How are overtones produced by plucking a string?

I read the following from wikipedia:

When a string is plucked normally, the ear tends to hear the fundamental frequency most prominently, but the overall sound is also colored by the presence of various overtones (frequencies greater than the fundamental frequency).

When I pluck a string, I just notice a node at each end and an antinode at the middle. How can we have overtones in addition to the fundamental frequency? It seems counterintuitive for me.

• Consider harmonics (on a guitar, bass, harp, etc.) -- when you lightly touch the vibrating string at its midpoint, you damp all vibrations without a node at L/2, yet you still hear a sound. These are the overtones with a node at L/2, and the one with the largest amplitude is usually 2*f, so you hear the octave above f. – Sam Sep 18 at 22:19
• The second harmonic does have a node in the middle, you just can't see it because the first harmonic is moving the middle of the string up and down. If you could somehow subtract the sinusoidal movement of the base tone from the vibration of the string, it would probably be clearer. But then again there is the third harmonic moving it up and down. – Arthur Sep 19 at 10:37
• because the shape of the plucked string is not sinusoidal but more triangular. – robert bristow-johnson Sep 19 at 20:16

The only way to avoid overtones would be to pluck the string in such a way that its initial shape is sinusoidal. However, that would be nearly impossible. In practice, the initial shape is almost always triangular.

If you are familiar with Fourier transforms, consider how you would do a discrete Fourier decomposition of the string's initial shape. The Fourier components correspond to the overtones.

• Do you have more information about this effect? The waveform doesn't stay triangular after the string has been plucked, does it? – Eric Duminil Sep 19 at 6:54
• An example of the string movement is shown in youtube.com/watch?v=_X72on6CSL0 (not for a guitar, but its a good example that is relevant). – Tricky Sep 19 at 8:10
• @EricDuminil No it doesn't, since the high harmonics get filtered out rather quickly. You may enjoy playing with Karplus–Strong string synthesis. It's very easy to implement (eg, in Python), and gives a surprisingly realistic plucked string sound for such a simple algorithm. – PM 2Ring Sep 19 at 17:34
• @EricDuminil: Even in the absence of friction and hysteresis, the string will not stay triangular at all. After all, the different harmonics are oscillating at different frequencies. Furthermore, the Fourier decomposition is not 100% correct for a plucked string, since the tension in the string is actually varying over time, and at any single point in time (except the start) the tension is not even equal at every point along the string. It is a good approximation, but the real situation is always far more messy. – user21820 Sep 20 at 11:02
• That is easy. Try touching the string exactly in the middle, and pluck it halfway between your finger and the end. You will get a note 1 octave above the fundamental. Then try the same, but touch the string 1/3 of the way from the end and pluck halfway between there and the end. Depending on the quality of your instrument, you can get pretty clear tones all the way up to the fourth or fifth harmonic. – S. McGrew Sep 30 at 12:43

Eigenmodes of a string have sinusoidal spatial form $f_m(x) = C_m \sin(\pi m x/L)$, where $x$ is the parallel coordinate and $L$ is the length of the string. Plucking a string at a fixed location $x_0$ means giving it a non-sinusoidal initial perturbation, e.g., something like a piece-wise linear function, $f(x) = A x/x_0$ for $x \le x_0$ and $f(x) = A (L-x)/(L-x_0)$ for $x \ge x_0$. Expanding the initial perturbation $f(x)$ in eigenmodes $f_m(x)$ shows how much each harmonic is excited initially, in general it would be a full spectrum of eigenmodes.

• You need to multiply that by a square function; strings are very finite compared to room they occupy. – aquirdturtle Sep 19 at 5:24

You start with a triangular form, which has its fourier series. Let's say the initial shape is $f(x)$:

$$f(x)=\sum_n a_n \sin \frac{\pi n x}{L}$$ where $n$ counts the modes ($1$ is fundamental). So initial shape determines the harmonic content $a_n$ of certain modes. If you pluck in the middle, you will put more of the fundamental into the initial spectrum, and in every odd mode, but no even modes at all. If you pluck near the end of the string (like on a guitar), you get all modes, with higher modes still less prominent.

This shape then evolves in time. Every mode has a frequency, related to the wavelength:

$$f(x,t)=\sum_n a_n \sin \frac{\pi n x}{L}\cos \omega_n t$$ $\omega_n$ is proportional to $n$ (determined by $\omega=kc$, where $k=2\pi/\lambda$ the wavevector of the mode and $c$ is the speed of propagation of waves on the string).

This would make motion completely periodic and the shape would return to initial form after one cycle of the fundamental frequency. However, vibration is always damped: some small amount by radiating sound into the air, and mostly, by conversion into heat (the string not perfectly elastic). Higher frequency modes are always much more damped: if you demand very fine-detailed curly vibrations, this will get damped a lot because curvature of the string is high. Usually, you get something like that: $$f(x,t)=\sum_n a_n e^{-t/\tau_n} \sin \frac{\pi n x}{L}\cos \omega_n t$$

where $\tau_n$ is the characteristic damping time of $n$th mode. You could say that the intensity of spectral components falls off faster for higher overtones and the tone is getting more sinusoidal. Very high harmonics associated by the sharp triangular kink at the plucking position just produce a "plunk" sound and disappear almost instantly.

Damping higher frequencies always has a smoothing effect on the shape: with time, the shape gets rounded, without sharp corners.

You are not hearing the string but the resonating body and the string. An acoustic guitar resonates with its plucked strings. The colour of the sound is determined by its formant, and what you hear as the fundamental frequency may in fact be some multiple of that, with both under and overtones, depending on the type of resonating body. The resonating bodies of a guitar and a cello (for example) have different formants and therefore result in a different spectral envelope and a different timbre, for the same fundamental frequency.