## Demonstration Highlight: Guitar & Oscilloscope

Our Guitar and Oscilloscope demonstration is a fun hands-on way to show a visualization of a waveform while students also hear it. It illustrates how waves on a string become sound waves in air, and how the decay times of different components of a complex waveform affect its sound over time. You can see it in action in our new demonstration video, starring physics student Alana Dixon.

A guitar produces sound from the vibration of the strings. When you strum or pluck a string, the string vibrates. The frequency of this vibration is determined by the string’s length, its tension, and its weight. The body of the guitar, and the air chamber within, can couple and resonate with these vibrations. The energy therein is passed to the air, creating the sound waves we hear across the room. A pickup has been attached to the guitar; this pickup uses magnetic induction with the strings to detect the vibrations and transform them into an electrical signal. This is then amplified by the amplifier and displayed by the oscilloscope. An oscilloscope displays a changing electrical voltage as a moving point on a graph. It allows us to display visually how the signal changes over time.

When we pluck a string, we can see the resulting sound wave reflected as a waveform trace on the oscilloscope. There is a slight difference, though: the pickup is showing the vibrations from the string, and the vibration of the body as well since it is connected to the body, but this is not always exactly identical to the wave as transmitted through the air! The particular shape of the guitar body and sound hole can emphasize slightly different elements of the sound as they couple with the outside air. But for the purposes of our experiments, it’s close enough.

The frequency produced by a vibrating string is determined by three factors: the length of the string, the tension in the string, and the linear density of the string. These define boundary conditions for the waveforms. Guitar strings have different linear densities (the weight per unit length) to help them produce a wider range of sounds. Then we adjust the tension in each string to tune the guitar to the exact frequencies we want. You’ll notice, though, that the waveforms we see here are not simple sine waves. The sound of a guitar is a very complex waveform that changes over time. The complexity of the waveform is in part due to the design of the instrument. There are several modes of coupling within the guitar, as energy passes from the vibrating strings to the surface of the guitar (the soundboard), from there to the air inside the body, and between the body and the outside air. And the complex shape of the instrument creates multiple possible resonances at various frequencies. All of these can reinforce certain harmonics of the fundamental wave, and these components add up to form the complex waveform we see on the oscilloscope screen. Additionally, the different components of the wave last different amounts of time. After the string is plucked, its vibration slowly dies down, but the vibrations it has set up within the instrument also last for different amounts of time – each component has its own decay time. As these components change in amplitude, the shape of the overall waveform changes, and that gives the guitar its complex sound and varying sound.

To understand this better, we can examine some simulations of how a string responds to being plucked, and its behaviour over time.

This simulation from Falstad http://www.falstad.com/loadedstring/ lets you pluck a string and see how the resulting wave in the string gradually decays. It will display graphs of both the amplitude and phases of various harmonics that make up the wave. Try adjusting the damping to see how that changes the decay over time!

If you’d like to learn more, check out this breakdown https://www.acs.psu.edu/drussell/Demos/Pluck-Fourier/Pluck-Fourier.html  by Dr. Daniel Russell of Penn State with both graphical and mathematical treatments of the initial conditions of a plucked string and its evolution over time.

Fred W. Inman. A Standing-Wave Experiment with a Guitar

The Physics Teacher 44, 465 (2006); https://doi.org/10.1119/1.2353595

Michael C. LoPresto. Experimenting with Guitar Strings

The Physics Teacher 44, 509 (2006); https://doi.org/10.1119/1.2362942

Polievkt Perov, Walter Johnson and Nataliia Perova-Mello. The physics of guitar string vibrations

American Journal of Physics 84, 38 (2016); https://doi.org/10.1119/1.4935088

Michael Sobel. Teaching Resonance and Harmonics with Guitar and Piano

The Physics Teacher 52, 80 (2014); https://doi.org/10.1119/1.4862108

Scott B. Whitfield and Kurt B. Flesch. An experimental analysis of a vibrating guitar string using high-speed photography

American Journal of Physics 82, 102 (2014); https://doi.org/10.1119/1.4832195