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Exercise about the wavefunction

  1. Jun 13, 2016 #1
    1. The problem statement, all variables and given/known data

    Consider an ideal rope where there is a wave moving at velocity ##v=20 m/s##. The displacement of one end of the rope is given by
    $$s(t)=0.1 \mathrm{sin}(6 t)$$
    a) Find the wavefunction ##\xi(x,t)##, knowing that it is progressive
    b) Find the distance ##\delta## (in absolute value) between two points of the rope, that, at a certain time istant, are displaced from Equilibrium position of ##0.02 \mathrm{m}##
    ##[\mathrm{Result} \, \delta=1.34 m]##
    2. Relevant equations

    A general wavefunction is
    $$\xi(x,t)=\xi_0 \mathrm{sin}( k x-\omega t +\psi)\tag{1}$$
    3. The attempt at a solution

    a) My boundary condition for ##(1)## is that
    $$\xi(0,t)=0.1 \mathrm{sin}(6 t)\tag{2}$$
    Now is it correct to conclude that ##\xi_0=-0.1 m##, ##\omega=6 rad/s## and ##\psi=0##?
    If so, then, considering also ##k=\frac{2 \pi}{\lambda}=\frac{\omega}{v}=0.3 \frac{1}{m}##
    $$\xi(x,t)=0.1 \mathrm{sin}(6t-0.3x )\tag{3}$$

    b)Here is the problem. I would say that

    $$0.02=0.1 \mathrm{sin}(6t-0.3x )\implies (6t-0.3x)=\arcsin(0.2)+2n\pi \vee \pi-\arcsin(0.2)+2n\pi \implies |x_2-x_1|=\frac{\pi-2\arcsin(0.2)}{0.3}$$

    But this does not give the correct result.

    Where did I go wrong in this problem? Any suggestion is highly apprectiated.
     
  2. jcsd
  3. Jun 13, 2016 #2

    TSny

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    Apparently they want the smallest Δx between two points for which the absolute value of the displacement ##\xi## is 0.02 m.
     
  4. Jun 16, 2016 #3
    Thanks for the reply!

    Yes that's part of the question I guess, could you suggest where I went wrong in my derivation? I still do not see the mistake..
     
  5. Jun 16, 2016 #4

    TSny

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    It looks like your solution only considers values of ##x## where ##\xi = +0.02## m. What about points where ##\xi = -0.02## m? It helps to make a sketch of the sine wave at some instant of time and mark the points on the x axis where ##\xi = \pm 0.02## m.
     
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