The main parameters of tone wood that are directly related to the vibroacoustic behavior of the soundboard are the density, the stiffness, the radiation coefficient, and the damping. Let’s have a close look at each single one, and how to use this data to improve the sound and performances of the instruments we build.
Resonance frequency of the first longitudinal bending mode, number of vibrations per second.
The frequency at which the sample, excited by the energy supplied by the marble, will resonate at its first longitudinal bending mode. The stiffness of the sample will be calculated at this resonance frequency, that has a strong influence on the character of its tone.
Density, the mass of an object divided by its volume.
Density is defined as the ratio between the mass and volume of a substance, and the Greek letter rho is used to describe it. The calculation of the density of a tone wood sample is simple: once the length, height and width of a sample have been measured, the volume is derived; by using a balance it is measured, and dividing the weight by the volume the density value is obtained, usually expressed in kg per cubic meter.
The first general quality of tone wood is lightness, therefore a low-density tone wood for the soundboard is almost always preferred. The strings can supply a limited amount of energy to produce sound, so that reducing the equivalent mass of the soundboard is improving the responsiveness of the guitar, and the perceived volume of sound.
Stiffness, the resistance to deformation under the action of an external force.
The rigidity of a material is the measure of its elasticity, and defines its resistance to deformation under the action of an external force. For our purposes we are mainly interested in two types of moduli:
The Young's modulus (E) describes axial elasticity, the tendency of an object to deform along an axis when opposing forces are acting along a direction normal to the object’s faces; it is often referred to simply as the elastic modulus.
The shear modulus or modulus of rigidity (G ) describes an object's tendency to shear (the deformation of shape at constant volume) when opposing forces are acting along a direction parallel to the object’s faces.
The Young’s Modulus is calculated as the ratio between the stress (the force applied) and the axial strain (the deformation of the body): E=σ/ε
The guitar maker is usually looking for a high Young’s modulus tone wood, as a stiff soundboard allows to reduce the thickness while keeping similar resonance frequencies, obtaining a lower equivalent mass and a more responsive instrument.
Sound radiation coefficient, how much energy is needed to excite the plate in vibrations.
In search of a single parameter that could summarize the ability of tone wood to transform vibrational energy into sound, already in 1963 John C. Schelleng presented in a seminal paper the simplified circuit of the main resonances of the violin, and suggested to consider the squared ratio between the speed of sound in the material and its density as a generic indicator of quality for soundboards, expressed in formula as: √(E/ρ^3 )
The acoustic radiation coefficient is a useful indicator of how much energy is needed to excite the monopole in vibrations, both above and below its resonance frequency. It is proportional to the stiffness of the sample, and to the inverse of the cube of density through a square root operation. This means that the density assumes the most important weight in the expression, and that in order to have a high radiation coefficient we need to look for lightweight tone woods.
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