2. Standing Sound Waves
• A long, narrow column of air, such as the air in a tube or pipe, can
support a longitudinal standing sound wave.
• Closed end of a column of air must be a displacement node because
the air molecules cannot move.
• Open end air molecules are free to move therefore the open end of a
column of air must be a displacement antinode.
3. • In a pipe open at both ends, the natural frequencies of oscillation form a
harmonic series that includes all integral multiples of the fundamental frequency.
fm = v/ λm m = v/2L m = 1,2,3…
f1 = v/2L f2 = 2f1 f3 = 3f1
• In a pipe closed at one end and open at the other, the natural frequencies of
oscillation form a harmonic series that includes only the odd integer multiples of
the fundamental frequency.
fm = v/ λm m = v/4L m = 1,3,5…
f1 = v/4L f3 = 3f1 f5 = 5f1
4. A vuvuzela (a horn used during the 2010 World Cup in South Africa) is a
tube that is open at one end and closed at the other.
1. A typical vuvuzela is 60 cm long. What is its fundamental frequency?
2. What are the frequencies for the next three harmonies?
3. By how much you would have to change the length of a vuvuzela if its fundamental frequency
has to be decreased by 100 Hz?
Hint: Speed of air = 343m/s
