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Mechanical Waves velocity

  1. Dec 6, 2015 #1
    1. The problem statement, all variables and given/known data
    The linear mass density of a non-uniform wire under constant tension gradually decreases along the wire while ensuring an incident wave is transmitted without reflection. The wire has constant density μ for x ≤ 0. In this region a transverse wave has the form y(x,t) = 0.003cos(30x -60t), where x,y are given in meters and t in seconds. From x=0 to x=20, the linear mass density decreases from μ to μ/9. For x > 20, the density remains constant at μ/9.

    i) What is the wave velocity for large values of x (>20m)?
    ii) What is the amplitude of the wave for large values of x? You should be able to determine this using conservation of mechanical energy,
    iii)Give y(x,t) for x>20.


    2. Relevant equations

    y(x,t) = Acos(kx-wt)
    w/k = v
    v= sqrt(T/μ)
    P = 1/2sqrt(μT)w2A2

    3. The attempt at a solution
    i) Comparing given wave equation and y=Acos(kx-wt) for x < 0
    k = 30 w = 60
    v = 60/30 =2
    v2μ = T
    4μ=T
    Tension remains constant so
    v20 = √4μ/μ/9 = 6m/s

    ii) E = 1/2mv2. At max kinetic energy potential energy = 0. Total energy = Kinetic energy
    P = ½√(μF)w2A2
    Energy before = Energy After
    ½√(μ4μ).6020.0032 =½√(μ/9.4μ).v2k2A2
    0.0027=k2A2

    I have no idea if this is correct and if i can take k to be constant or not.
    Any help would be really appreciated!
     
  2. jcsd
  3. Dec 6, 2015 #2

    TSny

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    Welcome to PF!

    What stays constant as a wave travels from one medium to another: the frequency or the wavelength?
     
  4. Dec 6, 2015 #3
    Thank you!
    The frequency stays constant and as the wave travels the wavelength would increase i think. That would mean that k wouldn't be constant.
     
  5. Dec 6, 2015 #4
    Could I work out the frequency initially as f = v/λ = vk/2pi = 30pi and then as frequency stays the same work out k when x=20 by
    k=2fpi/v = 60pi2/6 ?

    Thank you for helping me out!
     
  6. Dec 6, 2015 #5

    TSny

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    Yes, frequency remains constant. What can you say about ω?

    How does this help when thinking about the power P?
     
  7. Dec 6, 2015 #6
    ω = 2pif so would also be constant. This means I now know all the variables in the power equation apart from the Amplitude so by setting power before to power after I should hopefully be able to get the correct amplitude. Thank you so much!
     
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