Wave Energy and Superposition Physics 202 Professor Lee Carkner Lecture 7

PAL #5 Waves  =2  /k so A = , B =  /2, C =  /3  T = 2  /  so  Which wave has largest transverse velocity?  Wave C:  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 #5 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  t =  Maximum velocity when y = 0  0 =  2x -2t = arcsin 0 = 0  t = x  t =

Velocity and the Medium  The speed at which a wave travels depends only on the medium   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

String Properties  How does wave speed depend on the string? 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  We input this energy when we start the wave by stretching the string   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  The average power (energy per unit time) is thus: P=½  v  2 y m 2   v and  depend on the string  y m and  depend on the wave generation process

Superposition  y r = y 1 +y 2  Traveling waves only add up as they overlap and then continue on   Waves can pass right through each other with no lasting effect

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  Fully constructive --  No offset or offset by a full wavelength   Destructive Interference -- when the resultant has a smaller amplitude than the originals  Fully destructive --  Offset by 1/2 wavelength 

Next Time  Read:

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