#### dRic2

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- When solving the wave equation for an oscillating string we look for a solution that is the linear combination/superposition of an infinite serie of "harmonics", i.e. sine and cosine functions (classical Fourier series). But is that a mere mathematical trick or do these states have physical meaning ?

This question reminds me of the interpretation of the "wave packet" in QM for a free particle moving freely in the whole space; but in QM it is obvious that plane waves can't be of physical meaning. Now consider a guitar string (fixed at both ends) and suppose the solution to the wave equation is some function ##u(x, t)##. Assuming ##u(x, t)## to be a "nice" function, a Fourier serie representation could look like this ##u(x,t) = \sum_{n} c_n \sin(\frac { \pi n}{L} x)##. From Fourier's work we know that ##{c_n}## is a monotonically (fast) decreasing succession, so we hear mainly the first (principal) harmonic which define the note we are hearing. The other contributions give a sort of "background" which characterizes the particular sound of an instrument (in this case a guitar). Up to this point it doesn't really matter if each harmonic really exists or if it is just a mathematical representation, but now consider the following technique:

- play an open string, for example the low E

- gently put you finger on the metal strip corresponding to the 5th, 7th, or 12th fret.

- you'll hear a very high pitched tone, much higher then the tone you would expect for the corresponding fret. (I'll try to record something or post a video if I have the time... it's 2:28 am here now )

Basically the most common explanation (also reported by wikipedia) is that, by doing that "movement" with your finger you are muting all the various harmonics of the vibrating string

Then my question is: if you "mute" all the harmonics except for those having a node at that particular point, it means that they really do exist and that they're not just a mathematical trick. On the other hand this seems very strange to me because the string oscillates according to the function ##u(x, t)## which is something ##per sé##, something real and physically observable: it is the "true" motion of the string.

Sorry if it took me so long to get to my question, I hope I didn't bother you too much. If you'll find my question too silly (it might be...) at least I hope you'll find this as an interesting fact.

Thanks

Ric

**string harmonics**. For those of you who are not familiar with this beautiful technique it goes like this (there are tones of different ways of doing it, if you're curious just refer to the wikipedia page, I'll use a specific version which suits my problem better):- play an open string, for example the low E

- gently put you finger on the metal strip corresponding to the 5th, 7th, or 12th fret.

*Be careful, you don't press the fret, you simply lay your finger on the metal strip!*- you'll hear a very high pitched tone, much higher then the tone you would expect for the corresponding fret. (I'll try to record something or post a video if I have the time... it's 2:28 am here now )

*Explanation*Basically the most common explanation (also reported by wikipedia) is that, by doing that "movement" with your finger you are muting all the various harmonics of the vibrating string

*except those having a node at that particular point*. Since all the higher harmonics have higher frequencies that the fundamental one the high pitched tone is no more surprising. You can also do the math (it's super easy) and check that this explanation correctly predicts the right note (frequency). For instance the harmonic on the fifth fret corresponds to play the open string 2 octave higher (check with the math or with your ear... it's fun!)Then my question is: if you "mute" all the harmonics except for those having a node at that particular point, it means that they really do exist and that they're not just a mathematical trick. On the other hand this seems very strange to me because the string oscillates according to the function ##u(x, t)## which is something ##per sé##, something real and physically observable: it is the "true" motion of the string.

Sorry if it took me so long to get to my question, I hope I didn't bother you too much. If you'll find my question too silly (it might be...) at least I hope you'll find this as an interesting fact.

Thanks

Ric