The physics of the Les Paul Google Doodle
In honor on of the 96th birthday of Les Paul, inventor of the electric guitiar, Google created this great doodle that lets you actually experiment and play the guitar.
I’m going to try to pull a Rhett Allain and see what I can figure out about the physics of this new Google guitar.
Finding the notes on the Google guitar
First, can I figure out the notes that are being played? This seems like something Tracker video analysis might be able to help with. I recorded a video of the doodle playing a single note using Snapz Pro, imported it into tracker, and then marked the motion of the center of the string. Here’s a plot of the vertical motion of one of the strings.
Because the movie is taken at 30 frames per second (and probably also there is a limit to the frame rate of the animation), you don’t really have enough data points here to make a nice smooth curve. But you can still can measure the time between two successive peaks and get the period of this oscillation, which I found to be 0.133 seconds. If we get one oscillation every 0.133 seconds, we can invert this to find the frequency in Hz.
7.5 Hz? That’s below the range of human hearing.
Ok, there’s always more than one way to find the frequency. This time, let’s record the sound and see what happens when we analyze it.
Here’s the note being played:
And here’s the waveform of that note in Audacity, a great piece of free sound editing software.
The note produced by a real instrument like a guitar isn’t a single tone, it’s made up of a combination of many different frequencies. This combination of frequencies is what makes two instruments, say a cello and a guitar, sound so differently when playing the same note.
One other super cool feature of Audacity is that it can plot the spectrum of a waveform, and show you all the individual frequencies and their relative strength that go into creating that waveform.
It seems like we’ve figured out the frequency of the base note on the guitar. A little searching of the internet gives us the notes on the guitar, which form a G major scale.
Here’s a list of each of the notes and their corresponding frequencies:
- G: 98 Hz
- A: 110 Hz
- B: 123 Hz
- C: 131 Hz
- D: 146 Hz
- E: 165 Hz
- F-sharp: 185 Hz
- G: 196 Hz
- A:220 Hz
- B:247 Hz
The question remains, why is the sound produced by the guitar in the hundreds of Hz, while video analysis of the motion of the string is less than 10 Hz? Let’s think about a real guitar—whenever you watch a real guitar string vibrating, you don’t see it moving as you do in the Google doodle, it’s oscillating far too fast to be seen. I think this is one of those liberties the folks have Google have taken to present us with a more engaging animation, and additionally, it really isn’t possible to draw an animation that oscillates at hundreds of Hz, since the refresh of the screen is limited far below that number.
Finding the size of the google guitar
For a guitar string, the pitch, or frequency depends on two things: the length of the string and the speed of waves in the string. You can reason to the relationship of these quantites just by remembering that the one wavelength () passes an observer in a time of one period (). So the velocity () must be
knowing that the frequency is the inverse of the period.
The speed of wave in the string has its own dependencies, but we’ll get to that in a bit.
Since the string is fixed at both ends, we know that the lowest note this string can produce is the note that corresponds to a wavelength twice the length of the string, since this will allow the string to remain fixed at both ends of the string, as shown in illustration below.
Now we can try to find the length of this string, to do that, we’ll need to figure out the scale of this illustration. The good folks at Google did include a bridge in their guitar, and after some googling, I was able to track down the dimensions on the standard string spread on an electric guitar, which is 52.4mm (0.052 m).
And using tracker to make measurements according to the scale, I find that the length of this guitar string is 16 centimeters, or about 6 inches. Uh-oh—this seems like a pretty small guitar. But let’s keep going and figure out what this tells us about the speed of waves in this string.
If we go ahead and assume that string is, as the internet tells us, producing a G-note at 98 Hz, and use the length of the string we just found, we can measure the wave speed in the string, using our work above.
, which is pretty slow for a guitar string.
Are the strings on the Google guitar real?
I said previously that the speed of the wave in string has its own dependencies. We could determine this theoretically, by considering a small piece of the string, analyzing the forces on it, and ultimately solving a second order differential equation. I’ll skip that and just reproduce the result
where is the tension in the string in Newtons, and is the mass per unit length of the string. Unfortunately, for a real guitar, both of these quantities are manipulated in order to produce the a wider range of possible notes on the guitar. Each string in a guitar has a different mass per unit length. The string that produces the highest notes also has the smallest mass per unit length, so that it can produce even higher notes that it would if it were identical to the other strings.
This creates a bit of a problem, since both the tension and the mass per unit length of the strings are unknown, we can’t go much further without making some assumptions. Typical electric guitars are kept at a tension of 13-15 pounds. For our purposes, we’ll assume Google uses a middle value of 14 pounds of tension (63 N) and see what that tells us about the mass per unit length of this string.
Solving the above expression for , we get
Using our values of and , we get
That’s a mass per unit length of 6 grams per meter, which seems to be inline with some of the reasonable values for mass per unit length for guitar strings as reported on this page.
So, to recap, while the great Google guitar has been altered a bit so that you can see the strings vibrating, and this guitar is much smaller than any real guitar, it does seem to follow the same laws of physics as a real guitar, if we assume that the Google elves at Mountain View are keeping it tuned to a standard tension.
This was my first attempt at pulling of a Rhett Allain style analysis of some everyday (or everyweb) phenomena. My knowledge of guitars is strictly limited to the “air” and “hero” varieties, so there are problem a host of errors here to accompany the usual calculation mistakes you’re sure to also find.
At our GPD meeting this past Wednesday, which featured Rhett discussing his work, he made a nice statement (which I’m paraphrasing) about how much fun it is to explore the physics of angry birds, and other videos to see how they correspond to our physical laws, since what you are doing in essence, is exploring brand new worlds, where the laws of physics might be totally different from our own. It’s the thrill of discovery, and after a few hours spent this morning trying to study Google’s guitar, I’d have to agree.