1. PHY 231 1 PHYSICS 231 Lecture 32: interference & sound Remco Zegers Question hours:Tue 4:00-5:00 Helproom

2. PHY 231 2 example A pendulum with a length of 4 m and a swinging mass of 1 kg oscillates with an maximum angle of 10 o. What is the gravitational force parallel to the string, perpendicular to the string, the total gravitational force and the centripetal force when the mass passes through the equilibrium position and when it reaches its maximum amplitude? equilibriummaximum ampl. gravitation // to string 1gcos(0 0 )=9.8N1gcos(10 0 )=9.65N gravitation perpendicular 1gsin(0 0 )=0N1gsin(10 0 )=1.7N total gravitational9.8N9.8N (vectors!!) centripetal0.3N (mv 2 /L)0 N

3. PHY 231 3 describing a traveling wave While the wave has traveled one wavelength, each point on the rope has made one period of oscillation. v=  x/  t= /T= f : wavelength distance between two maxima. On a string: v=  (F/  )

4. PHY 231 4 quiz (extra credit) A person is tuning a guitar string. He makes the tension in the string 4 times larger than it originally was. If the wavelength of the oscillations through the string remains constant, by what factor does the frequency of the produced sound change? a)¼ b)½ c)1 d)2 e)4 v=  (F/  ) if Fx4 then vx2 v=  x/  t= /T= f if vx2 then fx2 if =constant given: v=  (F/  ) v=  x/  t= /T= f

5. PHY 231 5 Interference Two traveling waves pass through each other without affecting each other. The resulting displacement is the superposition of the two individual waves. example: two pulses on a string that meet

6. PHY 231 6 Interference II constructive interference destructive interference

7. PHY 231 7 Interference III constructive interference waves in phase demo: interference + = destructive interference waves ½ out of phase + =

8. PHY 231 8 Interference IV Two interfering waves can at times constructively interfere and at times destructively interfere If the two interfering waves always have the same vertical displacement at any point along the waves, but are of opposite sign: standing waves later more!!!

9. PHY 231 9 Interference holds for any wave type The pulses can be sine-waves, rectangular waves or triangular waves

10. PHY 231 10 Interference in spherical waves maximum of wave minimum of wave positive constructive interference negative constructive interference destructive interference if r 2 -r 1 =n then constructive interference occurs if r 2 -r 1 =(n+½) the destructive interference occurs r1r1 r2r2 r 1 =r 2

11. PHY 231 11 Interference of water waves

12. PHY 231 12 Example 0.7m direction of walking person two speakers separated by 0.7m produce a sound with frequency 690 Hz (from the same sound system). A person starts walking from one of the speakers perpendicular to the line connecting the speakers. After what distance does he reach the first maximum? And the first minimum? v sound =343 m/s d1d1 d2d2 v= f so =v/f=343/690=0.5m 1 st maximum: d 1 -d 2 =1 =0.5 d 2 =  (0.7 2 +d 1 2 ) d 1 -  (0.7 2 +d 1 2 )=0.5 d 1 =(-)0.24m 1 st minimum: d 1 -d 2 =½ =0.25 d 1 -  (0.7 2 +d 1 2 )=0.25 d 1 =(-)0.855

13. PHY 231 13 Reflection of waves. F rope on wall = -F wall on rope FIXED END: pulse inversion FREE END: no inversion demo: rope on wall

14. PHY 231 14 Connecting ropes If a pulse travels from a light rope to a heavy rope (  light <  heavy ) the boundary is nearly fixed. The pulse is partially reflected (inverted) and partially transmitted. before after If a pulse travels from a heavy rope to a light rope (  light <  heavy ) the boundary is nearly free. The pulse is partially reflected (not inverted) and partially transmitted. before after A in ARAR ATAT ARAR ATAT |A R |<|A in | |A T |<|A in | |A R |<|A in | |A T |>|A in |

15. PHY 231 15 Sound: longitudinal waves

16. PHY 231 16 The speed of sound Depends on the how easy the material is compressed (elastic property) and how much the material resists acceleration (inertial property) v=  (elastic property/inertial property) v=  (B/  ) B: bulk modulus  : density The velocity also depends on temperature. In air: v=331  (T/273 K) so v=343 m/s at room temperature

17. PHY 231 17 Quick question The speed of sound in air is affected in changes in: a)wavelength b)frequency c)temperature d)amplitude e)none of the above answer c)

18. PHY 231 18 Intensity Intensity: rate of energy flow through an area Power (P) J/s A (m 2 ) I=P/A (J/m 2 s=W/m 2 ) example: If you buy a speaker, it gives power output in Watts. However, even if you put a powerful speaker in a large room, the intensity of the sound can be small.

19. PHY 231 19 Intensity Faintest sound we can hear: I~1x10 -12 W/m 2 (1000 Hz) Loudest sound we can stand: I~1 W/m 2 (1000 Hz) Factor of 10 12 ? Loudness works logarithmic…

20. PHY 231 20 decibel level   =10log(I/I 0 ) I 0 =10 -12 W/m 2 y=log 10 x inverse of x=10 y (y=ln(x) x=e y ) log(ab)=log(a)+log(b) log(a/b)=log(a)-log(b) log(a n )=nlog(a)

21. PHY 231 21 decibels  =10log(I/I 0 ) I 0 =10 -12 W/m 2 An increase of 10 dB: intensity of the sound is multiplied by a factor of 10.  2 -  1 = 10 10=10log(I 2 /I 0 )-10log(I 1 /I 0 ) 10=10log(I 2 /I 1 ) 1=log(I 2 /I 1 ) 10=I 2 /I 1 I 2 =10I 1

22. PHY 231 22 Frequency vs intensity 1000 Hz

23. PHY 231 23 example A machine produces sound with a level of 80dB. How many machines can you add before exceeding 100dB? 1 machine 80 dB=10log(I/I 0 ) 8=log(I/I 0 )=log(I/1E-12) 10 8 =I/1E-12 I 1 =10 -4 W/m 2 ?? machines 100 dB=10log(I/I 0 ) 10=log(I/I 0 )=log(I/1E-12) 10 10 =I/1E-12 I ?? =10 -2 W/m 2 I 1 /I ?? =10 -4 /10 -2 =1/100 The intensity must increase by a factor of 100; one needs to add 99 machines.