Soph. Sem. 210:
Notes for class 15

Sound Waves in Pipes

Sound travels in pipes at the same speed that it travels in open air. It is not significantly affected by the cross-section shape of the pipe nor by bends or corners in the pipe.
At a closed pipe end the sound must bounce, be reflected, from the closed end.
At an open end the sound is also reflected. In order for the sound to diffract out of the open end of the tube a reflection must be generated travelling back up the tube.

Waves on Trays and in Pipes

Trays

Pipes

At open end
Slope is 0
Position varies most
Wave reflected in phase

At closed end
Movement is 0
Slope varies most
Wave suffers 180° phase change

At open end
Pressure is atmospheric so change in pressure is 0
Air can move freely, velocity is maximum.
Wave reflected in phase.

At closed end
Air cannot move because wall is in way.
Pressure can change freely, varies maximally.
Reflected waves suffers 180° phase change.

Standing Waves

NOTE in the text below L is the length of the tube and l the wavelength of the sound. HTML has problems mixing Greek letters and the fixed format text needed to get the formulae to line up right.

A standing wave occurs when the phase of the wave after 1 round trip is the same as the phase of the initial wave.

Change in phase = 360° Trip Time + diff. at reflect.
                       Period

 Trip Time = Dist. Traveled = 2L ,    1    = f
               Velocity       v    Period
For standing wave
Change in phase = 360° 2fL + diff. = 360° n
                        v
where n is an integer.

Tube open at both ends

Reflections take place without phase shift.
Total phase shift in 1 trip round tube

Phase shift = 360° 2fL = 360° n
                    v

For standing wave f = n v or l = v = 2L
                       2L       f    n

Thus, in standing wave an integral number of half wavelengths                    
fit in the tube, L = n l
                       2

The frequencies form a complete harmonic series.

Tube closed at one end

One reflection takes place without phase shift, other adds 180° phase shift.
Total phase shift in 1 trip round tube

 Phase shift = 360° 2fL + 180° = 360°n
                     v
For standing wave f = (2n-1) v or l = v =  4L 
                          4L          f   2n-1
Thus, in standing wave an odd integral number of quarter wavelengths
fit in the tube, L = (2n-1) l 
                         4

The frequencies form an incomplete harmonic series. The fundamental is 1 octave below that for an open tube of the same length and all even harmonics are missing.