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Are these travelling waves?

  1. Jan 25, 2015 #1
    1. The problem statement, all variables and given/known data
    Consider the following expressions:
    (a) y(z,t)=A{sin[4π(t+z)]}^2
    (b) y(z,t)=A(z-t)
    (c) y(z,t)=A/[bz^(2)-t]

    Which of them describe travelling waves? Prove it. Moreover, for the expressions that represent waves find the magnitude and direction of wave velocity.

    2. Relevant equations
    y(z,t)=A(kx-wt)

    v=w/k

    3. The attempt at a solution

    image.jpg
     
    Last edited: Jan 25, 2015
  2. jcsd
  3. Jan 25, 2015 #2

    Simon Bridge

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    Did you want to know something specific or are you just inviting comment? It is important to say.

    I think you have worked harder than you needed to.
    Definition: A shape of form ##y=f(z)## travelling in the +z direction with speed ##v## has form ##y(z,t)=f(z-vt)##
    ... this will be a travelling wave if it also satisfies the wave equation. (Do all such functions satisfy the wave equation?)

    ... a definition like that allows positive values of v to mean that the waveform propagates in the positive z direction - making it easier to keep track of minus signs.

    For (a): ##y(z,t)=A\sin^2 4\pi(t+z)## ... this is a travelling wave with form ##f(z)=A\sin^2 4\pi z##
    This means that ##z-vt = t+z \implies v=1\text{ (unit)} ## ... i.e. the wave propagates in the +z direction.
    See how that somes easily?

    There is also no need to go into wave numbers and angular frequencies.
    You don't need the ##\pm## sign in your definitions unless you insist that the constants ##\omega## and ##k## can only take positive values.

    Fortunately you don't have to prove that (c) is not a travelling wave.
     
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