# The SHM of a guitar string

1. Sep 18, 2011

### jaumzaum

Guitar strings behave like a spring when tuned:

F = k.x is the tension in the string, where k is the contant of the string and x the displacement (when tuned). So by the equation :

$v = \sqrt{ F/u}$

where u is the linear density of the string.

$v = \lambda.f -> f = \sqrt{ F/u}/\lambda$

The first string of a guitar is E and has a frequency f1, when tuned, that is proportional to the square root of the string tension.

But when we play the first string weakly, we seem to hear the same E note (sure, more weak) and when we play strongly, we seem to hear this same E stronger. So the force we apply in the string does not seem to change the frequency, only the amplitude. But in a spring, when we make a vertical displacement

[PLAIN]http://img716.imageshack.us/img716/6427/sgfhdf.jpg [Broken]

The tension do change

So why don't we have a change in the frequency?

Last edited by a moderator: May 5, 2017
2. Sep 18, 2011

### Philip Wood

Even for quite large displacements, the string tension doesn't change enough to affect the wave speed significantly. Indeed, constant tension is one of the assumptions we make when deriving the wave speed formula you've quoted. The derivation involves the transverse acceleration of the string due to transverse components of the tension which arise when the string is displaced.

For very large displacements, you're absolutely right: the note will change. It will also be far from pure!

3. Sep 18, 2011

### Ken G

That's the right answer, but just to make it perfectly clear, there's a big difference between a spring that is not stretched, for which plucking causes all the stretching, versus a spring that is already stretched when you pluck it. The guitar string is the latter case, so that plucking really doesn't increase the already significant tension. Indeed, you know that when you "tune" the guitar, you do so by altering the initial tension, so that's what is key in producing the tone.

4. Sep 18, 2011

### olivermsun

Actually you do hear a slight frequency shift when you pluck hard on a stringed instrument.