1. Musical Instruments

2. Standing Waves Waves that reflect back and forth interfere.
Some points are always at rest – standing waves.

3. Closed Ends Both ends fixed limits the standing wave.
A whole number of half-wavelengths must fit in the space.
A whole number of nodes and antinodes.
node
antinode

4. Harmonics
The lowest frequency that can fit is the fundamental frequency.
Higher frequencies are called harmonics.

5. Percussion
Percussion instruments strike an object with an impulsive force.
The object oscillates with a fundamental frequency (pitch) and harmonics.
Bulk modulus
Mass density F

6. Strings
String instruments use friction to cause a resonant oscillation.
The frequency is controlled on the instrument string.
Length
Tension
Mass density v
Ffr

7. Open Ends Columns of air can be directly vibrated.
Displacement is maximum at the open end.
Where displacement is maximum, pressure is minimum.

8. Single End
One end open and one end closed creates a different set of harmonics.
These patterns are the basis for wind instruments.

9. Clarinet
The clarinet has an air column 32.8 cm long and acts like a column open at one end.
What is the fundamental frequency?
The first harmonic is
The speed of sound is 343 m/s.
f = (343 m/s) / 4 (0.328 m)
f = 261 Hz

10. Speaker Systems
The sound from a source reaches each ear at a separate time.
Brain uses the time difference to tell direction.
Hard with short wavelength.
Stereo speakers reconstruct timing differences.
Constructive and destructive interference

11. Beats
When two wavelengths are at nearly the same frequency the superposed waves create beats.
The separation of peaks is the period T = 1 / f
The difference in frequencies is the beat frequency Df.
The beat period T = 1 / Df.

12. In Tune
A 440 Hz-tuning fork is held near a vibrating piano string. The tuner hears 20 beats in 5 s.
What is the pitch of the piano string?
Will the tuner ever get it tuned perfectly?
The beat frequency is 20 beats / 5 s = 4 Hz.
The piano string could be either at 444 or 436 Hz.
Note: a beat period of 1 min would get a frequency accurate to Hz.