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Phase constant and reflection of waves

  1. Mar 25, 2014 #1
    1. The problem statement, all variables and given/known data
    If two waves are created at x = 0 and t = 0 (and are in phase to begin with), and are then sent traveling along a string in the positive x direction, and they get reflected at the end of the string, there will be a similar pair of waves traveling in the negative x-direction. The two waves have a differing frequency.
    Now my question is whether there a phase angle associated with these wave (namely the reflected waves).
    Like for the waves traveling in the positive direction:
    [itex]ψ_{1}(x, t)[/itex] = Asin(([itex]ω_{1}/v[/itex])x - ([itex]ω_{1}[/itex])t + θ)
    [itex]ψ_{2}(x, t)[/itex] = Asin(([itex]ω_{2}/v[/itex])x - ([itex]ω_{2}[/itex])t + θ), would θ = 0 (the phase constant) in both these cases ?

    And the reflected waves:
    [itex]ψ_{1}(x, t)[/itex] = Asin(([itex]ω_{1}/v[/itex])x + ([itex]ω_{1}[/itex])t + θ)
    [itex]ψ_{2}(x, t)[/itex] = Asin(([itex]ω_{2}/v[/itex])x + ([itex]ω_{2}[/itex])t + θ), would θ = 0 in these cases as well?
    I'm really confused about this??
     
  2. jcsd
  3. Mar 26, 2014 #2

    Simon Bridge

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    So you are pretty sure of the initial (incident) wave, and are unsure about the reflected wave...
    [at some time t> 2L/v ...]

    the incident wave travels in the +x direction at speed v:
    ##y_{i}(x,t)=A\sin k(x-vt)##

    the reflected wave travels in the -x direction at speed v:
    ##y_{r}(x,t)=B\sin [k(x+vt)+\phi]##

    (where B and \phi are the unknowns ... though we expect |B|=|A| from your description.)

    we require that ##y_{r}(L,t)=-y_{i}(L,t)## ... if the wave inverts on reflection.
    The time and space derivatives also have a similar relationship.
    So you can solve the simultaneous equations.


    Off your description - both incident waves will have the same initial phase ##\theta##.
    The value of the initial phase depends on when you started your stopwatch ... it is usually convenient to set it to zero or pi/2 unless you have some reason to believe it is something different.

    You should be aware that the phase of the wave at x and t is given by the entire argument of the sine function. The \theta in there is the phase at (x,t)=(0,0). You'll also find it easier to represent the waves in terms of wave numbers ##k=\omega/v##.
     
    Last edited: Mar 26, 2014
  4. Mar 26, 2014 #3
    So I should just compute θ for the reflected waves by setting (x, t) = (0, 0) ?
    Assuming the end of the string is not fixed, then the reflections are not 180 deg out of phase. But will there still be a phase constant θ present?
     
  5. Mar 26, 2014 #4

    Simon Bridge

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    Why not try it and see?
    But why not just follow the suggestion you were given?
    i.e. relate the reflected wave to the incident one and use the boundary conditions?
     
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