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Resonance of a String under Tension
When a wave passes through a point on a string, that point oscillates.  
If the wave later returns to the same point (for example, after reflection), interference will occur between the existing oscillation and that due to the reflected wave.  
The interference will be constructive if the "effective distance" moved by the wave is equal to a whole number of wavelengths.  
When a wave is reflected at a fixed end of a string, a phase change occurs.  
A phase change of 180 makes a wave "look as if" it has travelled an extra half wavelength.  
So an effective distance of λ could be due to the wave actually moving λ/2 but also experiencing a phase change of 180 or πradians (see here for illustration)  
A given string at a given tension can resonate at a (theoretically infinite) series of frequencies.  
The lowest frequency of resonance is called the fundamental frequency or the first harmonic of the string.  
   
Fundamental Frequency of Resonance (First Harmonic)  
 
 
As the animation shows, all points oscillate in phase but with different amplitudes of oscillation.  
Consider the string to be disturbed at A (the centre).  
Waves travel towards B (and C) and are reflected with with 180 phase change.  
If the effective distance travelled by the waves is λ, then resonance occurs.  
This means that, for the first resonance, the distance A→B→A (or A→C→A) must be equal to λ/2.  
Therefore, at the fundamental frequency  
 
We conclude that the distance between adjacent nodes of a stationary wave is equal to half the wavelength of the travelling waves which are producing the stationary wave.  
The frequency at which this first resonance occurs is given the symbol fo  
 
Relation Between fo and the Length of the String  
If the distance A→B→A (= L) is λ/2 then the time taken by the waves to move this distance is half of a time period, T/2.  
This follows from the definition of one wavelength.  
   
If the wave velocity is v then  
 
and, remembering the relation between frequency and time period  
 
this gives us  
 
It can easily be shown that the velocity of propagation, v of waves on a string under tension is given by  
 
where Te is the tension and m is the mass per unit length of the string.  
So, the fundamental frequency, fo is given by  
   
and the frequencies of the higher harmonics are given by  
   
where n = 1, 2, 3 etc
   
The diagram below represents the second harmonic (also sometimes called the first overtone)  
 
Assuming that this represents the same string under the same tension as the first diagram, we can immediately see that the wavelength is half that of the fundamental so f = 2fo (as shown in the animation).  
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