Engineering and the Guitar (Part 2)
In this series, we’ll try to de-mystify some engineering terms by using something familiar to many of us: the acoustic guitar.
We need to use care when discussing any engineering term, to make sure the reference is not ambiguous. It’s all part of being a good observer. For instance, take the word amplitude. It could mean the distance described by the string’s side-to-side excursion (for the acoustic guitar, the maximum amplitude of excursion is a few millimeters). Or it could refer to sound volume.
We’ll use it to mean sound volume, which in this case is also related to the string excursion. When the string moves, it in turn moves the top of the guitar (in the case of the acoustic guitar). When the top vibrates in sympathy with the string, it moves the column of air inside the guitar and that in turn creates a sound pressure wave that ultimately reaches our eardrums. So the amplitude of the sound is directly related to how hard we pluck the string, but pluck the string too hard and it crashes against the fret board, severely altering the waveform and causing “buzzing”, or clipping.
If you cover the sound hole with a large, flat piece of rubber, you inhibit the air from moving inside the guitar and this prevents the guitar top from flexing, resulting in less amplitude. This effect is most noticeable at low frequencies.
Get a large sink stopper, not the kind that goes inside the sink drain, but the flat kind that goes over the whole drain. Make a clean stopper over the hole in the guitar. Now pluck the open low “E” string (the fattest one). As the string resonates, remove the sink stopper from the hole. The difference should be audible.
Amplitude is also related to a number of other things like type of wood, size and placement of opening in the top, type of string and age of the string. It’s pretty obvious why the acoustic guitar, with its hollow body and thin top, has a very different sound from an electric guitar, which has a solid body and relies on a magnetic pickup to detect the total string excursion. The electric guitar relies on an electronic amplifier, while the acoustic guitar gets its amplification from the motion of the guitar’s top and the resultant movement of a large column of air.
To see the guitar top’s influence, strike a tuning fork on a hard surface and listen to it ring. Now strike the tuning fork again, and lightly hold the base of the tuning fork directly on the guitar’s top. It sounds much louder, because the tuning fork isn’t just moving the air around its fork—it’s moving the whole column of air that is inside the guitar. The top acts as an acoustic amplifier.
An electric guitar has an advantage in this regard. For the acoustic guitar, the entire sound system is built into the guitar. But the electric guitar has a solid body, so there is no column of air to amplify the sound. Instead, the string vibration is picked up by a magnetic coil and sent to an amplifier where all sorts of digital signal alteration can take place. Unlike the acoustic guitar, the body of the electric guitar becomes only one small chain in the system.
Fig 1. While the acoustic guitar is the biggest part of the overall system, the electric guitar's transfer function comprises only a small part of the chain.
Sometimes you see an "electrified" classical guitar, and it sounds very different from an "electric" guitar. That's partly because the acoustic classical guitar has non-metallic strings, and must rely on an acoustic pickup, not a magnetic pickup.
Strain, Tensile Strength
Have you ever considered how difficult it is to construct a good acoustic guitar? Each string is under many kilograms of tension, and an ordinary guitar has six strings. The guitar top (the part with the circular hole) has to be very strong to withstand the string tension, yet very thin and light so that the sound will be amplified. Spruce and cedar are the woods of choice for guitar tops, since they have fine, straight grain and high tensile strength for their weight. If you've ever heard a "National" guitar, you immediately realize the impact that the body material can make on the sound produced. ( 'National' guitars have all-steel bodies. )
When you tune the guitar using the pegs, you create a “moment arm” with the peg. Your hand and wrist create torque in the form of a “couple”, or rotational force about the peg-- and the string is stretched. You are putting stress on the string. For an idea of how much tension is involved, suspend an old, castoff guitar string from a strong closet rod and hang a sack of sand from the string. Keep adding sand until the plucked string is near its proper frequency. Then measure the weight of the sand on a bathroom scale. The number might surprise you.
If we add the string tension from all six strings together, we’ll have a good idea of the total force on the guitar’s bridge. Notice that some folk guitars—acoustic guitars with steel strings—don’t terminate the strings at the bridge. They use a yoke that extends past the bridge to pull against the guitar’s side. That gives added strength to the guitar and lets the bridge serve only to transfer acoustic energy to the guitar top. The bridge doesn’t have to support the string tension, except in a vector that points almost perpendicular to the guitar top.
Stress and Strain
The amount of stretch relative to the string’s length is called strain, and it is an indication of the stress inside the string. If you put tension on something (pull it), it gets longer and skinnier. What we as engineers really want to know is the stress (force/unit area) that’s being put on the string when we pull it. But since we can’t go inside the string to measure its stress, we do the next best thing: measure the elongation and hope that it's directly related to stress.
Fortunately for us, within a certain tension range the percentage of elongation (strain) is related to stress by a constant, for most materials. As long as the string is not close to its breaking point, we can measure the elongation and have a good idea of the actual stress inside the string.
When the string is tuned, the stress due to tension causes it to get longer and thinner, much as our waist does when we stretch our hands high in the air. Since this is an important phenomenon, there is a special number assigned to the ratio between the elongated “waist” and the “waist” at rest. It’s called the Poisson Ratio.
Engineers measure strain via a "strain gage". (Note that in the specific instance of strain, mechanical engineers spell it a 'gage', not 'gauge'.) A strain gage is usually just a deposited film resistor on a substrate that is glued to specimen to be measured. Since the specimen (airplane wing, I-beam, etc.) stretches under tension, the resistor does also. Resistance is a function of length divided by cross-sectional area, and since the resistor's length increases with tension and its area gets smaller with tension, the resistance increases proportionally with tension. Measure the resistance, and you know the strain. Then you can infer stress from the strain measurement. It's a pretty indirect way to measure stress, but it's simple and straightforward. The hardest part of the process is getting the gage to stick to the metal so that it gives reliable measurements over a long period of time. To put it in perspective, preparing the surface and gluing a single strain gage to an airframe can take up to one whole day per gage.
Fig 2. This strain gage is a resistor with a typical resistance of 100 ohms. The resistance increases as the gage is stretched. Reading the resistor's value with an ohmmeter gives us an indication of the stress inside the test specimen.
Stress theory is taught in a course typically called “Strength of Materials”. One concept taught in this course is that materials, especially metals, exhibit a linear region where stress is indeed related to strain by a constant. That constant is called “Young’s Modulus”, and is the slope of the curve below.
It is possible for the guitar string to pass its elastic limit, much like pulling taffy. If the string doesn’t recover its original tension at rest, it has stretched and must be re-tuned. The musician wants the guitar string to act in its linear region so that when the tuning peg is turned the vibration frequency (tone) goes up or down proportionately. If the string is in its plastic region, as strings often are when they are new, then this relationship is nonlinear; in other words, Young's Modulus is no longer a constant. If a guitar with new strings is played vigorously, the strings will stretch past their linear range and the guitar will require re-tuning.
Fig 3. Linear Region: string stretches, then recovers. Plastic Region: string stretches permanently; time to re-tune before you sound bad. Failure: string breaks, concert's over, crowd demands refund; groupies leave, career ends.
In the design of aircraft, the study of stress is one of the single most important enabling factors. Think about how important it is for every single part of an aircraft to be designed for minimum weight without sacrificing strength. Stress analysis is the science that makes this possible. The same principles that apply to good airframe design also apply to the guitar.
If you use a dentist’s mirror to look inside the sound box of the guitar, you’ll see all kinds of ingenious bracing, all designed to create the “best” sound reproduction. Building a guitar is similar to building an airframe: In order for the top to be most responsive, the bracing has to be minimal and it has to allow maximum energy transfer without degrading the acoustic performance of the guitar.
The next time you hold a guitar or tune it, think about the stresses involved. Why does it go out of tune when left alone in the case for too long? What effect does temperature have on the tuning? Why are older strings not as "bright" as new strings? How loud is an electric guitar when you play it without an amplifier? Why do guitarists take all the tension off the strings when they put an expensive guitar back in its case? What would happen if electric guitar strings were made of copper instead of steel?