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, f_{o}
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 antinodes 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, f_{o} in which we see two
antinodes and one node. 



If we continue to increase the frequency, we find other
resonances corresponding to shorter wavelengths. 

Remembering that there must be an antinode 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 antinodes 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 
