1. Sound – Beats
Superposition leads to beats – variation in loudness

2. Resonance Occurs in fixed media length L
Depends on boundary conditions
Occurs only for certain frequencies for which an integral number of ½ or ¼ wavelengths fit in medium length L
Affected by propagation speed in medium

3. Boundary Conditions Ends of medium are either a node or antinode
Determined by the configuration of the system
Stringed instruments → both ends are nodes
Woodwinds, brass, organs → one end is always an antinode, the other end can be either

4. Procedure Identify physical situation (string or air column)
Identify boundary conditions (nodes or antinodes)
If the same use ½ wavelengths
If different use ¼ wavelengths
Sketch waveforms → determine how many ½ or ¼ λ fit in length L
For each allowed n, solve for λn
Determine propagation speed v
Use v = f λ to find fn

5. Harmonic Series of Standing Waves on a Vibrating String
Both ends are nodes (N) – use ½ λ s
Sketch
Solve for λn
If v not known use
Solve for fn

6. Harmonic Series of a Pipe Open at Both Ends
Both ends are antinodes (A) – use ½ wavelengths
Sketch
Solve for λn
If v not known use
Solve for fn

7. Harmonic Series of a Pipe Open at One End
One end A, other end N – use ¼ wavelengths
Sketch
Solve for λn
If v not known use
Solve for fn

8. Harmonics for Same Ends
Note that when ends are both nodes or both antinodes, n = 1, 2, 3, 4, …
f2 = 2f1 → 2nd harmonic or 1st overtone
f3 = 3f1 → 3rd harmonic or 2nd overtone
f4 = 4f1 → 4th harmonic or 3rd overtone
f5 = 5f1 → 5th harmonic or 4th overtone
And so on …
fn = nf1

9. Harmonics for Different Ends
Note that when one end is N and the other is A, n = 1, 3, 5, 7, …
f3 = 3f1 → 3rd harmonic or 2nd overtone
f5 = 5f1 → 5th harmonic or 4th overtone
f7 = 7f1 → 7th harmonic or 6th overtone
f9 = 9f1 → 9th harmonic or 8th overtone
And so on … fn = nf1
→ only odd harmonics are allowed

10. Propagation Speed
In a string v depends on the tension T and linear mass density μlin
In an air column v depends on temperature T

11. Sound Levels Decibel or dB scale I0 is the threshold of hearing
Threshold of pain is 120 dB

13. The Doppler Effect
An observer of the wave crests behind or away from the direction of motion would identify a lower frequency due to a greater travel distance from the vibration source.
This apparent change in frequency is called the DOPPLER EFFECT after Christian Doppler (1803 – 1853)
The greater the speed, the greater the Doppler effect
The police and weather reporters use the Doppler effect of radar waves to measure the speeds of cars or water in clouds.

14. The Doppler Effect
Vibrating in a stationary position produces waves in concentric circles. (because the wave speed is the same in all directions)
Vibrating while moving at a speed less than the wave speed produces nonconcentric circles. The wave crests in the direction of motion occur more often (have a higher frequency) due to a shorter travel distance.

15. Doppler Effect
If source (s) is moving, detected frequency (fd) changes
If detector (d) is moving, detected frequency changes
Top sign when moving away from
Bottom sign when moving towards

16. Astronomy and the Doppler Shift
Astronomers can measure the Doppler Shift of a moving object.
The radio wave has a known frequency for stationary objects.
If target is moving away, waves in the signal will be stretched, and will have a lower frequency.
The Doppler shift is called
“Red shift” if v > 0 (moving away) since the wavelength is getting longer and the frequency shorter
“Blue Shift” if v < 0 (moving closer) since the wavelength is getting shorter and the frequency higher (recall that blue light is higher in frequency than red)

17. TWO CLIPS ON THE DOPPLER EFFECT

18. Bow Waves
When the vibration source is moving at the same speed as the wave front, a bow wave is produced.
In aircraft this is called the “sound barrier”. It is not a real barrier, but is where the wave crests build up on each other at the speed of sound.
Speedboats and supersonic aircraft
travel in front of their bow waves.

19. Shock Waves
BOOM !
Shock waves produced by supersonic aircraft are the 3-dimensional version of the V-shaped bow wave produced by boats.
On the ground this is noted as a sonic boom.

20. SIMULATIONS FOR DOPPLER EFFECT AND BOW WAVES