface following the guidelines given in ISO 3745:2012 [29]. The violin’s simulated sound power for the six increasing f-hole sizes appears in Figure 4. Overlain vertical dashed lines indicate the open strings of the violin which sound at G3, D4, A4, and E5. Labels indicate the A0, CBR, B1⁻, and B1⁺ modes. In agreement with the results from Bissinger et al. [1], the A0, B1⁻, and B1⁺ modes radiate efficiently while the CBR mode does not. Table 2 additionally reports the modal peak levels and resonance frequencies for the extremal cases to highlight the range of values occurring for different f-hole sizes.

Figure 4 Simulated sound power curves over increasing f-hole size based on a 1 N input force.

Changes in sound power due to f-hole size reveal several significant trends. First, A0 frequency and radiated sound power increased with enlarged f-hole area, consistent with basic theoretical assertions and previous violin research [16]. For example, the modal frequency falls at 238 Hz (∼B♭3) with a radiated sound power of 94 dB for the smallest f-hole size, whereas the largest f-hole size has a modal frequency at 308 Hz (∼E♭3) with a radiated sound power of 103 dB. Consequently, the f-hole size enlargement led to a frequency shift of over 4 semitones and an increased sound power of nearly 10 dB.

Similar to the A0 mode, increasing the f-hole shape significantly impacted the B1⁻ mode. For the smallest size considered, the B1⁻ peak appeared to be buried by the B1⁺ mode. However, by the largest shape variation, the mode radiated more strongly than the A0 mode at 108 dB. The increase in sound power between the second smallest to the largest size was over 15 dB. However, unlike for the A0 mode, the f-hole size did not appear to affect the B1⁻ modal frequency, decreasing its value by less than 1 semitone.

Increases in the f-hole size influenced the radiation from CBR and B1⁺ as well, although their changes were not as dramatic as for A0 or B1⁻. With increasing f-hole size, the B1⁺ mode’s radiated level increased by less than 3 dB from the smallest to largest f-hole size. For CBR, the radiated power increased by almost 5 dB. However, the shape of the sound power curves suggests that this effect did not result from changes in the CBR mode itself but rather from the increasing radiated levels of A0 and B1⁻.

Although most works primarily focus on the A0, A1, CBR, and B1 modes, interesting behavior appears beyond this region. For example, in the range of 660 Hz (E5) to 1 kHz, at least three modes appear which have increasing radiated levels with enlarged f-hole size. For example, the enlarged f-holes decreased the modal frequency of the mode nearest to B1⁺ by over a semitone and increased its radiated power by over 10 dB. These results suggest that f-hole shape is important even for the instrument’s higher modes.

Additional insights followed by considering changes in radiated levels over important frequency ranges for the violin rather than for isolated modal peaks. Over the frequency range from 200 Hz to 660 Hz, which encompasses the modes A0, CBR, B1⁻, and B1⁺ and covers the notes on the G to A strings in first position, the energetically averaged increase (averaging mean-squared pressure values rather than decibel quantities) in radiated sound power from smallest to largest f-hole size was 2.7 dB. However, over the range of 200 Hz to 520 Hz, which excludes the strongest-radiating but minimally affected mode B1⁺, the energetically averaged value increased to 12.2 dB. Over the frequency range of 660 Hz to 1 kHz, the energetically averaged increase was 8.2 dB. Thus, changes in f-hole shape appeared significant to increasing the violin’s overall sound in the signature mode region, particularly for the lowest two strings of the instrument but also for higher tones as well.

2.3 Near-field analysis

While the sound power curves can reveal how changes in f-hole shape influence overall radiated levels, near-field analysis can provide complementary insights into underlying acoustic and structural mechanisms. For example, Bissinger et al. [1] used pNAH to isolate radiation due to volume flow through the f-holes and that due to vibrations of the violin’s body. Although a detailed investigation incorporating structural-acoustic wavelength analysis is not pursued here, evaluations of the pressure field just above the violin top plate highlighted trends observed both in the sound power curves and from previous works.

For example, radiation from the A0 mode is primarily associated with volume flow through the f-holes [1, 16]. Figure 5 plots pressure levels of the front evaluation surface calculated at the A0 modal peaks for the six f-hole sizes. The color scale on all plots remains the same to enable equitable comparisons. The white rectangular region in the upper center of the plots corresponds to the area obstructed by the fingerboard. The surfaces show that increasing f-hole size corresponds to more substantial pressure around the f-holes. This result is in agreement with assertions from Nia et al. [16], who showed that increasing volume flow through the f-holes is correlated with higher pressure due to the strong monopole moment at this frequency. The pressure levels around the violin body also suggest that sound radiation for the A0 mode is primarily due to acoustic radiation from the violin cavity through the f-holes rather than structural vibrations, in good agreement with the results of Bissinger et al. [1]. From the smallest to largest f-hole size, the maximum pressure level increased by 7.9 dB.

Figure 5 Simulated sound pressure level in the plane directly above the top plate at the A0 resonance peak for the six different f-hole sizes.

In contrast to A0, the CBR mode is not considered a strong radiator. The pressure evaluated at the peak of the CBR mode for the six f-hole sizes appears in Figure 6. The pressure in this plane has strong nulls falling near the center line of the instrument as well as regions of reduced levels. These complex patterns may be associated with CBR’s bending motion, which leads to multiple in- and out-of-phase regions of corpus vibrations [1]. Unlike for A0, enlarging f-hole size appears to have minimal effect on the radiated levels, consistent with the sound power results in Figure 4. Over the six f-hole sizes, the maximum level in this plane varied by less than 1.5 dB.

Figure 6 Simulated sound pressure level in the plane directly above the top plate at the CBR resonance peak for the six different f-hole sizes.

Similar to the A0 mode, the increased f-hole size corresponded to increased frontal pressure levels for the B1⁻ mode (Fig. 7). However, several important trends differ. For the A0 mode, the increased radiation may be assumed to be directly attributed to increased volume flow through the f-holes. However, while increased pressure levels around the f-holes appears, particularly for the 5th and 6th size variations (500 Hz and 496 Hz), there is also substantial radiation from the bass-bar side lower bout. The B1⁻ mode shape showed an anti-node in this region. Consequently, this result suggests that the increasing sound power for the B1⁻ mode may be due to both increased radiation through the f-holes and stronger vibrations of the violin body. The change in maximum radiated level between these smallest and largest size was 18 dB, again highlighting the significance of the f-hole’s shape on the violin’s bending modes.

Figure 7 Simulated sound pressure level in the plane directly above the top plate at the B1-resonance peak for the six different f-hole sizes.

The results for the B1⁺ mode appear in Figure 8. The changes between f-hole sizes had less effect on the radiated levels, similar to the results seen in the sound power curve of Figure 4. For this mode, the maximum pressure level change from the smallest to largest f-hole size was 3.4 dB. Unlike B1⁻ but similar to the other bending mode CBR, the pressure distribution over the front surface remained relatively unchanged over f-hole size variations. Thus, changes in f-hole shape do not appear to impact structural-air mode coupling as appeared to be the case for the B1⁻ mode. Consequently, these near-field results suggest that f-hole size may be less influential for the B1⁺ and CBR modes compared to A0 and B1⁻, as the former modes had changes of only 1.5 dB and 3.4 dB, respectively, compared to 7.9 dB and 18 dB for the latter modes.

Figure 8 Simulated sound pressure level in the plane directly above the top plate at the B1+ resonance peak for the six different f-hole sizes.

As evidenced by the sound power curves and near-field results, changes in f-hole shape do not uniformly impact the violin’s signature modes. The strong coupling between the A0 and B1⁻ modes observed in previous works [1, 9] may in some part explain increasing radiated levels for the B1⁻ mode which were not as pronounced for the B1⁺ mode. For example, Bissinger et al. [1] compared relative contributions from corpus-induced flow through the f-holes to that produced from structural vibrations, finding that the former contributed to over 50% of the radiation of the B1⁻ mode. This relative acoustic-structural contribution was the strongest of any of the violin’s normal modes when excluding A0. Because Weinreich et al.’s [9] modal veering experiments identified a strong coupling between A0 and B1⁻, it seems plausible that increasing the radiated level and modal frequency of A0 could increase the radiated level of B1⁻, primarily due to increased volume flow through the f-holes. That the curves in Figure 4 showed an increase in the radiated sound power with increasing f-hole size over the entire frequency range between A0 and B1⁻ and that the near-field pressure shown in Figure 7 showed increasing pressure levels around the f-holes supports this claim.

However, Weinreich et al.’s [9] modal veering results predicted that an increasing A0 modal frequency should increase the modal frequency of B1⁻, while the simulation results showed contrary movements. The discrepancy may arise because Le Gruyère did not have variable f-hole size but rather plug-able holes in the ribs. Consequently, changes in radiating aperture area derived from changes in the ribs, not from changes in the f-hole shape. Gough’s FEM results [5] suggested that the introduction of the f-holes led to a decrease in top-plate modal frequencies, which is more consistent with the simulation trends. Decreasing modal frequency with increasing f-hole size may be further substantiated by noting that the three structural modes CBR, B1⁻, and B1⁺ lowered their modal frequencies in agreement with Gough’s results. Interestingly, the change’s magnitude was similar for all three of these modes, roughly 50 cents for CBR and B1⁻ and closer to a full semitone for B1⁺.

3 Measurements

3.1 Coupled Simulation Comparison

While coupled FEM-BEM models can provide insights into underlying behavior, they are inherently limited due to the approximations necessary for practical implementation. For example, the FEM-BEM model does not account for material property inhomogeneity nor the impact of the tensioned strings and bridge on the violin top plate. Consequently, comparisons of radiativity measurements (radiated SPL relative to an input force) between two different f-holes applied to the same violin complement observed trends from the simulations.

First, a set of radiativity measurements on the 3/4 size violin with and without the bridge and strings gave insights into the validity of the FEM-BEM simulations. As suggested in Figure 9, the measurements took place in an anechoic environment (f_c = 80 Hz) with the violin suspended by elastic strings at the neck and scroll to approximately realize free boundary conditions. For the condition without strings or bridge, which best represents the FEM-BEM simulation settings, a force hammer tapped the same location on the top plate as the simulations, a point a few millimeters away from the treble-side f-hole. For the bridge and strings condition, which best represents the instrument as played in practice, the force hammer tapped the G-string side of the bridge. In both cases, a 12.7 mm (0.5 in) microphone placed 15 cm from the center of the bridge measured the acoustic output. The coupled-model simulation results imitated the experimental procedure by numerically evaluating the pressure 15 cm from the center of the bridge based on a 1 N excitation force.

Figure 9 Violin radiativity measurement set-up.

Across the signature modes, agreement between the measurement and simulation results remained reasonable. Figure 10 compares the measured and simulated acoustic output, while Table 3 reports the predicted and measured modal frequencies. The modal frequencies agreed well for A0, CBR, B1⁻ and B1⁺, achieving deviations less than 50 cents for the case without the bridge. This result highlights the utility of coupled FEM-BEM models for predicting the violin’s modal response. Even when the bridge was included, the largest deviation among these modes was 52 cents for B1⁺, which occurred due to the bridge lowering the frequency of this mode.

Figure 10 Radiativity curves for the simulated and measured violin.

The A0 and B1⁺ modal peaks had 2 and 3 dB deviations, respectively, between the simulation output and measurement results without the bridge. For the A0 mode, including the bridge decreased the peak deviation to less than 0.5 dB, although this occurs because exciting the violin on the top plate without the bridge apparently excited a mode falling near A0, leading to a split peak. This mode may be due to the imperfect free boundary conditions that led to entire corpus bending motion, including the neck. This motion would not have been as strongly excited with the impact on the side of the bridge as compared to the impact in the normal direction of the violin top plate. The CBR and B1⁻ modal peaks had 6 and 5 dB deviations, respectively, between the simulation output and measurement results without the bridge. However, it should also be noted that the material properties used in the simulation and estimated from tonewood samples varied from those used in the actual violin’s construction, which would lead to differences in radiated levels between the two sets of results.

While incorporating the bridge only appeared to have a minor impact on A0, CBR, and B1⁻, it did lower both the modal frequency and radiated level of B1⁺. In addition, it had a significant effect on higher-frequency modes. For example, for the mode just beyond B1⁺, the deviations for the modal peak magnitude and frequency between simulation and measurement results were just 2 dB and 23 cents without the bridge. However, including the bridge caused a significant null in the radiated level. In general, the bridge with tensioned strings lowered the radiated level of the modes B1⁺ and beyond, similar to the lowering of modal frequencies of a piano soundboard with increased downbearing [30].

3.2 f-hole shape enlargement

Because it is unfeasible to exactly reconstruct the same violin with the same material properties, comparisons of radiativity measurements between two different f-hole shapes applied to the same violin helped validate observed trends from the simulations. However, cutting different f-hole designs sequentially on the same violin severely restricts the possible range of shapes. Figure 11 shows the reference 3/4 size Yamaha violin with its original f-hole shape and with an enlarged shape that increased the f-hole’s total surface area by nearly 87%. Although this shape may be atypical for traditional f-holes, it does allow validation of some simulation trends.

Figure 11 Comparison of the violin with its original f-hole shape and after the f-hole enlargement.

The measured radiativity before and after the f-hole shape change appears in Figure 12 for the case of measurement without the bridge and strings. Many of the trends predicted by the simulation appear. First, the enlarged f-hole shape significantly alters the A0 mode, increasing both its resonance peak and maximum radiated level. For the original size, the resonance frequency falls at 291 Hz (∼D4) with a radiated level of 107 dB. However, for the enlarged f-hole shape, the resonance frequency increases roughly two semitones to 331 Hz (∼E4) with a radiated level increase of 7 dB to 114 dB.

Figure 12 Measured radiativity curves between original and enlarged f-hole shape without the bridge and strings at tension.

When played with the bow, this change led to an audibly stronger sound on the D string, although the tones on the G string sounded significantly weakened due to the loss of modal support over its notes. In fact, the energetically averaged radiated pressure level loss over the frequency range between the G and D string (∼196 Hz to ∼294 Hz) was 8.5 dB. However, for the frequencies falling between the D and A string (∼294 Hz to 440 Hz), the energetically averaged radiated pressure level gain was 5.9 dB. Consequently, instrument makers must take care to control the modal behavior of A0 to obtain a balanced sound. This is particularly the case for fractional-sized instruments, whose modal frequencies fall higher than those of a full-sized violin.

Second, the enlarged f-hole shape increased the B1⁻ modal strength, similar to the simulation results. From the original to enlarged f-hole shape, the radiated level increased 4 dB from 111 dB to 115 dB. As in the simulation results, the region between A0 and B1⁻, which encompasses the CBR mode, saw increased radiated levels. The energetically averaged increase in radiated pressure levels was 4.7 dB for the frequency range extending between the modal peaks of A0 and B1⁻.

Another similarity included the minor effect of the f-hole shape on the modes CBR and B1⁺. The modes CBR and B1⁺ increased the radiated pressure level at their modal peaks by only about 1 dB. Over the frequency range from 200 Hz to 660 Hz (including A0, CBR, B1⁻, and B1⁺), the enlarged f-hole on average increased the radiated level by 2.0 dB, while over the range of 200 Hz to 520 Hz (including only A0, CBR, and B1⁻), the enlarged f-hole increased the radiated level on average by 4.3 dB. Consequently, increases in f-hole shape primarily augmented radiated levels for the A0 and B1⁻ modes.

While the measured results followed the predicted trend of increasing A0 frequency with enlarged f-hole size, the three bending modes CBR, B1⁻, and B1⁺ saw rising modal frequencies contrary to the simulation results. This trend of increasing bending-mode modal frequencies due to an increasing A0 modal frequency is in agreement with Weinreich et al.’s experimental modal veering observations [9] but contrasts with results from Gough’s results showing a lowering of top-plate modal frequencies with increasing f-hole size [5] as discussed in Section 2.3. The discrepancy between simulation and measurements may be due to differing f-hole shapes (the simulation used the same scaled shape whereas the measurement used an enlarged shape), material property variations, or the impact of the bridge. Because the magnitude of the change in the measured modal frequency for B1⁻ and B1⁺ was very slight (<2 Hz or 1% of the initial value), repeating the experiment multiple times could provide insight into whether this change was statistically significant or rather due to measurement inconsistencies.

Other similarities between measurements and simulations included increased radiated pressure levels and decreased modal frequencies for the two modes beyond B1⁺. The nearest mode lowered its modal peak frequency by about 33 cents and increased its radiated level by about 4 dB, whereas the next nearest mode lowered its modal frequency by about 24 cents and increased its radiated level by about 2 dB. Over the range of 660 Hz to 1 kHz, the average increase in radiated pressure level was 0.9 dB. Additionally, measured results from 2 kHz to 4 kHz, encompassing the bridge-island region, had an averaged increase in radiated pressure levels of 2.1 dB. Consequently, f-hole size does impact the violin’s radiated levels at higher frequencies.

Incorporating the bridge and strings does change the modal behavior compared to the free vibrations of the isolated body, as evidenced by the curves appearing in Figure 13. While the trend of increased radiated pressure levels for A0, CBR, and B1⁻ appeared, the level for B1⁺ decreased with an enlarged f-hole shape, in contrast to the simulation results and the measurements without the bridge. In addition, radiated pressure levels for the modes beyond B1⁺ appeared less impacted by the shape change. In fact, the frequency-averaged change in radiated pressure levels was less than 0.1 dB from 660 Hz to 1 kHz. However, the bridge-island region still had increased radiation even with inclusion of the bridge. The average radiated pressure level increase from 2 kHz to 4 kHz was 2.6 dB. Consequently, the measurement results suggest that f-hole shape size enlargements not only increase radiation from the signature modes of the instrument but also increase averaged levels in the bridge-island region.

Figure 13 Measured radiativity curves between original and enlarged f-hole shape including the strings and bridge.

4 Discussion

Precisely understanding how the f-holes influence the violin’s acoustics presents several challenges. First, because constructing a violin is time consuming, expensive, and difficult to exactly reproduce due to varying wood characteristics, testing many f-hole shapes and sizes through experiments alone may be impractical. While using numerical methods such as a coupled FEM-BEM simulation can provide valuable information, there are limitations to this approach as well, including the difficulty in obtaining exactly calibrated results with respect to modal frequencies and amplitudes for the assembled instrument [19, 20, 22].

Another difficulty in understanding the f-hole’s role is the strong acoustic-structural coupling of the instrument. Although previous lumped-element predictions were successful in describing A0 behavior through resonator analogies, these simple but informative models are more difficult, if not impossible, to directly apply to all of the violin’s signature modes. This leaves a significant knowledge gap in understanding how changes to one mode coupled to others may influence the overall response of the instrument. Because the f-holes change both the acoustical and structural properties of the violin, better understanding of air-wood mode coupling in stringed instruments would help explain experimentally observed behavior.

It is also interesting to observe changes in measured results due to modified boundary conditions such as including the bridge with strings at tension. Although violin makers commonly tune free plates, Bretos et al. [18] and Gough’s [5] works already showed significant changes which occur when assembling the instrument. The measurement results from this work further demonstrate that incorporating the bridge and strings modifies some modal behavior compared to the freely vibrating body. Perhaps more significantly, even with strung strings, the violin is never played with truly free boundary conditions but is rather clamped between the chin rest and the shoulder rest. How this additional boundary condition could modify the violin’s modal behavior has generally been overlooked, but perhaps deserves greater scrutiny. In either case, the results demonstrate that the most faithful numerical modeling requires full consideration of the instrument under natural playing conditions, including its assembly and support.

Despite these limitations, results of this work and previous research confirm that the f-holes are important to the violin’s sound, in particular for the fundamental frequencies of the lowest notes. However, properly tuning these modes using f-hole shape variations requires careful consideration. For example, while increasing the f-hole size increased radiated levels between A0 and B1⁻, it also led to reduced levels for the lower tones of the instrument due to the A0 resonant peak shift. This fact is particularly consequential for fractional-sized violins, which almost entirely rely on the A0 mode for support on the G string. Nonetheless, the experimental results of this work suggest that f-hole modifications may be a useful tool to tune the instrument’s sound.

5 Conclusions

The f-holes influence the modal characteristics of the violin, both for the A0 air-resonance mode and its wood modes. Through simulations and measurements, this study considered how changes in f-hole size impacted a fractional-sized violin’s overall acoustic response. Both results confirmed that enlarging f-hole size increased A0 frequency and loudness. The results also suggested that an enlarged f-hole size increased radiated B1⁻ levels and improved radiation in the region between A0 and B1⁻. Future work should include evaluating whether these results generalize to other stringed instruments, such as the cello or full-sized violin. Other areas of investigation could also include further analysis on inter-mode coupling between the signature modes of the instrument or the perceptual influence of the A0 mode on the instrument’s timbre.

Conflicts of interest

The authors declare no conflict of interests.

Data availability statement

Data are available from the authors upon reasonable request.

References

All Tables

All Figures

	Figure 1 CAD model of the 3/4 size violin used in the coupled FEM-BEM simulations.
In the text

	Figure 2 Violin cavity BEM mesh used for the coupled simulations.
In the text

	Figure 3 Smallest (70% of original size) and largest (110% of original size) pair of f-holes on the fractionalsize violin model.
In the text

	Figure 4 Simulated sound power curves over increasing f-hole size based on a 1 N input force.
In the text

	Figure 5 Simulated sound pressure level in the plane directly above the top plate at the A0 resonance peak for the six different f-hole sizes.
In the text

	Figure 6 Simulated sound pressure level in the plane directly above the top plate at the CBR resonance peak for the six different f-hole sizes.
In the text

	Figure 7 Simulated sound pressure level in the plane directly above the top plate at the B1-resonance peak for the six different f-hole sizes.
In the text

	Figure 8 Simulated sound pressure level in the plane directly above the top plate at the B1+ resonance peak for the six different f-hole sizes.
In the text

	Figure 9 Violin radiativity measurement set-up.
In the text

	Figure 10 Radiativity curves for the simulated and measured violin.
In the text

	Figure 11 Comparison of the violin with its original f-hole shape and after the f-hole enlargement.
In the text

	Figure 12 Measured radiativity curves between original and enlarged f-hole shape without the bridge and strings at tension.
In the text

	Figure 13 Measured radiativity curves between original and enlarged f-hole shape including the strings and bridge.
In the text