1. Standing Waves
Music to my ears ? II

2. Standing Waves in Air Columns
Examples: organs, pipes, trumpet
Depends on the reflective end
Is it open or closed

3. Standing Waves in Air Columns
Pipe open at both ends
Fn = n (V/2L)
n = 1,2,3,…
Placement of nodes and antinodes

4. Both Ends Open
The relationship is the same as that for strings. Instead of a node at each end, an antinode exists at each end. Therefore, f1 = 2L, f2 = L etc.

5. Standing Waves in Air Columns
Pipe open at only one end
Only odd harmonics are present
Fn = n (V/4L)
n = 1,2,3,…
Placement of nodes and antinodes

6. Closed at One End
Ask students why the harmonics only include the odd-numbered ones.
In order to have a frequency that corresponded to n = 2, the air column would need to have one-half a wave. This is not possible because one end is a node and the other an antinode. Therefore, you can only have 1/4 of a wave or the odd multiples of 1/4.

7. Standing Waves in Air Columns
The shape of the instrument can effect the harmonic series

8. Wind Instruments Wind instruments are not as simple as organ pipes.
The shape is not always cylindrical.
The holes change the wave patterns as well.
The size of the “pipe” varies along the length.

9. Standing Waves in an Air Column
Wind instruments also use standing waves.
Flutes, trumpets, pipe organs, trombones, etc.
Some instruments have pipes open at both ends while others have one end closed.
Air is free to move at open ends so antinodes occur.
Closed ends are nodes.
The velocity of the wave is now the velocity of sound in air (346 m/s at 25°C).
At this point use the web site:
and choose “Standing Longitudinal Waves”
This is an excellent demonstration showing the particles in the air column as well as the wave pattern.
Ask students what they expect the wave to look like when switching to higher harmonics.
Ask them how it will appear if you close one end.
This simulation also allows the calculation of the frequency for the various harmonics by changing the length of the tube.

10. Problem #1
What is the fundamental frequency of a 0.20m long organ pipe that is closed at one end, when the speed of sound in the pipe is 352 m/s?

11. Problem #2
A flute is essentially a pipe open at both ends. The length of a flute is approximately 66.0 cm. What are the first three harmonics of a flute when all keys are closed, making the vibrating air column approximately equal to the length of the flute? The speed of sound in the flute is 340 m/s.

12. Problem #3
What is the fundamental frequency of a guitar string when the speed of waves on the string is 115 m/s and the effective string lengths are as follows:
a cm
B cm
C cm

13. Problem #4
A violin string that is 50.0 cm long has a fundamental frequency of 440 Hz. What is the speed of the waves on this string?