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Create a wave equation with the following properties.

  1. Sep 22, 2011 #1
    1. The problem statement, all variables and given/known data
    Write down an equation to describre a wave [itex]\psi[/itex](x,t) with all of the following properties

    a) It is travelling in the negative x direction
    b) It has a phase velocity of 2000ms-1
    c) It has a frequency of 100kHz
    d)It has an amplitude of 3 units
    e) [itex]\psi[/itex](0,0)= 2 units

    2. Relevant equations
    f=[itex]\frac{\omega}{2\pi}[/itex]
    [itex]\lambda[/itex]=[itex]\frac{2\pi}{k}[/itex]
    vp=[itex]\frac{\omega}{k}[/itex]
    [itex]\psi[/itex](x,t)=Aexp(kx+[itex]\omega[/itex]t)

    3. The attempt at a solution
    a) For it to be travelling in the negative x direction the equation would have to be:
    [itex]\psi[/itex](x,t)=Aexp(kx-[itex]\omega[/itex]t)

    c) For it to have a frequency of 100kHz:
    [itex]\omega[/itex]=2[itex]\pi[/itex]f=2*105[itex]\pi[/itex]rads-1

    b) For it to have a phase velocity of 2000ms-1:
    vp=[itex]\frac{\omega}{k}[/itex]
    k=[itex]\frac{\omega}{vp}[/itex]=[itex]\frac{(2*105)[itex]\pi[/itex]}{2000}[/itex]=100[itex]\pi[/itex]radm-1

    d) For it to have an amplitude of 3 units:
    [itex]\psi[/itex](x,t)=3exp[(100[itex]\pi[/itex])x-(2*105[itex]\pi[/itex])t]

    e) This part I can't work out since I thought if x=0 and t=0 then the answer would just be the amplitude as exp0=1 but the amplitude is 3 units not 2 so were have I gone wrong here?

    Is any of the above correct or have I got it completely wrong? Also if anyone can help me out with part e) that would be great. Finally I don't know why its not including the greek symbols so sorry if it makes anything harder to read like the greek symbols next to the units. Thanks in advance.
     
  2. jcsd
  3. Sep 22, 2011 #2

    PeterO

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    The amplitude gives the variation from the mean. Who says the mean is 0

    For example a wave on the surface of a swimming pool. The amplitude may be 10 cm, but the depth of water is varying from 1.9 to 2.1 metres.
     
  4. Sep 22, 2011 #3
    I understand this but how can I reflect part e) in the wave equation without tampering with the amplitude since as far as I can remember the "A" in the equation is just amplitude, no need to do any equation seperately for it and with x=0 and t=0, it cant be any other number. Any extra hints would be grateful as I know we can't give direct answers in this forum. Thankyou for replying also.
     
  5. Sep 22, 2011 #4

    PeterO

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    Are you allowed to just add or subtract a number at the end? Like with sine graphs

    like y = a.sin(x - j) + c
     
  6. Sep 23, 2011 #5
    It doesn't say you can't. Didn't think about using sine, been so used to the format of exp wave equations that involve finding out or using the phase velocity that it slipped my mind. Thankyou.
     
  7. Sep 23, 2011 #6

    PeterO

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    I wasn't recommending sin - though it may be suitable - just using it as an example of having a number added on to the rest of your function, in order to change the mean from zero.
     
  8. Sep 23, 2011 #7
    I think I will stick with exp then, it doesn't really indicate what type of wave eqution to use so I think any type would suffice as long as it contained the given properties, especially considering the question before it uses sin and the question after uses exp.
     
  9. Sep 23, 2011 #8

    ehild

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    Gold Member

    The wave has to be periodic both in space and time. If it travels along the x axis it is periodic in x.
    [itex]\psi(x,t)=Ae^{kx+\omega t}[/itex] is not a periodic function. It would be periodic with an imaginary exponent: [itex]\psi(x,t)=Ae^{i(kx+\omega t)}[/itex]. Or, to describe a real wave, you can choose a sin or cosine function or the linear combination of a sine and cosine to fulfil a given initial condition. You can also incorporate a phase constant in the phase of a single sine or cosine.
    When the wave travels in the positive x direction, its phase is kx-ωt: a wavefront (place of the wave with a certain displacement) will shift to higher x values with increasing time. A wave travelling in the negative x direction has a phase kx+ωt.
    So you can choose your waveform as [itex]\psi(x,t)=A\sin(kx+\omega t+\alpha)[/itex]

    The amplitude is 3 units, so A=3, and ψ(0,0)=2, that means:

    [itex]3\sin(\alpha)=2[/itex]

    You can find an appropriate alpha from here.

    ehild
     
  10. Sep 24, 2011 #9
    Thankyou ehild for going in detail through this question, very useful.
     
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