How do standing waves form on strings without a vibrator?

[Cheman]

Standing waves on strings...

I understand how to produce standing waves on a string when we have a vibrator at one end, as you usually use in demonstrations in physics classes. ie - increase frequencies until we achieve the correct kind of interference to produce standing wave.

However, it i when we apply this to talking about how a stringed instrument, let's say a guitar, works that I begin to ask questions. Here we go:

i) We pluck the string - how does that initiate the standing wave? Does it set of transverse waves in both directions (ie - towards both fixed ends) which then reflect to form standing wave?

ii) What frequency does the string vibrate at? In the demonstration I mentioned, we chose the frequency the string vibrates at, so what frequency does this "ungoverned" string vibrate at? And how do we know this should produce at least the fundamental frequency?

iii) Why do we get complex wave forms? ie - not just fundamental frequency? Is it because we do not get perfect interference so the standing waves are not "perfect"? And how come we can describe these as if they are just mixtures of the harmonic frequencies?

If anyone can answer these questions i would be most greatful.

Thanks in advance.

 

[Integral]

The fundamental frequency of a string is determined by the length, tension and mass/unit length of the string. So you change the frequency of guitar string when tuning by changing the tension. When playing you change the length.

Some very interesting things occur around how the vibrations are induced. A string plucked with a pick has a different sound from one plucked with a finger, also a bowed string has yet a different sound.

The reason lies in the fact that the initial displacement of the sting is shaped by the object causing the displacement. A pick causes an initial shape with very sharp edges, while your finger has much softer edges. These edges translate into a range of frequencies. Sharper edges require higher frequencies. This is seen well when Fourier analysis is applied to the problem. Any frequency which is not a harmonic of the fundamental string length will be damped out very quickly, those that are a harmonic with resonate and produce the sounds you hear.

 

[Cheman]

Ok thanks - would it be possible to go into a bit more detail in answer to my questions? Its just that I cannot find anything anywhere to answer them - textbooks don't go into a great deal of detail about it.

Thanks. :-)

 

[brewnog]

When you pluck the string, you set the string vibrating. The vibration with which you are concerned is a transverse wave, - it turns out to be a standing wave because the string is constrained at both ends.

As Integral already pointed out, the fundamental frequency is determined by the length, tension and mass per unit length of the string. My Latex is poor, equations relating the above can be found here.

Complex wave forms do occur in instruments such as guitars. There is no reason why harmonics should not occur, but it should be noted that they are always of a lesser amplitude than the fundamental. Also (in addition to Integral's comments regarding plucking) are affected by such things as the type of wood which the guitar is made from, the shape of the guitar body, any artificial harmonics which the player has introduced, and ambient conditions. It is partly due to the mixture of amplitudes of fundamental and harmonic frequencies which gives a guitar its characteristic sound.