How do guitar strings produce harmonics?

[Guus van Aarle]

Hi all,

I've been reading up on the physics of waves in order to better understand what goes on with sound. I'm having difficulties understanding how harmonics are produced in guitar strings. It's probably not as complex as I'm making it in my head, and it's really starting to frustrate me. This is the most important question that I have:

- In order to create a standing wave in a string, you need resonance. How does resonance occur in the strings? From what I understand, resonance occurs when an object is set into vibrational motion at one of its natural frequencies by another object that's vibrating at that same natural frequency. I don't understand how guitar strings can resonate by just plucking them.

"As mentioned earlier, all objects have a frequency or set of frequencies with which they naturally vibrate when struck, plucked, strummed or somehow disturbed. Each of the natural frequencies at which an object vibrates is associated with a standing wave pattern. When an object is forced into resonance vibrations at one of its natural frequencies, it vibrates in a manner such that a standing wave is formed within the object."

I hope the answer to this question will answer all the questions that I have, or at least highlight what's going wrong with my thinking. Thanks in advance.

 

[sophiecentaur]

Hi and welcome to PF.
There are two parts to this answer.
Not all resonances involve waves as on a string. A mass on a spring or a pendulum will resonate in a very simple way and the analysis doesn't involve waves. But many objects behave very much like the simplest wave example which is a taught string
Assume the string is taught and supported firmly at each end
1. Excited oscillations: Firstly, if you take a string and start to vibrate it at one end (at a single frequency), a wave will start off, traveling along the string and get reflected by the rigid mounting at the other end. If the length of the string length is a half wavelength of the excitation wave, the returning wave will meet the vibrating end and be exactly in step with the vibrations and the energy in the wave reflected off that end will be augmented by more energy, put in by the excitation. You will have constructive interference and this is a resonance. Each journey from end to end, the energy in the wave will build up until the losses (friction, air drag etc) are the same as the energy injected each cycle. The same thing will happen if you make the string a whole wavelength or exactly any number of half wavelengths. If the frequency of excitation is slightly different, there will be no build up and low level waves will travel along the string but there will be no build up and the energy will just dissipate as it is supplied. A string of fixed length will support overtones (approximately harmonically related) where there are 1,2,3,4 etc half waves along the string. These are called the Normal Modes of vibration.
2. Plucking: When a string is pulled to one side, the shape it takes up will consist of a whole set of spatial frequencies (The Fourier Transform of its triangular shape will only consist of these). They are also the normal modes mentioned above and the string will vibrate, when released, with some or all those frequencies. The proportions of each mode will depend upon the actual shape of the string. If you hold your finger half way along, when you pluck the string, the fundamental mode will not be present and you will hear the first overtone (second harmonic) with two antinodes on the string. Partially stopping the string at various places will produce different number harmonics.
Harmonics and Overtones: A string is a very ideal oscillator and the Normal Modes tend to be very near harmonic frequencies but most vibrating objects (e.g. the air column in an wind instrument or a 2 Dimensional object like a drum membrane) do not resonate at harmonics because the overtones are very different and they can sound very unmusical because of this. If a trumpet player tries to play notes on a simple length of pipe, the high notes will come out all wrong. A trumpet has a Bell on the end, to deal with this problem.

 

[Guus van Aarle]

Thanks a lot for taking the effort to answer my question. Your explanation has really cleared things up!

Two more questions though: how come all of the harmonics other then the first harmonic have lower volume than the first harmonic?

And, if I understand correctly, the repeated interference of source waves and reflected waves in the string causes all of the standing wave patterns to occur. So, if I for instance pluck a guitar string, it will vibrate for one second, and during this one second, all standing wave patterns of all the harmonics could theoretically be seen for different amounts of time?

 

[sophiecentaur]

Thanks a lot for taking the effort to answer my question. Your explanation has really cleared things up!

1.Two more questions though: how come all of the harmonics other then the first harmonic have lower volume than the first harmonic?

2.And, if I understand correctly, the repeated interference of source waves and reflected waves in the string causes all of the standing wave patterns to occur. So, if I for instance pluck a guitar string, it will vibrate for one second, and during this one second, all standing wave patterns of all the harmonics could theoretically be seen for different amounts of time?

1. When you get an harmonic by damping the centre of the string, most of the energy of that harmonic has been absorbed with your finger as well as the other, lower frequencies. If you could use a different method for exciting a harmonic (letting the string go from multiple points at the same time), you would probably find the harmonic would be louder.
2. All the modes that have been excited will be seen, at various levels. The higher order modes tend to die down sooner than the fundamental mode.