1. Standing Waves Physics 202 Professor Lee Carkner Lecture 8

2. PAL #7 Wave Energy  How do you find linear density?  v = f = (  /  ) ½ or  =  /f 2 2  Get frequency from function generator or by timing oscillator (f = 27.76 Hz)  Get wavelength by measuring on string ( = 70 cm = 0.7 m)  Get tension from hanging weights (hanging mass is 100g so  = mg = (0.1)(9.8) = 0.98 N)   =0.0026 kg/m or 2.6 g/m

3. What kind of string propagates waves the fastest? a)Heavy and tight b)Heavy and loose c)Light and loose d)Light and tight e)We can’t know wave speed without knowing the input frequency

4. How would you modify the wave generator to input the maximum amount of energy? a)Increase frequency, increase amplitude b)Increase frequency, decrease amplitude c)Decrease frequency, increase amplitude d)Decrease frequency, decrease amplitude e)Input energy is independent of frequency and amplitude

5. What kind of string transmits energy the fastest? a)Heavy and tight b)Heavy and loose c)Light and loose d)Light and tight e)All strings transmit energy at the same rate

6. Consider a wave traveling along a string that can be combined with three otherwise identical waves with phase shifts of 0.5 , 1.0 , and 1.9  radians. Rank the resulting wave by amplitude, largest first. a)0.5 , 1.0 , and 1.9  b)1.9 , 1.0 , 0.5  c)1.0 , 0.5 , 1.9  d)1.9 , 0.5 , 1.0  e)0.5 , 1.9 , 1.0 

7. Exam #1 Friday  About 1/3 multiple choice  Study notes  Study Quizdom questions  Look at textbook “Checkpoint” questions  About 2/3 problems  Study PAL’s and SuperPALS  Study old homework  Do new practice homework questions  Try to do this with just equation sheet  Need (real) calculator and pencil

8. Standing Waves  The two waves will interfere, but if the input waves do not change, the resultant wave will be constant   Nodes --  Antinodes -- places where the amplitude is a maximum (only place where string has max or min displacement)  The positions of the nodes and antinodes do not change, unlike a traveling wave

9. Standing Wave Amplitudes

10. Equation of a Standing Wave  y 1 = y m sin (kx -  t) y 2 = y m sin (kx +  t)  Then the sum is:  The amplitude varies with position  e.g. at places where sin kx = 0 the amplitude is always 0 (a node)

11. Nodes and Antinodes  Consider different values of x (where n is an integer)   Node: x=n ( /2)   For kx=(n+½) , sin kx = 1 and y=2y m  Antinode:  Antinodes also occur every 1/2 wavelength, but at a spot 1/4 wavelength before and after the nodes

12. Resonance Frequency  When do you get resonance?   Since you are folding the wave on to itself   You need an integer number of half wavelengths to fit on the string (length = L)  In order to produce standing waves through resonance the wavelength must satisfy: = 2L/n where n = 1,2,3,4,5 …

13. Resonance?  Under what conditions will you have resonance?   n is the number of loops on a string   v = (  ) ½ = f   Can find new in terms of old and see if it is an integer fraction or multiple

14. Harmonics  We can express the resonance condition in terms of the frequency (v=f or f=v/ ) f=(nv/2L)   Remember v depends only on  and   The number n is called the harmonic number   For cases that do not correspond to the harmonics the amplitude of the resultant wave is very low (destructive interference)

15. Generating Harmonics  Many devices are designed to produce standing waves   Frequency corresponds to note   Can produce different f by  changing v   Changing L 

16. Next Time  Test #1  Next class, Monday, January 7  Read 17.1-17.4  Homework: 17.1, 17.4, 17.8