Waves on a Guitar String Caught on Camera

My students saw a video where a cell-phone was place inside an acoustic guitar with the camera facing up, and the strings plucked. The strings appeared to 'vibrate' in wave-shapes, slow sinusoidal for the lower strings and almost-sawtooth for the upper ones. Is this real, or just an illusion created buy the camera's recording app's parameters?
- David Schmidt (age 59)
Bangkok, Thailand

Both! The sinusoidal vibrations are real, and an illusion created by the camera allows you to see them.

The strings on a guitar create sound by vibrating, and each string vibrates at different frequencies to produce a different pitch. (The strings also transfer their vibration to the body of the guitar and the air inside the guitar, which is important for making an audible sound.) There are three factors which control the pitch of a string:

1. The length of the string. Longer strings can vibrate at lower frequencies than shorter strings.

2. The weight of the string (per unit length). Heavier strings vibrate at lower frequencies than lighter strings.

3. The tension on the string. This is why you can tune the guitar by turning a knob to tighten the string—tighter strings produce higher frequencies than looser strings.

The lowest frequency a string can vibrate at is called the fundamental frequency. This is the main pitch that you hear. When a string is vibrating at the fundamental frequency, the whole string moves back and forth, like a jump rope moving straight up and down. Guitar strings can also vibrate at higher frequencies than the fundamental frequency, and these are called higher harmonics or overtones. At higher harmonics the string is "wiggling" in a sine wave shape. There are some nice pictures on this site: http://www.bsharp.org/physics/guitar

When a guitar is played, many of these modes of vibration are happening all at once, which creates the rich sound of the guitar. The different harmonics combine to determine the exact shape of the total vibration of the string, and could create something like a sawtooth wave. The shape of the vibration also depends on how and where the string is plucked.

The reason these vibrations aren't obvious to the naked eye is because they happen too fast to see. For example, the fundamental frequency of the A string is at 110 Hz, so the string is vibrating back and forth more than one hundred times a second. The higher harmonics vibrate even faster.

A cell phone camera (or any video camera) can make these rapid oscillations visible because it records images at a certain frame rate. 30 frames per second is a typical frame rate for a phone camera—that means the camera is taking a picture 30 times per second. If you're recording the motion of a guitar string, the camera gets a snapshot of the string's position 30 times a second. If the string is vibrating at a frequency which is a multiple of 30 Hz, it will always come back to the same position at the moment the camera captures a frame, so in the video recording it looks frozen in place. If the frequency is close to a multiple of 30 Hz but not quite, the string will appear to oscillate slowly as it gets caught in a slightly different position in each frame. You can also see this effect with your eyes if you use a strobe light to illuminate the strings. The flashing light gives you a snapshot of the string position at regular intervals, similar to the camera's frame rate.

Rebecca H.

(published on 08/28/2016)