# Frequencies and overtones

1. Apr 11, 2004

### broegger

it is often said that the rich sounds produced by e.g. a guitar is due to the string vibrating at several different frequencies at once.. does this mean that there is both the fundamental frequency and several overtones present at the same time.. if so, I don't get it.. how can there be more than one frequency present at a time - seems to me that this would correspond to the string having several different shapes at every instant in time (since every frequency corresponds to a unique wave pattern), which is absurd..

2. Apr 11, 2004

### MiGUi

I think that the answer maybe this: if we have a string whose extremes are fixed, if the lenght of the string is not an exact multiple of the wavelenght, the wave goes from one extrem to another and it is reflected back causing the superposition of the original wave with this new. This effect is replied many times so at the end, we have many different waves interfering, so then, the sound produced don't cames from only one perfect wave.

Bye

3. Apr 11, 2004

### broegger

so "several frequencies" simply means that the there is no unique frequency because the wave pattern is constantly changing, which in turn is because there can't be a perfect standing wave because of the physical properties of the string (the length) or...?

4. Apr 11, 2004

### Staff: Mentor

Absolutely! The unique sound of a guitar (or any instrument) depends on the particular blend of harmonics that are produced.
I'm not sure I'm getting your difficulty. The vibrating string has a particular shape, which is a composite of the individual simple wave shapes of each harmonic that exists.

5. Apr 11, 2004

### jdavel

broegger,

On a guitar string the fundamental is one half of a sinewave. So it starts at 0 at one end, goes up (or down), and then back to zero at the other end. So when the string vibrates at its fundamental frequency, every place along the string is going up and down (in synch), with the amplitude of the vibration greatest in the center.

All the overtones (or harmonics) are odd multiples ( n=3, 5, 7....) of one half a sinewave. So the first harmonic starts at 0, goes up, then down, then up, then back to zero at the other end. And so on. So when the string vibrates at say the n=3 harmonic, there are 2 nodes (no vibration) at 1/3 and 2/3 along the string, and in between is a shape (going up and down) just like the fundamental except 1/3 as long, and going 3 times as fast (which is why it's at a higher pitch).

Taken all together (1, 3, 5, 7....) these possible individual vibrations are called the string's normal modes of vibration. The string can only vibrate as some linear combination of these modes. That means the shape of the string when it's at its maximum displacement looks like:

fundamental: a1*sin(1*2*pi*x/L)

or fundamental and the n=3 harmonic: a1*sin(1*2pi*x/L) + a3*sin(3*2*pi*x/L)

or n=1, n=3 and n=5: a1*sin(1*2pi*x/L) + a3*sin(3*2pi*x/L) + a5*sin(5*pi*x/L)

and so on.

Each of those sums is a single function of x, the distance from one end of a string with length L. So the string only has one shape at a time.

By the way you can play around with this on a spread sheet by defining a function as the sum of the first five or ten modes, and then make of chart (graph) of the function. Be sure to have the a1, a3, a5.... in your function be references to cells on the sheet so you can easily change their values and see what happens. For example, give this one a try: a1= 1, a3= 1/3, a5= 1/5....

6. Apr 11, 2004

### jdavel

Doc Al said: "The vibrating string has a particular shape, which is a composite of the individual simple wave shapes of each harmonic that exists."

I think broegger's problem is with words like "composite". Anyway that's what I was trying to clarify in my previous post.

7. Apr 11, 2004

### broegger

thanks again.. it's just the phrase "several frequencies at the same time", which to me seems analagous to e.g. a particle being at several positions at the same time or having several different velocities..
isn't it more accurate to say something like "the strings vibration is a sum (superposition) of several vibrations with the fundamental frequency and/or some of the harmonic frequencies".. i mean, the string must have a well-defined frequency at all times or is it constantly changing or...?

8. Apr 12, 2004

### turin

Actually a quite decent analogy (the one with velocity).

How is saying that the string "has a frequency" meaningful? The string vibrates. This vibration can have some certain period which repeats at the fundamental frequency, so I suppose that might be what you mean. But it also has resonant modes. These modes each have their perfectly well-defined periods, and the string "has" all of these modes after it is plucked.

Here's something interesting that you can try with your guitar. Pluck a string and then pinch it down at exactly the halfway point. What happens to the pitch? This shows that the other even modes are independent of the fundamental (and so it is meaningful to say that they exist, and that the string "has" them). This may not be possible without a fretless guitar, though. I haven't tried it on a guitar.

Last edited: Apr 12, 2004
9. Apr 12, 2004

### mmwave

You can indeed do this on a guitar. Pluck the string then touch your finger were the string passes over the 12 th fret (don't press down - just lightly touch your finger to the string so it has zero amplitude vibration at that point). You will hear the pitch double. There are other locations that do this on a guitar but I don't remember them.

10. Apr 13, 2004

### broegger

by "the strings frequency" i mean the frequency with which the particles of the string vibrates, which must be the same for all particles if the wave is sinusoidal.. or am I missing something??

if you for example pluck a guitar string at some point, then the resultant vibration can be represented as a sum of resonance mode-vibrations.. what is the frequency associated with this motion - or is it ambiguous to associate a single frequency or...?

11. Apr 13, 2004

### turin

I would say so. The string does not "have" a frequency. The vibration has a frequency. You could say that the frequency of the vibration is the fundamental, since that tells how often the string returns to some state (disregarding decay). But you can also consider the state as a superposition of states. Each of these states has a repeat frequency.

Why should all of the particles vibrate at the same frequency? As a simple model, consider three heavy beads connected by rubber bands in a line, and the two end beads each connected to a wall by a rubber band. Try to imagine the center bead moving up and down, and for each up-down oscillation, the two outside beads move up and down twice (twice the frequency). If you buy that, then why should the particles of the string be any different? Even one particle can vibrate at two different frequencies simultaneously (though I will neglect the physical mechanism). Imagine a particle moving back and forth between two points once every second (1 Hz). And then, as it does so, it "jiggles" at 10 Hz. What's wrong with saying that it is vibrating at 1 Hz? What's wrong with saying that it is vibrating at 10 Hz?