1. Standing Waves Physics 202 Professor Lee Carkner Lecture 8

2. PAL #7 Wave Energy  How do you find linear density?   Get frequency from function generator or by timing oscillator (f = 40 Hz)   Get tension from hanging weights (hanging mass is 250g so  = mg = (.250)(9.8) = 2.45 N)   =

3. PAL #7 Wave Energy (cont.)  Can you maximize P by maximizing input energy and wave speed?   Can maximize  and minimize  to increase wave speed  Since P = ½  v  2 y m 2, this does not maximize P    Slow waves on a massive string transfer more energy than fast waves on a light string

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

5. 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

6. Standing Wave Amplitudes

7. Equation of a Standing Wave  If the two waves have equations of the form: y 2 = y m sin (kx +  t)  Then the sum is: y r = [2y m sin kx] cos  t   e.g. at places where sin kx = 0 the amplitude is always 0 (a node)

8. Nodes and Antinodes  Consider different values of x (where n is an integer)  For kx = n , sin kx = 0 and y = 0  Node:  Nodes occur every 1/2 wavelength   Antinode:  Antinodes also occur every 1/2 wavelength, but at a spot 1/4 wavelength before and after the nodes

9. Standing Wave on a String

10. Reflecting Waves  How are standing waves produced?   The wave will reflect off of the boundary and travel back up the string   The sign of the pulse will be reversed after the reflection

11. Reflecting Standing Waves  After reflection the string then contains 2 waves traveling in opposite directions   The wave bounces back and forth between the boundaries reinforcing itself as it goes

12. Allowed Standing Waves

13. 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 …

14. Standing Wave on Tacoma Narrows Bridge

15. 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    n=1 is the first harmonic, n=2 is the second etc. 