Announcements

1. Announcements • Final exam is next Monday, April 28, 3:30-5:30 PM • Must earn at least 25% to pass class • Section 001 – S BEH AUD • Section 006 – FMAB AUD • Format: 10 T/F (3 pts); 10 MC3 (5 pts); 20 MC5 (7 pts) = 220 points (200 is perfect score) • Bring pencils, calculator, and UID card • No cheat sheets allowed! • Study session on Friday, April 25 in 2Creek coffee shop(WBB) • Prof. Gerton will be around all day long (7:30A - 5P) • $2 discount on all drinks during that time (for 2210 students only!!) • 3 Review Sessions: 9-11 am (Ruth); 12-2 pm (Vikrant); 3-5 pm (Lauren or Nolan). I will announce the room shortly.

2. Announcements Remaining Assignments: • New and old Homework:All homework through Unit 24 is due on Monday, April 28 at 3:30 pm • Optional Assignments: PL/CP/HW for Units 25 & 26 are due on Friday, May 2. 20 points total. • Course survey: Will also be due on Friday, May 2 (available now). 10 points total. Window for reporting errors in the Canvas gradebook: • I’ll post final scores for all assignments in Canvas by Sunday, May 4. • I’ll send an email when this is done. • You’ll have through Wednesday, May 7 to report any errors.

3. Error in smartPhysics book • On page 259 and 267, there is an error: • Velocity of a wave on a string should be: • Book says: • Prelecture gets it right! • Formula sheet gets it right!

4. Lecture 24: Waves and Superposition Today’s Concepts: A) Superposition B) Standing Waves

5. CheckPoint: Wave Pulse y v Case A x y Case B x v Suppose a pulse in Case A described by the function y(x,t) = P(x-vt). Which of the following functions described the pulse in Case B? A) y(x,t)=P(x +vt) B) y(x,t)= -P(x +vt) C) y(x,t) = -P(x -vt)

6. CheckPointDiscuss: Wave Pulse y v Case A x y Case B x v A) y(x,t)=P(x +vt) B) y(x,t)= -P(x +vt) C) y(x,t) = -P(x -vt) A) Everything is the same except the direction of the velocity is in the opposite direction. B) x+vt makes the wave go in the negative x direction, -P flips the wave upside down C) This is the only equation that the waves will move to the left.

7. Superposition Q: What happens when two waves “collide?” A: They ADD together!

8. Superposition A(ω1t) B(ω2t) C(t)= A(t)+ B(t) The wave equation we discussed last time is linear. (It has no terms where the variables x, tare squared.) For linear equations, if we have two separate solutions, A and B, then A + B is also a solution! ω2 > ω1 DESTRUCTIVEINTERFERENCE CONSTRUCTIVEINTERFERENCE

9. Beats Can we predict this pattern mathematically? Of course! Just add two cosines and remember the identity: where and cos(ωLt) cos(ωHt)

10. Standing Waves What happens when two waves having the same frequency but moving in the opposite direction “collide”? Wave 1 Wave 1 Wave 1 Wave 2 Wave 2 Wave 2 zero (Nodes) Superposition

11. How it Works Stationary wave Changingamplitude How to make it:

12. CheckPoint: Guitar Suppose the strings on your guitar are 24” long as shown. The frets are the places along the neck where you can put your finger to make the wavelength shorter, and appear as horizontal white lines on the picture. When no frets are being pushed the frequency of the highest string is 4 times higher than the frequency of the lowest string. Is it possible to play the lowest string with your finger on any of the frets shown and hear the same frequency as the highest string? A) Yes B) No 24 20 frets 16 12 8 4

13. ACT If you want to increase the frequency of a 24” string by a factor of 4while keeping the tension the same, how long should the string be? A) 4”B) 6”C) 8”D) 12” 24 20 Hint: A guitar string tends to vibrate as a standing wave with the minimum number of wiggles (i.e., the smallest possible number of nodes). frets 16 12 8 4

14. ACT 6 Is there a fret that you can push on this guitar to make the string 6”long? A) Yes B) No 24 20 frets 16 12 8 4

15. CheckPoint Results: Guitar Suppose the strings on your guitar are 24” long as shown. The frets are the places along the neck where you can put your finger to make the wavelength shorter, and appear as horizontal white lines on the picture. When no frets are being pushed the frequency of the highest string is 4 times higher than the frequency of the lowest string. Is it possible to play the lowest string with your finger on any of the frets shown and hear the same frequency as the highest string? A) Yes B) No 24 20 frets 16 12 8 4

16. ACT The highest string vibrates with a frequency that is 4 times that of the lowest string. Compare the speed of a wave on the high string and the low string: A) vhigh = vlow*2 B) vhigh = vlow*4 C) vhigh = vlow*16 λ is set by the length of the string: it’s the same for both strings! vhigh vlow

17. ACT The speed of a wave on the high string is 4 times the speed on the lowest string. If the tensions are the same, how does the mass per unit length of the strings compare: A) μlow =μhigh*2 B) μlow = μhigh*4 C) μlow = μhigh*16 vhigh vlow

18. ACT Two cylinders have the same length and are made from the same material. If the mass of the bigger one is 16 times the mass of the smaller one, how do their diameters compare? A) dbig =dsmall*2 B) dbig = dsmall*4 C) dbig = dsmall*16 L dsmall dbig

19. CheckPoint: Guitar Strings dhigh dlow The 6 strings on a guitar all have about the same length and are stretched with about the same tension. The highest string vibrates with a frequency that is 4 times that of the lowest string. If the strings are made of the same material, how would you expect the diameters of the lowest and highest strings to compare? A) dlow =dhigh*2 B) dlow = dhigh*4 C) dlow = dhigh*16

20. .012” dhigh .050” dlow We just found that the frequency of a string is inversely proportional to its diameter if the tension is the same. This is really what you find if you examine a guitar: Why is it important for the strings to have about the same tension?

21. Different Example – 12 String Guitar The lighter string in these pairs is one octave higher (2x frequency). .028 .052 .022 .042 .014 .030 Their diameters are about half of the adjacent strings.

22. Tacoma-Narrows Bridge