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Engineering and the Guitar (Part I)

In this series, we’ll try to de-mystify some engineering terms by using something familiar to many of us: the acoustic guitar.

Observing

Have you ever been on a hike, just walking along and daydreaming, when your partner points out an eagle soaring in the sky? Suddenly your attention raises skyward, and you begin to see eagles and hawks and other birds as well. It’s not that they haven’t been there, and it’s not as if your hiking partner is better at spotting wildlife than you are (no duh…he couldn’t even find the car keys.); it’s simply that you weren’t looking.

One thing you learn in engineering school is to begin looking—looking at anything—bridges, ships, video games, cars and radio antennas. As engineers, we look at things through hypercritical eyes, always asking ourselves, “How did they do that?” or “What a crummy design! I know I can do better.” We can learn a lot about engineering principles simply by looking around, sometimes right in our midst. The guitar is a great example.

Study the guitar and watch the musician who plays it. The guitar touches on a rich set of engineering principles, among them: resonant frequency, period, amplitude, distortion, harmonics, wavelength, stress & strain, elastic limit, am, fm, damping coefficient, Doppler effect, step response, coupled oscillations, fft’s and signal processing. Let’s play around with some of these and see what we can learn from the simple acoustic guitar.

Engineering Tip: OBSERVE. THEN OBSERVE AGAIN. QUESTION EVERYTHING. LOOK AT WAYS TO MAKE EVERYTHING BETTER…IT’S THE ONE MAJOR TRAIT OF SUCCESSFUL INVENTORS.

Resonant Frequency

Stuff resonates. Virtually everything, from skateboards to the molecules that make up our bodies to the geologic plates under our feet, has some favorite resonant frequency. Pluck a guitar string, and the string moves back and forth at a predetermined resonant frequency. It’s always the same—every single time, unless the string’s length or tension changes.

The string is a spring. It is under tension, and that tensile force acts in line with the longitudinal axis of the string. When you crank on the tuning peg, you tune the guitar by increasing or decreasing the string tension. If you wail too hard on the guitar, you can de-tune it by stretching the strings beyond their elastic limit. Strings can also change tension when they relax with age or temperature.

When the string is plucked, it has two “nodes”, one on either end (the nut and the bridge). At a node, the string is captured and not allowed to move. The string naturally moves the most in the point at the center of these two nodes:

Engineering and the Guitar Part I Figure 1

Fig 1. The string is held captive at two nodes: the “nut” (to the left) and the “bridge” (to the right). As shown here, the open “A” string vibrates when plucked. If the guitar is perfectly tuned, the “A” string will vibrate at 440Hz.

If you observe the guitar strings, you’ll notice they all have different diameters. They get progressively heavier, starting with the high “E” string (the smallest) and moving all the way to the low “E” string, the one with the most mass. That leads us to take a guess at the string model.

Engineering Models

Any time you want to study an engineering phenomenon, you need a model of some kind. If you follow the procedures learned in a Statics class for example, you might draw a diagram to represent the guitar string as a device held captive between two stationary reference “nodes”: (Fig 2)

Engineering and the Guitar Part 1 Figure 2

Fig 2. Two diagrams for a guitar string held captive by the nut and the bridge. The guitarist initiates the vibration by plucking the string. When s/he lets go, the string eventually reaches its normal mode of vibration. (Note: If it appears to you that the two reference points above are not vertical, it’s an optical illusion.)

This is just a first attempt at a model. Of course in a real guitar, the bridge isn’t perfectly still. It moves slightly as the guitar is played (as the string tension increases and decreases). In the “ideal” case shown in the diagram, the endpoints don’t move. But that’s an engineering approximation. It’s virtually impossible to create such reference points in real life. This isn’t just true of mechanical devices. The same problem happens when we try to model a circuit in electrical engineering.

For example, the term “ground” (or ‘low’ or ‘common’) is used by electrical engineers to denote a point of zero voltage. But if there is any current in a circuit, and any resistance in the wire, there must be some voltage drop. Hence, no two places in the “ground” wire can both be at “zero” volts. Grizzled old instrument engineers are fond of saying, “There’s no such thing as ground.”

That’s not to say that engineering models don’t work—we just need to make sure that the approximation is close enough to real life to make the analysis meaningful. We could try to refine our model by using the observation that we made about the strings. Since the larger strings vibrate more slowly, having mass must have something to do with the vibration frequency. And we know that the string acts in tension, somewhat like a spring. So let’s try another, more refined model:

Engineering and the Guitar Part I Figure 3

Fig 3. The guitar string is being modeled as a set of springs, with a mass in the middle. As the mass of the string increases, the vibration frequency gets lower.

This is a closer approximation, but it’s not perfect either. We know that the string’s weight isn’t all at one point—it is distributed throughout the string. We could get fancy and model the string as a whole series of springs and a whole series of weights. That’s what’s known as a distributed system. But that’s enough for now. Let’s talk about the “signal” that the guitar emits.

Engineering Tip: THE ANALYSIS IS ONLY AS GOOD AS THE MODEL. TRY TO MAKE THE MOST REALISTIC MODEL YOU CAN, BUT LEARN TO RECOGNIZE THE PART THAT CAN BE IGNORED. SUCCESSFUL ENGINEERS DON’T USUALLY HAVE TIME TO SPEND YEARS TRYING TO MAKE THE MODEL PERFECT—THEY MAKE THE MODEL AS CLOSE TO REALITY AS POSSIBLE, THEN GET ON WITH THE DESIGN.

Frequency Components

Pluck the “A” string on a guitar and listen to the sound. Is it a pure tone? If you measure the sound with a microphone and display it on a spectrum analyzer or an oscilloscope, you’ll find that the guitar puts out a lot more than just one pure 440 Hz tone. The signal is full of other sounds, or frequency components. That’s because the guitar’s string transfers energy to the whole guitar “system”. First, it moves the bridge and that moves the guitar’s top. That’s where most of the energy goes. But the top isn’t the only part of the guitar that affects its sound pattern. When the top moves, it causes the other strings also vibrate slightly, each at their own natural resonant frequency (a coupled oscillation). And the guitar’s top, back and sides vibrate and interact to add to the mixture. These other sound sources are smaller in amplitude than the 440Hz main signal, but they are responsible for an acoustic guitar’s “rich” sound. A guitar that produced only perfect, pure sine waves would sound pretty boring.

Engineering and the Guitar Part 1 Figure 4

Fig 4. A tuning fork signal converted by a microphone into a voltage waveform would look something like this on an oscilloscope. If the fork is tuned to an “A” note, the oscilloscope displays a signal frequency of 440 Hz, the period of which is: T= 1/440 = 2.273 ms

In Fig. 4, the frequency is 440 Hz. In other words, the string completes a vibration (one side to the other and back) 440 times/second. The period is 1/f = 1/440th second, or 2.273 milliseconds. If we called the magnitude of the peak excursion (at the top of the sine wave) X, the mathematical description would be: v(t) = X* sin 2 pi * 440 t

Changing Frequencies

You don’t have to be an engineer to figure out that if you push the string all the way down to the fretboard just behind any one of the frets, and pluck the string, you can play a note. Do this along the entire fretboard, and you can play the whole scale. Moving the point obviously changes the vibration frequency of the string. * So it’s more than just string tension that changes the sound—it also has to do with the length of the string.

An expert guitarist who plays the scale over the full range of the fretboard can detect changes in loudness (amplitude). One challenge for the master guitar builder is to make a guitar that is equally “bright” over its entire range of notes, with no dead spots or unusually loud notes.

Engineering and the Guitar Part 1 Figure 5

Fig. 5. Unlike an electric guitar, the acoustic guitar itself comprises most of the sound "system". It is difficult to change what the audience hears, since the guitar body and strings determine the characteristics of the sound.

The guitar itself acts as a whole sound system. It has a frequency response just as a “boom box” does. The guitar can only remain “true” to a limited range of frequencies, and it is optimized for the frequencies produced by the strings. The guitar acts as a bandpass filter with a transfer function that passes unequal amounts of energy to different parts of the frequency spectrum. The chores of amplification, filtering, signal generation and pitch control all lie within the instrument itself.

A Zillion Variables

It doesn’t take much imagination to see that there are all kinds of variables in this system: sound box air volume, hole size, top thickness, wood type, humidity, bracing, crowd size, room size, floor covering, hand position and more. Each one of these variables could be analyzed by an engineer to help optimize the audience’s experience. Maybe you can think of ways to improve the “system”.

Part I
©2001
Keysight Technologies

*Note: What engineers call "frequency", musicians call "pitch".