Harmonic frequencies of a guitar string

I'm studying harmonic frequency at the moment but I'm just a bit confused about something. How are more than one different frequencies able to be produced from plucking a guitar string (fundamental frequency, 2nd harmonic, 3rd harmonic etc.)? Isn't this impossible for a guitar to vibrate at more than one frequency at the same time?

If a string has multiple waves expressed in it, this is done by adding the waves individually. Each frequency in the harmonic series can be expressed by a wave, a guitar string is the sum of these waves in different proportions. The resulting wave is significantly different than the others.

See below for the sum of the first three frequencies in the harmonic series (black is the sum):

Here is the graph, if you are interested: https://www.desmos.com/calculator/wpy1kovuso

Basically Oscars answers says it all, but I just want to add a few more things.

When a string is plucked its motion need to follow the wave equation $$\frac{d^2}{dt^2}y(x,t) - c^2 \frac{d^2}{dx^2}y(x,t) = 0$$ with Dirichlet boundary conditions (the ends of the string are fixed). $c$ is the speed of sound of the string's medium. The function $y_n(x,t) = \sin(n \pi x / L) \cos( 2 \pi f t)$ is a solution to that equation. Here, $n$ is an integer which enumerates different solutions corresponding the to waves shown in Oscar's post. They correspond to different harmonics. The $\sin$ part gives a standing wave while the $\cos$ part lets it vibrate with time at frequency $f$. The quantities $f$ and $L$ (the length of the string) are related through the speed of sound of the string $$f = (n c)/(2 L).$$

The important bit is that any sum of different $y_{n_1}, y_{n_2}, \dots$ is also a solution to the wave equation, the the overall motion is described by $$y(x,t) = c_1 y_1(x,t) + c_2y_2(x,t) + \dots,$$ and the string can follow this shape. But because every $y_n$ corresponds to a certain frequency, the string effectively vibrates with many frequencies.

Now to the practical part. You can easily convince yourself that this is true by taking a guitar yourself. If you pluck a string you will hear all the harmonic frequencies. However, the loudest will be the fundamental which corresponds to $n=1$ (little exercise: draw the standing wave). If you pluck a string and then put put a finger on top of the 12th fret you will hear that the sound gets more quite but there is still a high pitched sound ringing. What you hear is the second harmonic ($n=2$) and all other even harmonics, because you muted the odd ones (including $n=1$). (exercise #2: draw the standing wave for $n=2$ and figure out why you mute $n=1$ but not $n=2$ when you place your finger on the 12th fret).

• The wave equation on a string is derived by looking at the forces on a given dx. To derive the wave equation, sin (theta) is approximated as theta. Cos (theta) is approximated as one. This only works for small angles. Hence when describing a string's motion the angles in the string must be small. Hence, the physical amplitude must be much smaller than the string length.
– user97957
Jul 24, 2016 at 16:46

If you've got an older web browser kicking around that still runs Java applets you should check out Paul Falstad's Loaded string simulation. You can add harmonics to your heart's content.