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Resonance in Air Columns
Resonance in Tubes Open at Both Ends
Stationary (or standing) waves occur when waves of identical frequency travel in opposite sense through the same medium.
Common examples include waves on a string under tension and waves in air columns (see also resonance in closed tubes).  
   
Consider a source of sound close to one end of a tube, of length L, as shown below.  
The frequency of the sound produced by the speaker is varied, starting from a very low frequency.  
At a certain frequency, the fundamental frequency or first harmonic, fo a louder sound is heard.  
This loud sound is due to constructive interference between the reflected waves and the oscillations produced by the speaker.  
 
 
   
Waves are reflected from an open end because sound travels a little more slowly inside a tube than in free air.  
At an open end, waves are reflected without a change of phase.  
So, if waves travel twice the length of the pipe in one time period, they will return to the source in phase and constructive interference (resonance) will occur.  
This means that, for this first resonance, twice the length of the tube must be equal to one wavelength, λ.  
So  
 
So, we see that the distance between two adjacent anti-nodes is equal to half the wavelength of the waves producing the stationary wave.  
This agrees with the result found when considering stationary waves on strings.  
The relation between frequency, wavelength and velocity gives us  
 
So the fundamental frequency is given by  
 
where v is the speed of sound inside the tube.  
   
As explained here, standing waves in tubes are often represented as shown in the next diagram.  
   
 
   
This represents the air column resonating at its lowest frequency, fo in which we see two anti-nodes and one node.  
   
If we continue to increase the frequency, we find other resonances corresponding to shorter wavelengths.  
Remembering that there must be an anti-node at each end and a node in the middle, we can drawn diagrams to represent these higher harmonics.  
   
Second Harmonic  
 
   
As the distance between two adjacent anti-nodes is equal to half the wavelength of the waves producing the stationary wave, we now have L equals one wavelength  
 
   
Third Harmonic  
 
   
 
   
Therefore, in general, for resonances in tubes open at both ends we can write  
 
and the relation between the frequencies of the harmonics is  
 
where n = 1, 2, 3 etc  
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