Wave Energy and Superposition Physics 202 Professor Lee Carkner Lecture 7

Consider mark made on a piece of string with a wave traveling down it. At what point does the mark have the largest velocity? : At what point does the mark have the largest acceleration? a)In the middle : At the top b)At the top : In the middle c)In the middle : In the middle d)At the top: At the top e)Velocity and acceleration are constant

Suppose you are producing a wave on a string by shaking. What properties of the wave do you directly control? a)Amplitude b)Wavelength c)Frequency d)Propagation velocity e)a and c only

PAL #6 Waves  =2  /k so A = , B =  /2, C =  /3  T = 2  /  so T A = , T B = 1/3 , T C = 1/4   Which wave has largest transverse velocity?  Wave C: largest amplitude, shortest period  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)

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  2 = 2 sin (2x-2t)  1 = sin (2x-2t)  arcsin 1 = 2x-2t   /2 = (2x - 2t)  t = [2x-(  /2)]/2  t = 4.4 seconds  Maximum velocity when y = 0  0 = sin (2x-2t)  2x -2t = arcsin 0 = 0  t = x  t = 5.2 seconds

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

Wave Tension in a String

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  There is a force balance between tension force:  and centripetal force:  Solving for v,  This is also equal to our previous expression for v v = f

String Properties  How do we affect wave speed? v = (  ) ½ = f   Wave speed is solely a property of the medium   The wavelength then comes from the equation above  The wavelength of a wave on a string depends on how fast you move it and the string properties

Tension and Frequency

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

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

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

Pulse Collision

Interference   The waves may be offset by a phase constant  y 1 = y m sin (kx -  t) y 2 = y m sin (kx -  t +  )  y r = y mr sin (kx -  t +½  )  What is y mr (the resulting amplitude)?  Is it greater or less than y m ?

Interference and Phase  y mr = 2 y m cos (½  )  The phase constant can be expressed in degrees, radians or wavelengths  Example: 180 degrees =  radians = 0.5 wavelengths

Resultant Equation

Combining Waves

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 

Next Time  Read:  Homework: Ch 16, P: 20, 30, 40, 83