# A functional approach of the Mersenne’s law

Guitar strings is a huge and one of the first topics of interest for every guitarist. There is a vast amount of options in the market along with promises for their effectiveness.

Let’s see what happens, looking form a physics point of view.

A string has to produce an acoustic frequency. The mathematical equation that gives as the frequency is:

f = (k/2L) √(F/μ)

where:

• f →the product frequency
• k →is a series of harmonic frequencies, produced from the oscillating string. Because of the string having fixed edges the first wave reflects on those edges and travels back colliding with the next, creating new combined waves, we call harmonies (k=1, k=2, k=3etc)
• L →the active length of the string
• F→ the string’s tension
• μ→the linear, mass per unit length of the string

At first, we notice that we can’t see the width of the string anywhere in that equation. That is because the factor is hidden within the , which with simple terms, indicates how much of the material, the string is made of, exists within one meter of that string. We can clearly understand that due to the length being constant, the only way to increase the mass of the string is to increase the width.

Does that mean that two strings with the same length and width have the same behaviour on the instrument?

The answer is no. The only way to do so is to have the same mass per unit length, in other words, to be constructed exactly the same. Having said that, if we have for example two 0.42’’ strings, that doesn’t mean that we can achieve the same note with the same tension and that is because they may be built by different materials. By experimenting with different materials, we can alter the  μ factor by keeping the length and width constant. That’s the way the ‘‘low-tension’’ strings are made.

In simple terms.

• Increasing the length of a string we lower the product frequency and vice versa.
• Increasing the width of a string we lower the product frequency and vice versa.
• Increasing the tension of a string we increasing the product frequency and vice versa.
• Different materials differentiate the behaviour of the string.

So, we see that the width of astring, works implicitly to the final sound and that is why we find such a great variety in the string market. The quality of the string is related mostly, almost entirely, to the quality of the production. Namely, we want the best possible accuracy of the alloys used, at the dimensions, homogeneity etc.

Materials, Alloys and manufacturing methods.

A very critical factor, that affects the behaviour of a string is the alloy/alloys used, along with the manufacturing method. This factor determines the tone, the clarity, the sustain, as well as the resistance to corrosion from the acidity of the sweat and fat from our fingers, along with the wearing caused by contacting/hitting the frets. There is also a great effort to produce fewer toxic alloys for the strings to enhance their anti-allergic behaviour. There is, yet another time, a large variety in the market. The factor is a large area of experimentation; therefore, a huge amount of research takes place every day by the string manufacturers in the core of the string, on the material of the core and the winding, the shape of the core etc. We can see many different construction methods, coatings and materials for the core and the windings, such as metallic alloys (steel, brass etc), composite strains (carbon, Kevlar, fibre-glass) and a variety of anticorrosion/anti-allergic coatings.