1. 12-3 Harmonics

2. Harmonics: a set of natural frequencies which are related by being integer multiples of the natural (lowest or fundamental) frequency.

3. String Fixed at both ends
Fundamental (n=1)
2nd harmonic
3rd harmonic

4. Waves in an Open-Open Pipe

5. Waves in an Open-Closed Pipe
For closed pipe - no "even harmonics”: fundamental frequency is a half-loop or ¼ L. Since every harmonic represents the addition of a complete loop, which contains two half-loops, we can never add just one more half-loop. Thus, we cannot generate even harmonics.

6. Example: The Length of an Organ Pipe
An organ pipe open at both ends sounds its 2nd harmonic at a frequency of 523 Hz (one octave above middle C).
What is the length of the pipe from sounding hole to end?

7. Complex (Real) Sound Waves
Harmonics account for the quality of sound or timbre.
Most sounds that we hear are not pure tones (single frequency )
But are superpositions of many frequencies with various amplitudes
For example, when a note (tone, frequency) is played on a musical instrument, we actually hear all of the harmonics (f1, f2, f3, …), but usually the amplitudes are decreased for the higher harmonics
This is what gives each instrument it’s unique sound

8. For example, the sound of a piano is dominated by the 1st harmonic while for the violin, the amplitudes of the 1st, 2nd, and 5th harmonic are nearly equal – gives it a rich sound
Violin wave form

9. Beats Interference from two waves with slightly different frequency.
Alternating loud and faint sound can be heard. This periodic variation in loudness is called beats.
Beats are useful for tuning instruments.

10. Beats f1 f2
fbeat= |f1 - f2|

11. Example Problem The beat frequency is The frequencies become ``tuned’’
When a guitar string is sounded along with a 440-Hz tuning fork, a beat frequency of 5 Hz is heard. When the same string is sounded along with a 436-Hz tuning fork, the beat frequency is 9 Hz. What is the frequency of the string?

12. Solution:
Given: f1=440 Hz, f2=436 Hz, fb1=5 Hz, fb2=9 Hz
But we don’t know if frequency of the string, fs, is greater than f1 and/or f2.
Assume it is:

13. If we chose fs smaller 