Advances in Acoustics and Vibration

Volume 2016 (2016), Article ID 6084230, 10 pages

http://dx.doi.org/10.1155/2016/6084230

## Mathematical Modelling and Acoustical Analysis of Classical Guitars and Their Soundboards

School of Engineering, Taylor’s University, No. 1 Jalan Taylor’s, 47500 Subang Jaya, Selangor, Malaysia

Received 8 September 2016; Revised 14 November 2016; Accepted 1 December 2016

Academic Editor: Kim M. Liew

Copyright © 2016 Meng Koon Lee et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Research has shown that the soundboard plays an increasingly important role compared to the sound hole, back plate, and the bridge at high frequencies. The frequency spectrum of investigation can be extended to 5 kHz. Design of bracings and their placements on the soundboard increase its structural stiffness as well as redistributing its deflection to nonbraced regions and affecting its loudness as well as its response at low and high frequencies. This paper attempts to present a review of the current state of the art in guitar research and to propose viable alternatives that will ultimately result in a louder and better sounding instrument. Current research is an attempt to increase the sound level with bracing designs and their placements, control of natural frequencies using scalloped braces, as well as improve the acoustic radiation of this instrument at higher frequencies by deliberately inducing asymmetric modes in the soundboard using the concept of “splitting board.” Various mathematical methods are available for analysing the soundboard based on the theory of thin plates. Discrete models of the instrument up to 4 degrees of freedom are also presented. Results from finite element analysis can be utilized for the evaluation of acoustic radiation.

#### 1. Introduction

Classical guitars are unique musical instruments as the acoustic response of each piece of a particular model is different from another one although they are dimensionally identical and are all made by the same luthier according to French [1]. Two reasons given for this lack of acoustic consistency are firstly the variations in the natural properties of wood and secondly the manual tuning process of the soundboard by experienced luthiers which is not well understood analytically. Borland [2] has determined that humidity of air and moisture content in the wood are important factors affecting how wood responds when it vibrates.

Technically, classical guitars have been modelled and analysed by using several mathematical models. These models were used for determining modal frequencies and frequency response function. Using these results, classical guitars can be objectively assessed by evaluating their acoustic radiation.

Throughout the evolution of the classical guitar since 1500 AD, it is generally agreed among luthiers that the type of wood, the design and placement of bracings on the soundboard, and the reinforcement of the back plate play important roles in the production of a good acoustically radiating instrument. This consensus among luthiers creates an aura of mysticism that surrounds the construction of the guitars and translates into an enormous respect for top quality concert instruments. This intuitive analysis of the luthier can be reinforced by scientific knowledge through collaboration with research scientists in the fields of Mechanics of Solids and Continuum Mechanics. Such collaboration creates an interdisciplinary research in the field of musical acoustics such as that existing at the research centre of Universitat Politècnica de Catalunya (BarcelonaTech). The research here centres on a combination of experimental and numerical research and the experience of a well-known luthier [3].

Studies by Richardson et al. [4], Siminoff [5], and Bader [6] show the relatively greater importance of acoustic radiation from the soundboard as compared to those from the back plate and the bridge at high frequencies. Based on these findings, research on increasing the loudness of the guitar by focusing on the soundboard alone is a potential in future research.

The objectives of this paper are categorically summarised under the following sections: Section 2: Mathematical Models Section 3: Acoustical AnalysisThe above two categories define the scope of review in this paper.

#### 2. Mathematical Models

Richardson et al. [4] and Siminoff [5] have shown that the soundboard is the single most important component affecting the sound pressure level of the classical guitar. Factors that affect the performance of vibrating soundboards in terms of acoustic radiation are design, placement and arrangement of bracings, and thickness. These factors contribute to the musical acoustics of the classical guitar.

An in-depth understanding of the dynamic characteristics of the classical guitar can be obtained by considering some mathematical models. In particular, the simplest two-mass model of Christensen-Vistisen [7] provides a simple understanding of the interaction between a vibrating air mass and the soundboard. This model and the three-mass and four-mass models are three classical examples of discrete mathematical models of this instrument. These models are based on the mass-spring-damper mechanism. As Richardson et al. [4] and Siminoff [5] have shown that the soundboard is the single most important component affecting the sound pressure level of the classical guitar, it can therefore be modelled separately as a vibrating thin plate and the theory of thin plates can be applied to study its dynamic behaviour with the aim of improving its contribution to the sound pressure level of this instrument. This component, complete with design and arrangement of fan strutting, can then be assembled with the ribs and back plate to study the effects on modes and natural frequencies due to noncoupling between the soundboard and back plate versus coupling between these two components via the air mass inside the guitar body as was carried out by Elejabarrieta et al. [8].

##### 2.1. Discrete and Continuous Systems

###### 2.1.1. Discrete Systems

Christensen and Vistisen [7] proposed a simple 2-degree-of-freedom model, also known as the “Christensen-Vistisen” lumped parameter model. This model consisted of an air piston and the soundboard. Christensen [9] proposed a 3rd degree of freedom in the form of a back plate while Popp [10] proposed yet a 4th degree of freedom in the form of ribs. These are the 3- and 4-degree-of-freedom models, respectively. These additional degrees of freedom provided more realistic representations of the guitar. The range of frequencies investigated was 80–250 Hz.

A hybrid mechanical-acoustic model proposed by Sali and Hindryckx [11] and Sali [12] was used to investigate the changes in loudness relative to the first peak (the first resonance) of the complete instrument. This model consisted of a mass, spring, damper, and a massless membrane rigidly attached to the mass. The membrane had a constant area equivalent to that of the radiating surface.

###### 2.1.2. Continuous Systems

As the number of degrees of freedom increases, modelling using discrete masses becomes cumbersome. To circumvent this problem, the complete instrument can be considered as a continuous system and its vibration characteristics can be effectively analysed using the finite element method as shown by Derveaux et al. [13] and Gorrostieta-Hurtado et al. [14]. The soundboard can also be shown to satisfy the criterion of a thin plate in flexure and the application of the theory of thin plates results in a fourth-order partial differential equation of the vibrating system. Attempts to solve this model analytically can be researched using current mathematical methods.

##### 2.2. Vibration of the Complete Instrument

Analytical models with 2-, 3-, and 4-degree-of-freedom have been formulated by Christensen and Vistisen [7], Christensen [9], and Popp [10], respectively, and are applicable to the complete instrument. Modal analysis of the 2- and 3-degree-of-freedom models by Caldersmith [15] and Richardson et al. [4], respectively, predicted two and three eigenvalues in the frequency range from 80 to 250 Hz. Similarly, modal analysis of the 4-degree-of-freedom model also predicted three eigenvalues in the frequency range of 80 to 250 Hz but a fourth eigenvalue was missing. It was concluded by Popp [10] that assigning a fourth-degree-of-freedom model in the form of a finite mass to the ribs does not introduce any new elastic restoring force and hence there is no fourth eigenvalue. Hence adding extra degrees of freedom beyond the fourth in this method of modelling the classical guitar would add unnecessary complications as even with the 4-degree-of-freedom model; the mechanics of the complete guitar body cannot be adequately represented [4].

Hess [16] conducted a parametric study with the two-mass model to identify a unique combination of physical parameters in an attempt to increase the sound level over a frequency range of 70 Hz to 250 Hz. Results showed that, by decreasing the stiffness and effective mass of the soundboard by 50% and decreasing the soundboard area by 28%, there was an increase of 3.2 dB in the sound pressure level per unit force over the entire frequency range. Although this investigation was performed on an acoustic guitar, there is no indication that these parameters could not be used to examine their influence on classical guitar soundboards.

The range of frequencies of a classical guitar investigated by Czajkowska [17] varies from 70 Hz to just under 2 kHz. However, there are also harmonic notes that the classical guitar can produce. To account for these higher frequency notes, Richardson [18] suggested that the range of frequencies is extended to 20 kHz, which is the upper threshold of human hearing. However, from ISO 226:2003, the minimum sound pressure level for any arbitrary loudness occurs within a bandwidth of 3 to 4 kHz. For practical purposes, experiments could be conducted up to 5 kHz.

Sakurai [19], a luthier, made some interesting video recordings of the vibration of the soundboard. He experimented with the traditional bracing structure and with diagonal braces and discovered that the soundboard could be made thinner and could vibrate with larger amplitudes without compromising on its structural integrity. However, there was no accompanying mathematical analysis.

Richardson et al. [4], in their revisitation of the 3-mass model, are of the opinion that the properties of the soundboard together with the design of bracings and the bridge play more important roles than results from the 3-mass model in relation to the fundamental top plate mode. They further concluded that low-order modes have significant controlling influence on the playing qualities of the guitar. This conclusion was based on informal listening tests. Studies by Richardson et al. [20] have shown that the noise components generated by these low-order modes are an important perceptual element in guitar sounds as perception is regarded as an important element in music.

###### 2.2.1. Two-Degree-of-Freedom Model

The simplest model consists of two masses representing the soundboard and an air piston as proposed by Christensen and Vistisen [7]. Hess [16] has shown that this model gives good agreement between theoretical and experimental results for sound pressure and acceleration frequency response at low frequencies (80 to 250 Hz) as shown in Figures 1 and 2.