STRING TENSION:
\(F = 4L^2f^2m\)
Where F = force in N/m2, L = length (m), f = frequency (Hz), m mass/length (kg/m)
The equation tells that a 10% longer scale increases the tension by approximately 20% and the same can be said about a 10% increase in diameter of a steel string while keeping same tuning of string.
RESONANCE FREQUENCY AIR VOLUME (HELMHOLTZ FREQUENCY):
The Helmholtz frequency for a guitar which has a very stiff back and top can be estimated by the folowing equation. The value will be lower in reality depending on the stiffness of the top and back of the instrument.
\(f = c/(2{ π}) \sqrt{( π r^2/( 1.7 Vr))}\)
where f=Helmholtz frequency (Hz), c=speed of sound in air(approximately 340m/s), r = radius sound hole (m), V = volume guitar body
For a classical guitar you will find a calculated value around 130 Hz. The measured value is likely to be in the range 100 to 110 Hz or even lower. Reducing thicknesses of top, back and bracing reduce the Helmholtz frequency and the peak frequencies of top and back as well.
MECHANICS BRIDGE (STATIC AND DYNAMIC)
Simple mechanics can be applied to calculate the forces acting on the top of a guitar. Dynamic calculation are much more complicated. It shows however that a bridge acts as a filter depending on mass, stiffness and damping properties and it should be possible to model the transfer of energy from strings to top. A higher bridge may increase projection while reducing sustain at the same time.
DENSITY AND MASS
These are important parameters. Density, mass, stiffnes and damping are the main parameters to quantify a resonating system. To accelerate mass takes energy and more energy for higher frequencies. A light top is more responsive than a heavy top and is better in reproducing the higher tones. See under tonewoods for more explanation.
DYNAMIC ELASTICITY MODULUS AND RESONANCE FREQUENCY OF A FLAT PLATE
\(f = C \sqrt{((h^2E)/(dL^4))}\)
where L= length plate(m), h = thickness(m), d=density(, E=elasticity modulus, C = 1.02 for metric units
It's pretty easy to measure resonance frequencies and to calculate the modulus of elasticity. The values are dependent on grain direction. This equation is very useful when changing dimensions of an instrument or making adjustments for different tone-woods. The equation is very helpful in making quantitative judgements about tonewoods.
BRACE DIMENSIONS AND STIFFNESS
A first estimate is provided by the equation for deflection of a beam under static load W(kg)
\(y = KWL^3/(Elh^3)\)
where K=0.25 for rectangular cross section, E elasticity modulus, L=length(m), l =width beam(m), h=thickness(m)
The equation quantifies sensitivity of deflection for length and thickness parameters. High narrow braces make sense to achieve more stiffness from lightweight bracing.
PROPERTIES OF COMPOSITE BRACES
Reduced weight while maintaining stiffness can be achieved by combining light and stiff materials.
An interesting example is a surprisingly good sounding violin made out of balsa wood and graphite (see YouTube). Use of double tops and lattice bracing are other examples.