1. Interference and Doppler
Questions?
Interference in Space
Interference in Time (Beats)
Doppler Shift
Source moving
Observer moving
Both moving
Examples

2. Interference in Space
Occurs when 2 waves - of same frequency and wavelength - combine constructively or destructively because of different propagation paths.
Two loudspeakers (stereo) animation
Example 12-12
FM car radio
AM car radio
(cellphone)
Noise-cancelling headphones
Diffraction gratings

3. Interference in Space Animation
Two loudspeakers
Two speaker animation
Note: this is Java, not Flash

4. Example 12-13 𝜆= 𝑣 𝑓 = 343 𝑚 𝑠 1150 𝑠 Wavelength
𝜆= 𝑣 𝑓 = 343 𝑚 𝑠 𝑠
=0.298 𝑚
For wavelength of 0.3 m,
must be 0.15 m (half
wavelength) further
for destructive
interference
4.15 m

5. Interference in Time
Occurs when 2 waves - of slightly different frequency and wavelength - combine constructively or destructively with the variation in time.
Produce “Beat” frequencies
Instrument tuning
Superheterodyne radio receiver
Doppler weather radar

6. Interference in Time Derivation
Combine 2 waves with different frequencies
𝑦=𝐴𝑐𝑜𝑠 𝜔 1 𝑡 +𝐴𝑐𝑜𝑠 𝜔 2 𝑡
Use Trig identify to get
𝑦=2𝐴𝑐𝑜𝑠 𝜔 1 − 𝜔 𝑡 𝐴𝑐𝑜𝑠 𝜔 1 + 𝜔 𝑡

7. Interference in Time Animation
Beats in time
Animation
(go to the last scene)

8. Example 12-3 Beat frequency is Guitar must be either 396 Hz or 404 Hz
∆𝑓= 20 𝑣𝑖𝑏𝑟𝑎𝑡𝑖𝑜𝑛𝑠 5 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 =4 𝐻𝑧
Guitar must be either 396 Hz or 404 Hz

9. Superheterodyne radio receiver
For FM, local oscillator always 10.7 MHz above desired station frequency
Beat frequency of 10.7 MHz emerges.
Can build very selective 10.7 MHz filters!

10. Doppler Shift (Source moving toward)
Wavelength compressed during wave creation
Source moving toward observer compresses wavelength
λ ′ =λ− 𝑣 𝑠𝑜𝑢𝑟𝑐𝑒 𝑇
But period T involves v and λ in still air
𝑇= 1 𝑓 = 1 𝑣 𝑠𝑜𝑢𝑛𝑑 𝜆 = λ 𝑣 𝑠𝑜𝑢𝑛𝑑
λ ′ =λ− 𝑣 𝑠𝑜𝑢𝑟𝑐𝑒 𝑣 𝑠𝑜𝑢𝑛𝑑 λ
That shortened wave passing you ear sounds like
𝑓 ′ = 𝑣 𝑠𝑜𝑢𝑛𝑑 λ ′ = 𝑣 𝑠𝑜𝑢𝑛𝑑 λ 1− 𝑣 𝑠𝑜𝑢𝑟𝑐𝑒 𝑣 𝑠𝑜𝑢𝑛𝑑
𝑓 ′ = 𝑓 1− 𝑣 𝑠𝑜𝑢𝑟𝑐𝑒 𝑣 𝑠𝑜𝑢𝑛𝑑
Moving away change “-” to “+”

11. Doppler shift
Frequency and/or wavelength changed due to motion of source and/or listener
Animation (moving source)
(Play scene 1)
s.html
(Break the sound barrier!)

12. Doppler Shift (Observer moving toward)
Frequency increased as wave passes your ear
Observer moving toward source increases frequency
𝑓 ′ = 𝑣 𝑠𝑜𝑢𝑛𝑑 + 𝑣 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑟 λ
𝑓 ′ = 𝑓 𝑣 𝑠𝑜𝑢𝑛𝑑 + 𝑣 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑟 𝑣 𝑠𝑜𝑢𝑛𝑑
𝑓 ′ =𝑓 1+ 𝑣 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑟 𝑣 𝑠𝑜𝑢𝑛𝑑
Summary
𝒇 ′ = 𝒇 𝟏∓ 𝒗 𝒔𝒐𝒖𝒓𝒄𝒆 𝒗 𝒔𝒐𝒖𝒏𝒅 𝑺𝒐𝒖𝒓𝒄𝒆 (−𝒕𝒐𝒘𝒂𝒓𝒅, +𝒂𝒘𝒂𝒚)
𝒇 ′ =𝒇 𝟏± 𝒗 𝒐𝒃𝒔𝒆𝒓𝒗𝒆𝒓 𝒗 𝒔𝒐𝒖𝒏𝒅 𝑶𝒃𝒔𝒆𝒓𝒗𝒆𝒓 (+𝒕𝒐𝒘𝒂𝒓𝒅, −𝒂𝒘𝒂𝒚)

13. Example 12-14 Toward Away Frequency shifts down as it passes you!
𝑓 ′ = 𝑓 1− 𝑣 𝑠𝑜𝑢𝑟𝑐𝑒 𝑣 𝑠𝑜𝑢𝑛𝑑 = 1600 𝐻𝑧 1− 25 𝑚/𝑠 343 𝑚/𝑠 =1726 𝐻𝑧
Away
𝑓 ′ = 𝑓 1+ 𝑣 𝑠𝑜𝑢𝑟𝑐𝑒 𝑣 𝑠𝑜𝑢𝑛𝑑 = 1600 𝐻𝑧 𝑚/𝑠 343 𝑚/𝑠 =1491 𝐻𝑧
Frequency shifts down as it passes you!

14. Example 12-15 Doppler shift 1 (listener moving toward sound)
𝑓 ′ = 1+ 𝑣 𝑜𝑏𝑠 𝑣 𝑠𝑛𝑑 𝑓= 𝑚/𝑠 343 𝑚/𝑠 𝐻𝑧
=5051 𝐻𝑧
Doppler shift 2 (listener re-emitting sound at shorter wavelength)
𝑓 ′′ = 𝑓′ 1− 𝑣 𝑠𝑜𝑢𝑟𝑐𝑒 𝑣 𝑠𝑜𝑢𝑛𝑑 = 5051 𝐻𝑧 1− 3.5 𝑚/𝑠 343 𝑚/𝑠
=5103 𝐻𝑧

15. Problem 54 - Bat squeaking
Doppler shift 1 (bat creating squeak while flying toward wall)
𝑓 ′ = 𝑓 1− 𝑣 𝑠𝑜𝑢𝑟𝑐𝑒 𝑣 𝑠𝑜𝑢𝑛𝑑 = 30 𝑘𝐻𝑧 1− 5 𝑚/𝑠 343 𝑚/𝑠
=30.44 𝑘𝐻𝑧
Doppler shift 2 (bat listening to returning echo while flying toward wall)
𝑓 ′′ = 1+ 𝑣 𝑜𝑏𝑠 𝑣 𝑠𝑛𝑑 𝑓′= 1+ 5 𝑚/𝑠 343 𝑚/𝑠 𝑘𝐻𝑧
=30.88 𝑘𝐻𝑧

16. Doppler Shift for Electromagnetic Waves
Doppler weather radar
Red shift
(Bell Labs – Cosmic background radiation)

17. Problem 55 - Tubas and Beats
Only one Doppler shift (tuba emitting sound while moving toward station)
𝑓 ′ = 𝑓 1− 𝑣 𝑠𝑜𝑢𝑟𝑐𝑒 𝑣 𝑠𝑜𝑢𝑛𝑑 = 75 𝐻𝑧 1− 10 𝑚/𝑠 343 𝑚/𝑠
=77 𝐻𝑧
Beat frequency with tuba at station
∆𝑓=77 𝐻𝑧 −75 𝐻𝑧=2 𝐻𝑧

18. Problem 56 – Blood flow
First Doppler shift (blood moving away from incoming sound)
𝑓 ′ = 1− 𝑣 𝑜𝑏𝑠 𝑣 𝑠𝑛𝑑 𝑓= 1− 0.02 𝑚/𝑠 1540 𝑚/𝑠 3.5 𝑀𝐻𝑧
= 𝑀𝐻𝑧
Second Doppler shift (blood moving away while re-emitting reflected sound)
𝑓 ′′ = 𝑓′ 1+ 𝑣 𝑠𝑜𝑢𝑟𝑐𝑒 𝑣 𝑠𝑜𝑢𝑛𝑑 = 𝑀𝐻𝑧 𝑚/𝑠 1540 𝑚/𝑠
= 𝑀𝐻𝑧
Beat frequency
∆𝑓= 𝑀𝐻𝑧 − 𝑀𝐻𝑧= 𝑀𝐻𝑧=91 𝐻𝑧

19. Diagnose heart-valve problems with Ultrasound
Medical Applications
Diagnose heart-valve problems with Ultrasound
Measure blood velocity before and after valve using Doppler shift.*
Get velocity difference very accurately using beat frequencies.
Get pressure difference before and after valve using Bernoulli’s equation.
𝑃 ℎ 𝜌 𝑣 ℎ 2 = 𝑃 𝑙 𝜌 𝑣 𝑙 2
Get valve area from pressure difference.
*Just like chapter 12, problem 56!
High pressure
Low pressure
flow

20. End of Waves and Sound
Next Thermodynamics