## The Physics of Guitar Strings

Friends, regular readers may be aware that I play the guitar.

And sleuths amongst you may have deduced that if I play the guitar, then I must occasionally change guitar strings.

The physics of tuning a guitar – the extreme stretching of strings of different diameters – has fascinated me for years, but it is only now that my life has become pointless that I have managed to devote some time to investigating the phenomenon.

What I have found is that the design of guitar strings is extraordinarily clever, leaving me in awe of the companies which make them. They are everyday wonders!

Before I get going, I just want to mention that this article will be a little arcane and a bit physics-y. Apologies.

What’s there to investigate?

The first question is why are guitar strings made the way they are? Some are plain ‘rustless steel’, but others have a steel core around which are wound super-fine wires of another metal, commonly phosphor-bronze.

The second question concerns the behaviour of the thinnest string: when tuned initially, it spontaneously de-tunes, sometimes very significantly, and it does this for several hours after initially being stretched.

Click image for a larger version. The structure of a wire-wound guitar string. Usually the thickest four strings on a guitar are made this way.

Of course, I write these questions out now like they were in my mind when I started! But no, that’s just a narrative style.

When I started my investigation I just wanted to see if I understood what was happening. So I just started measuring things!

Remember, “two weeks in the laboratory can save a whole afternoon in the library“.

Basic Measurements

The frequency with which a guitar string vibrates is related to three things:

• The length of the string: the longer the string, the lower the frequency of vibration.
• The tension in the string: for a given length, the tighter the string, the higher the frequency of vibration.
• The mass per unit length of the string: for a given length and tension, heavier strings lower the frequency of vibration

In these experiments, I can’t measure the tension in the string directly. But I can measure all the other properties:

• The length of string (~640 mm) can be measured with a tape measure with an uncertainty of 1 mm
• The frequency of vibration (82 Hz to 330 Hz) can be measured by tuning a guitar against a tuner with an uncertainty of around 0.1 Hz
• The mass per unit length can be determined by cutting a length of the string and measuring its length (with a tape measure) and its mass with a sensitive scale. So-called Jewellery scales can be bought for £30 which will weigh 50 g to the nearest milligram!

Click image for a larger version. Using a jewellery scale to weigh a pre-measured length of guitar string. Notice the extraordinary resolution of 1 mg. Typically repeatability was at the level of ± 1 mg.

These measurements are enough to calculate the tension in the string, but I am also interested in the stress to which the material of the string is subjected.

The stress in the string is defined as the tension in the string divided by the cross-sectional area of the string. I can work out the area by measuring the diameter of string with  digital callipers. These devices can be bought for under £20 and will measure string diameters with an uncertainty of 0.01 mm.

However there is a subtlety when it comes to working out the stress in the wire-wound strings. The stress is not carried across the whole cross-section of the wire, but only along its core – so one must measure the diameter of the core of these strings (which can be done at the ends) but not the diameter of the playing length.

Click image for a larger version. Formulas relating the tension T in a string (measured in newtons); the frequency of vibration f (measured in hertz); the length of the string L (measured in metres) and the mass per unit length of the string (m/l) measured in kilograms per metre.

Results#1: Tension

The graph below shows the calculated tension in the strings in their standard tuning.

Click image for a larger version. The calculated tension in the 6 strings of a steel-strung guitar. String 1 is the thinnest (E4) and String 6 is the thickest (E2).

The tension is high – roughly 120 newtons per string. If the tension were maintained by a weight stretching the string over a wheel, there would be roughly 12 kg on each string!

Note that the tension is reasonably uniform across the neck of the guitar. This is important. If it were not so, the tension in the strings would tend to bend the neck of the guitar.

Results#2: Core Diameter

The graph below shows the measured diameters of each of the strings.

Click image for a larger version. The measured diameter (mm) of the stainless steel core of the 6 strings of a steel-strung guitar and the diameter of the string including the winding. String 1 is the thinnest (E4) and String 6 is the thickest (E2).

First the obvious. The diameter of the second string is 33% larger than the first string, which increases its mass per unit length and causes the second string to vibrate at a frequency 33% lower than the first string. This is just the basic physics.

Now we get into the subtlety.

The core of the third string is smaller than the second string. And the core diameters of each the heavier strings is just a little larger than the preceding string.

But these changes in core diameter are small compared to changes in the diameter of the wound string.

The density of the phosphor bronze winding (~8,800 kg/m^3) is similar to, but actually around 10% higher than, the density of the stainless steel core (~8,000 kg/m^3). This is not a big difference.

If we simply take the ratios of outer diameters of the top and bottom strings (1.34/0.31 ≈ 4.3) this is sufficient to explain the required two octave (factor 4) change in frequency.

Results#3: Why are some strings wire-wound?

The reason that the thicker strings on a guitar are wire-wound can be appreciated if one imagines the alternative.

A piece of stainless steel 1.34 mm in diameter is not ‘a string’, it’s a rod. Think about the properties of the wire used to make a paperclip.

So although one could attach such a solid rod to a guitar, and although it would vibrate at the correct frequency, it would not move very much, and so could not be pressed against the frets, and would not give rise to a loud sound.

The purpose of using wire-wound strings is to increase their flexibility while maintaining a high mass per unit length.

Results#4: Stress?

The first thing I calculated was the tension in each string. By dividing that result by the cross-sectional area of each string I can calculate the stress in the wire.

But it’s important to realise that the tension is only carried within of the steel core of each string. The windings only provide mass-per-unit-length but add nothing to the resistance to stretching.

The stress has units of newtons per square metre (N/m^2) which in the SI has a special name: the pascal (Pa). The stresses in the strings are very high so the values are typically in the range of gigapascals (GPa).

Click image for a larger version. The estimated stress with the stainless-steel cores of the 6 strings of a steel-strung guitar. Notice that the first and third strings have considerably higher stress than the other strings. In fact the stress in these cores just exceeds the nominal yield stress of stainless steel. String 1 is the thinnest (E4) and String 6 is the thickest (E2).

This graph contains the seeds of an explanation for some of the tuning behaviour I have observed – that the first and third strings are tricky to tune.

With new strings one finds that – most particularly with the 1st string (the thinnest string) – one can tune the string precisely, but within seconds, the frequency of the string falls, and the string goes out of tune.

What is happening is a phenomenon called creep. The physics is immensely complex, but briefly, when a high stress is applied rapidly, the stress is not uniformly distributed within the microscopic grains that make up the metal.

To distribute the stress uniformly requires the motion of faults within the metal called dislocations. And these dislocations can only move at a finite rate. As the dislocations move they relieve the stress.

After many minutes and eventually hours, the dislocations are optimally distributed and the string becomes stable.

Results#5: Yield Stress

The yield stress of a metal is the stress beyond which the metal is no longer elastic i.e. after being exposed to stresses beyond the yield stress, the metal no longer returns to its prior shape when the stress is removed.

For strong steels, stretching beyond their yield stress will cause them to ‘neck’ and thin and rapidly fail. But stainless steel is not designed for strength – it is designed not to rust! And typically its yield curve is different.

Typically stainless steels have a smooth stress-strain curve, so being beyond the nominal yield stress does not imply imminent failure. It is because of this characteristic that the creep is not a sign of imminent failure. The ultimate tensile strength of stainless steel is much higher.

Results#6: Strain

Knowing the stress  to which the core of the wire is subjected, one can calculate the expected strain i.e. the fractional extension of the wire.

Click image for a larger version. The estimated strain of the 6 strings of a steel-strung guitar. Also shown on the right-hand axis is the actual extension in millimetres Notice that the first and third strings have considerably higher strain than the other strings. String 1 is the thinnest (E4) and String 6 is the thickest (E2).

The calculated fractional string extension (strain) ranges from about 0.4% to 0.8% and the actual string extension from 2.5 mm to 5 mm.

This is difficult to measure accurately, but I did make an attempt by attaching small piece of tape to the end of the old string as I removed it, and the end of the new string as I tightened it.

Click image for a larger version. Method for estimating the strain in a guitar string. A piece of tape is tied to the top of the old string while it is still tight. On loosening, the tape moves with the part of the string to which it is attached.

For the first string my estimate was between 6 mm and 7 mm of extension, so it seems that the calculations are a bit low, but in the right ball-park.

Summary

Please forgive me: I have rambled. But I think I have eventually got to a destination of sorts.

In summary, the design of guitar strings is clever. It balances:

• the tension in each string.
• the stress in the steel core of each string.
• the mass-per-unit length of each string.
• the flexibility of each string.

Starting with the thinnest string, these are typically available in a range from 0.23 mm to 0.4 mm. The thinnest strings are easy to bend, but reducing the diameter increase the stress in the wire and makes it more likely to break. They also tend to be less loud.

The second string is usually unwound like the first string but the stresses in the string are lower.

The thicker strings are usually wire-wound to increase the flexibility of the strings for a given tensioning mass-per-unit-length. If the strings were unwound they would be extremely inflexible, impossible to push against the frets, and would vibrate only with a very low amplitude.

How does the flexibility arise? When these wound strings are stretched, small gaps open up between the windings and allow the windings to slide past each other when the string is bent.

Click image for a larger version. Illustration of how stretching a wound string slightly separates the windings. This allows the wound components to slide past each other when the string is bent.

Additionally modern strings are often coated with a very thin layer of polymer which prevents rusting and probably reduces friction between the sliding coils.

Remaining Questions

I still have many  questions about guitar strings.

The first set of questions concerns the behaviour of nylon strings used on classical guitars. Nylon has a very different stress-strain curve from stainless steel, and the contrast in density between the core materials and the windings is much larger.

The second set of questions concerns the ageing of guitar strings. Old strings sound dull, and changing strings makes a big difference to the sound: the guitar sounds brighter and louder. But why? I have a couple of ideas, but none of them feel convincing at the moment.

And now I must get back to playing the guitar, which is quite a different matter. And sadly understanding the physics does not help at all!

P.S.

The strings I used were Elixir 12-53 with a Phosphor Bronze winding and the guitar is a Taylor model 114ce

### 2 Responses to “The Physics of Guitar Strings”

1. Nikola Stojanovic Says:

Dear Michael,
Thank you for the very nice article. As a fellow guitar player and a physicist I relate to this particular topic even more than your usual content. Out of curiosity, I would have few specific questions (maybe too specific for main audience):
1. What are gauge of the strings? In particular, high E, 1st string?
2. Are so-called plain strings (1st and 2nd on acoustic set) from stainless or from carbon steel?
I ask this because i noticed that on my acoustic guitar 3rd wound string is most prone to breaking, while bending. I use so-called 10’s and 3rd has thinnest core to the best of my knowledge. Of course, if you use different gauge, results would vary…
Stay healthy and be well.