1. Wave Energy and Superposition Physics 202 Professor Lee Carkner Lecture 7

2. PAL #6 Waves   T = 2  /  so T A = , T B = 1/3 , T C = 1/4   Which wave has largest transverse velocity?   Largest wave speed?  v =  f = /T, v A = 1, v B = 1.5, v C = 1.3  A: y=2sin(2x-2t), B: y=4sin(4x-6t), C: y=6sin(6x-8t)

3. PAL #6 Waves (cont.)  Wave with y = 2 sin (2x-2t), find time when x= 5.2 cm has max a  Happens when y = y m = 2   1 = sin (2x-2t)    /2 = (2x - 2t)  t = [2x-(  /2)]/2   Maximum velocity when y = 0   2x -2t = arcsin 0 = 0  t = x 

4. Velocity and the Medium   If you send a pulse down a string what properties of the string will affect the wave motion?  Tension (  )   If you force the string up, tension brings it back down  Linear density (  = m/l =mass/length)   You have to convert the PE to KE to have the string move

5. Wave Tension in a String

6. Force Balance on a String Element  Consider a small piece of string  l of linear density  with a tension  pulling on each end moving in a very small arc a distance R from rest  F = (   l)/R  F = (   l) (v 2 /R)  Solving for v, v = (  ) ½  This is also equal to our previous expression for v

7. String Properties  How do we affect wave speed? v = (  ) ½ = f   We set the tension by how hard we pull on the string    The wavelength of a wave on a string depends on how fast you move it and the string properties

8. Tension and Frequency

9. Energy  A wave on a string has both kinetic and elastic potential energy   Every time we shake the string up and down we add a little more energy  This energy is transmitted down the string   The energy of a given piece of string changes with time as the string stretches and relaxes   Assuming no energy dissipation

10. Power Dependency  P=½  v  2 y m 2  If we want to move a lot of energy fast, we want to add a lot of energy to the string and then have it move on a high velocity wave   y m and  depend on the wave generation process

11. Superposition  When 2 waves overlap each other they add algebraically  Traveling waves only add up as they overlap and then continue on   Waves can pass right through each other with no lasting effect

12. Pulse Collision

13. Interference  Consider 2 waves of equal wavelength, amplitude and speed traveling down a string  The waves may be offset by a phase constant  y 2 = y m sin (kx - wt +  )  From the principle of superposition the resulting wave y r is the sum of y 1 and y 2   Is it greater or less than y m ?

14. Interference and Phase  The amplitude of the resultant wave (y mr ) depends on the phase constant of the initial waves  The phase constant can be expressed in degrees, radians or wavelengths  Example:

15. Resultant Equation

16. Combining Waves

17. Types of Interference  Constructive Interference -- when the resultant has a larger amplitude than the originals   No offset or offset by a full wavelength   Destructive Interference -- when the resultant has a smaller amplitude than the originals   Offset by 1/2 wavelength 