# How can I calculate the number of nodes on a plucked string?

Given the length of the guitar string, can you predict the number of nodes that would exist on that string?

My textbook says that there are an infinite number of harmonics that can be created when a string is plucked, however upon observing guitar strings I find that there are a distinct number of nodes upon plucking. So how would I predict how many nodes would exist on a guitar string when it would be plucked?

• Curious - how did you find that there a distinct number of nodes? Jul 11 at 18:51

There are many harmonics excited one one plucks a string, but the distribution of energy in these harmonics is different. The lowest harmonic, corresponding to the oscillations of the whole string (i.e., with the nodes at its ends) is usually the dominant one, and the height of the dominant sound is determined by the length of the oscillating part of the string (which can be made different by pressing it at a specific fret).

The distribution of energy among the harmonics (partially) determines the timber of the sound, that is why the same pitch sounds differently on guitar, violine, ukulele etc.

However, there are some methods of play that expressly excite the modes of the string with multiple nodes, such as flageolets, where the string is held at a fret, plucked, and immediately released.

For a single frequency, there is a finite number of nodes, which you can find by fitting an integer number of half-waves along the string.

But since there are infinite frequencies, there are an infinite number of possible nodes.

Let's say the fundamental frequency is $$n_0$$ such that the length of the string fits 1/2 wave

$$n_0 = \frac{c}{2 \ell}$$ where $$c$$ is the wave speed on the string.

A string vibrating with an integer multiply of the fundamental frequency $$n = i\, n_0$$ has a total of $$i$$ nodes along its length. You can check the half-wavelength $$\lambda = c/(2 n)$$ fits $$i$$ times in the length $$\ell$$.

So for each frequency $$n = i\,n_0$$ there are $$i$$ nodes, but since $$i$$ can be anything integer positive $$i >0$$, there are inifnite possibliities.

Initially, as long as the damping is negligible, there are no nodes on a plucked string (Obviously apart from the two nodes at the ends). The motion is a linear combination of modes which are indeed standing waves with fixed nodes. But their sum is not a standing wave.

In the course of time, the modes decay with different typical times and I would tend to think that the lowest mode, with only two nodes at the ends is the one that remains. This is what I think I see in this video:

Motion of a plucked string

Hopefully this article might help. If you’re able to work out the frequency, you should be able to count the nodes of each string.

http://www.bsharp.org/physics/guitar

• Just to clarify: empty answers with only an external link aren't allowed and might be deleted. Please make an effort to answer the question and to write an answer that is as self-contained as possible. Thank you. Jul 12 at 15:19