1. PHY 231 1 PHYSICS 231 Lecture 37: standing waves & harmonics Remco Zegers Last tutorial: tomorrow at 5:30 pm in CEM138 Last lecture: Friday (Review)

2. PHY 231 2 standing waves 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

3. PHY 231 3 How to create standing waves: a rope The oscillations in the rope are reflected from the fixed end (amplitude is reversed) and create a standing wave. demo

4. PHY 231 4 we can produce different wave lengths 1 =2L 2 =L 3 =2L/3 4 =2L/4 5 =2L/5 both ends fixed n =2L/n or L=n n /2

5. PHY 231 5 standing waves both ends fixed n =2L/n or L=n n /2 F: tension in rope  : mass per unit length n th harmonics f 1 : fundamental frequency

6. PHY 231 6 example: the guitar n th harmonics: depends where and how the string is struck note that several harmonics can be present and that non-harmonics are washed out length can be chosen by placing fingers changes from string to string: bass string is very heavy tension can be varied by stretching the wire

7. PHY 231 7 example A guitar string is struck. Assume that the first harmonic is only excited. What happens to the frequency if: a)The player put a finger at half the length of the string? b)The player makes the tension 10% larger (by turning the tuning screw)? c)A string is struck in the same way, but its mass is 3 times higher? a)L x 0.5 then f x 2 b)F x 1.1 then f x  1.1=1.05 c)  x 3 then f x  (1/3)=0.58

8. PHY 231 8 Standing waves in air columns Just like standing waves in transverse oscillations, one can make standing waves in longitudinal oscillations as well.

9. PHY 231 9 An air column (pipe) A pipe can be open or closed on either or both sides. For now, let’s consider the air-displacements (anti-)nodes

10. PHY 231 10 Both ends open

11. PHY 231 11 One end open, one end closed even harmonics are missing!!!

12. PHY 231 12 example A simple flute is played by blowing air in on one side and the other end is open. The length of the tube can be varied manually (like a trombone). What are the frequencies of the first two possible harmonics if L=0.5m? If the length is made half of the original length, how will these change v=343m/s? f 1 =343/(4*0.5)=172 Hz f 3 =3*343/(4*0.5)=514 Hz f 1 =343/(4*0.25)=343 Hz f 3 =3*343/(4*0.25)=1028 Hz

13. PHY 231 13 example A simple flute is played by blowing air in on one side and the other end is closed. The length of the tube can be varied manually (like a trombone). What are the frequencies of the first two possible harmonics if L=0.5m? If the length is made half of the original length, how will these change v=343m/s? f 1 =343/(2*0.5)=343 Hz f 2 =2*343/(2*0.5)=686 Hz f 1 =343/(2*0.25)=686Hz f 2 =2*343/(2*0.25)=1372 Hz

14. PHY 231 14 beats Superposition of 2 waves with slightly different frequency The amplitude changes as a function of time, so the intensity of sound changes as a function of time. The beat frequency (number of intensity maxima/minima per second): f beat =|f a -f b | DEMO

15. PHY 231 15 example Someone is trying to tune a guitar. One of the strings is supposed to have a frequency of 500 Hz. The person is using a tuning fork which produces a sound of exactly this frequency, but while sounding the fork and the playing the guitar, hears a beat in the sound with a frequency of 3 Hz (3 beat per second). a) What is the real frequency of the guitar string? b) By what fraction does the person need to change the tension of the guitar string to tune it properly? a) f b =|f fork -f guitar | 3=|500-f guitar | f guitar =497 or 503 Hz b) so f~  F f current /f ideal =  (F current /F ideal ) 497/500=0.954 or 503/500=1.006 F ideal =F current /(0.994) 2 =1.012F current or F ideal =F current /(1.006) 2 =0.988F current

16. PHY 231 16 Resonances Realistically, oscillations are damped due to frictional forces. However, we can drive the oscillation via an external source. Example: mass on a spring: natural frequency f=1/(2  )  (k/m) If the frequency of the driving force equals the natural frequency: large oscillations occur: Resonance demo Resonances occur in many daily situations: shock absorber in car playing basketball resonating lecture room!! Famous example: Tacoma bridge

17. PHY 231 17 review Most important? midterm 1 midterm 2 midterm 3 midterm 4 new material 2 nd most important? midterm 1 midterm 2 midterm 3 midterm 4 new material type? a) Conceptual b) Numerical