School of Rock: The Physics of Waves on Guitar Strings

Perhaps the most iconic instrument in modern rock is the guitar. It’s really just a bunch of strings stretched across a board, which you can strum to make awesome tunes, thanks to the physics of waves and sound.

A Wave Pulse on a String

Let’s start with a demo you could probably repeat at home. Get a nice string—one that’s sort of thick—and lay it out in a straight line on the floor. Now grab one free end and give it a side-to-side shake. Here’s what it might look like:

There are some really important things we can see from this simple experiment. First, notice that with the shake of the hand, the motion displaces the string to the side. This displacement in the string then travels down its length. It’s important to realize that it’s just the displacement that travels, not the string itself. We can call this traveling displacement a “wave pulse” and the string “the medium.”

Second, we can see that the wave pulse has a finite speed, and this speed does not depend on the shape of the pulse.

Here is a second experiment to try. Pull the string tighter to increase the tension and then create another wave pulse. You should be able to see that with a higher tension in the string, the pulse has a greater speed. This is rather important: It says that the speed of a wave doesn’t depend on the source that made the wave. (In this case, the source is your hand.) Instead, it depends on the properties of the medium, which is the string.

You can also increase the wave speed by decreasing the linear density—or the thickness—of the string. A thinner and lighter string would make the wave pulse travel faster.

Now you can see why it’s better to use a heavy string for our demos instead of a light one. Yes, you can make a nice wave pulse travel down a thinner string, but it goes so fast that it’s difficult to see. (We will get back to this idea of wave speed when we look at guitar strings.)

Repeating Waves on a String

Are you ready for your next experiment with the string? Now instead of making a single motion with your hand, continuously move it back and forth to make repeating wave pulses. It should look something like this—except probably without the annotations:

For a repeating wave, we have a few more attributes that we can use to describe the displacement. It helps to think of these repeating waves as a series of high and low points, even if the oscillations are side to side.

We can use these high and low points to describe two things. First, there is the amplitude. This is the distance of the high point from the equilibrium position—the position of the string if it was lying flat and there wasn’t a wave at all. Essentially, the amplitude is the size of the disturbance.

Next, we have the wavelength. This is the distance it takes for the wave to repeat itself. One simple way to measure the wavelength is to measure the distance from one high point to the next—or from one low point to the next.

There’s one other important property of a wave: the frequency. Imagine that you look at one point on the string as waves pass by. That single point will be moving back and forth perpendicular to the direction of the wave. Now suppose you found that the point oscillated five times during a one-second interval. We would say that the wave has a frequency of five oscillations per second, or 5 hertz (Hz).

Now for some notation comments. We normally use the Greek letter λ for the wavelength. The frequency is f (I mean, that makes sense), and the wave speed is v, for velocity. For the amplitude, it depends on the type of wave—so we are free to use something obvious like A.

It’s useful to use symbols for these properties because they obey the following relationship:

Let’s take a moment to see how this equation makes sense. Remember that if I don’t change anything about the string, then the wave speed (v) will remain constant. Just to use simple values, we can use a wave speed of 1 meter per second.

Now suppose that I shake the end of the string back and forth one time every second. This would produce a wave with a frequency (f) of 1 Hz. With a time of one second between shakes, the wave pulse will travel 1 meter and make the wavelength (λ) equal to 1 meter. That’s simple.

But what happens if the frequency is increased to 2 Hz? Now the time between shakes is half a second, and the pulse will only travel 0.5 meter before the next pulse is created, producing a wavelength of 0.5 meter. You can see that increasing the frequency will decrease the wavelength. (This will be important for our guitar.)

Waves of Light

As a bonus, describing waves on a string works for other types of waves too—like our favorite wave, light. Light is an electromagnetic wave. So, instead of oscillations that travel down the length of a string, a light wave is an oscillation in both the electric and magnetic fields. The awesome thing is that the electric and magnetic fields depend on each other in a way that makes it possible for an electromagnetic wave to travel through empty space. This is how light can travel from a distant star all the way to your eyes on Earth—with mostly nothing in between.

Light also has a wave speed. We call this the speed of light, and it has a value of 3 x 10⁸ meters per second. The wave speed is still equal to the product of the wavelength and the frequency of the wave, just like a wave on a string.

Standing Waves

Now that we have a basic idea of the properties of a wave, it’s time for another demo. This one is easy. Take a rubber band and stretch it between your thumb and finger on one hand. Now pluck the rubber band. Just in case you don’t happen to have a rubber band, it should look something like this:

This is perhaps the simplest example of a standing wave. The pluck creates a wave pulse that travels down the rubber band and reflects off the ends, where your fingers are. This reflected wave then interferes with itself and creates a wave that seems to stay in place. That’s why it’s called a “standing” wave. (See, sometimes science makes sense.) The key is that the middle of the stretched rubber band just moves up and down, like the high and low points on our repeating wave from before.

But you can make other standing waves that are a little bit more complicated. Here’s a wave on a longer string:

An animation might make things clearer. Below is a numerical calculation using Python. It’s sort of complicated, but I have all the details in this video.

Both of these standing waves show something that you can’t see as easily with the rubber band demo. Notice that right in the middle of the string there is a point that doesn’t move up or down. It just remains stationary. We can then break a standing wave into parts that oscillate fully up and down (antinodes) and parts that stay stationary (nodes).

Having a node in the middle not only looks cool, but it’s much easier to see the relationship between the length of the string (we can call it L) and the wavelength (λ). Since the string goes all the way up and down over the length of the string, it’s one full wavelength.

The rubber band example does indeed have two nodes—they are at the ends of the rubber band where your fingers hold it. We only have half a wavelength in the standing wave, but there is indeed a relationship between the length of the rubber band and the size of the wavelength.

Guitar Strings

It’s time to put all these ideas together and look at a guitar string. Once you hit that string, it’s going to create a standing wave with an antinode in the middle and two nodes on the ends. This is called the first harmonic wave.

It’s possible to also produce a second harmonic wave (with a node in the middle) and even higher harmonics. However, because of drag forces on the string, these higher frequencies die out fairly quickly so that you are just left with a standing wave that has a wavelength equal to twice the length of the string.

But you don’t strum a guitar string to see a standing wave. No, you strum the guitar because you want to make a sound—maybe even some music. What we really care about is the frequency of that oscillating guitar string. Let’s use some realistic values. If you use the highest-frequency string, it could oscillate at 330 Hz. In terms of musical notes, that’s an E. Let’s also assume that the length of the string is 76.5 centimeters (30 inches). From this string length we can get a wavelength of 1.53 meters. Now using v = λf, we find a wave speed of 504.9 meters per second.

What if I want to play a G note, or 391 Hz, on the same string? I can do that by using my finger to push the string down on the fretboard. This effectively changes the length of the string and changes the wavelength. We can do the math and find that with an effective length of 64.6 centimeters (25.4 inches), the wavelength will decrease enough to cause the frequency to increase to 391 Hz. If you want an even higher-frequency note, just make the string even shorter.

How do you make a guitar note that’s lower than 330 Hz? You can’t do it with that same string. But you can get another string that has the same length but a higher linear density, or mass per unit length—which is why the strings on a guitar have different thicknesses. Remember that we can change the speed of the waves on the string by changing the properties of the string. With a higher density you get a lower wave speed, which means a lower frequency. The rest is just music.

What if your guitar doesn’t sound right, like if your E note is playing at 325 Hz instead of 330 Hz? You can solve this problem by tuning your guitar. At the end of each guitar string is a tuning peg. If you turn this, you will either increase or decrease the string’s tension. Increasing the tension will also increase the wave speed on that string, which increases the frequency. Now you aren’t just playing a guitar, you are a guitar hero. Wait, that’s a video game. Never mind.