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Homework Help: Wave propagation

  1. Jan 24, 2010 #1
    1. The problem statement, all variables and given/known data
    Consider
    y_1=Asin(5x)exp(-2t)
    y_2=Aexp(4ix)exp(-2it)
    y_3=Asin(2x-5t)exp(-2t)

    (i)which one represents a wave that propagates at constant speed with no change in its profile
    (ii)Confirm it satifies the wave equation and obtain the wave velocity
    (iii)Comment qualitatively on the behaviour descried by the other 2


    2. Relevant equations



    3. The attempt at a solution
    (i) i think it's the second one. I took the Re(y) and got Acos(3x-2t) so the amplitude is constant, while the other 2 will decrease in amplitude as exp(-2t) will tend to 0?

    (ii)just differeniate it and stick into equation

    (iii)They will oscillate with the same frequency but the amplitude will decrease?
     
  2. jcsd
  3. Jan 24, 2010 #2
    for (ii) i have just differentiated and subbed it into the wave equation and got v= 0.5i. can you have and imaginary wave speed?
     
  4. Jan 24, 2010 #3

    vela

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    No, you must have dropped a factor of i somewhere.

    Edit: Actually, you probably flipped a sign somewhere since the wave equation tells you what v^2 is.
     
    Last edited: Jan 24, 2010
  5. Jan 24, 2010 #4

    vela

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    Do you know what the effect is of the -5t in y3, in comparison to y1?
     
  6. Jan 24, 2010 #5
    not really. isn't 5 the frequency? actually no.. because that would mean y1 has a frequency of 0...
     
  7. Jan 24, 2010 #6

    vela

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    Actually, you're right. It's the frequency, so the first one doesn't oscillate at all. I'm not sure I'd call it a wave. The exponential factor in y1 and y3, as you mentioned above, just causes the amplitude to decrease over time.
     
  8. Jan 24, 2010 #7
    ahhh ok thanks, so which doesnt change its profile. i thought 2 as it would keep repeating, but the actual height would be constantly changing
     
  9. Jan 24, 2010 #8

    vela

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    Oh, I hope you don't mean you're changing your answer to (i). You're right that the answer to (i) is y2.

    The reason I asked about -5t was because in your answer to (iii) you said y1 and y3 both oscillated with the same frequency. Y3 oscillates, but y1 doesn't.
     
  10. Apr 18, 2011 #9
    is the Re(y2) not A cos(4x-2t) instead of A cos(3x-2t) (posted above) cus how would you get to that answer?
     
  11. Apr 18, 2011 #10

    vela

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    Yes, you're right. I think it was just a typo on Emma's part.
     
  12. Apr 18, 2011 #11
    but im not understanding how this tells u that the wave propagates at constant speed with no change in its profile compared to the other two wave disturbances?
     
  13. Apr 18, 2011 #12
    and how can u tell that y1 is a stationary wave whose amplitude is decreasing exponentially with time and that y3 is a travelling wave also decreasing exponentially with time?
     
  14. Apr 21, 2011 #13

    vela

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    What's the difference in how eiωt, e-iωt, e-ωt, and eωt behave? (Assume ω>0.) In particular, what do these functions do as t goes to ±∞?
     
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