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Need help with sine waves Equation!

  1. Apr 27, 2008 #1
    Hey friends and Sir's ,

    I am trying to understand simple concept that why sine waves are function of (t-(x/v))

    x= position in x direction
    v= velocity of wave
    t= is time at any instant

    although i have read many articles on it but still unable to understand , any help will be great and will be best , if you can help with diagrams !

    Thanks for taking your precious time for me!
     
  2. jcsd
  3. Apr 27, 2008 #2

    tiny-tim

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    Hi daemonakadevil! :smile:

    If a function f(x,t) has speed v, then that means that f looks the same a time T later, but shifted along by a distance vT.

    In other words: f(x,t) = f(x + vT, t + T) for all x and t.

    So, putting T = -x/v, so vT = -x, and x + vT = 0, we have:
    f(x,t) = f(0,t - x/v).​

    So f is obviously a function of t - x/v. :smile:
     
  4. Apr 27, 2008 #3

    HallsofIvy

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    Suppose you are "surfing" on top of the wave [itex]A sin(\theta(x,t))[/itex]. First, in order that that "look" like a regular sine wave, [itex]\theta(x,t)[/itex] must be linear (any non-linear function of x and t would "shrink" or "stretch" some parts of the sine graph more than others and so change the shape). That is, [itex]\theta(x,t)= Bx+ Ct[/itex]. We can further simplify by setting up a coordinate system so that when t= 0, x= x0. At that time, since we are on the top of the wave, the value must be A (the maximum value). Just a moment later you are at time t1 and at position x1 and, since you "riding" the wave the value must still be A. Since sin(Bx0)= sin(Bx1+ Ct1[\sub]), we must have Bx0= Bx1[/sup]+ Ct1 (or add a multiple of [itex]2\pi[/itex] but remember, the is "just a moment later".) Then B(x1- x0)= Ct1 or (x1- x0= (C/B)t1. That means that (C/B)= (x1- x0)/t1, the distance you have moved divided by the time in which you moved: your speed, v. So your function must be of the form sin(Bx+ Ct)= sin(B(x- (C/B)t))= sin(b(x- vt)), a "function of x- vt".

    I see Tiny Tim got in just ahead of me. We are saying basically the same thing.
     
  5. Apr 28, 2008 #4
    Here is what i understood from your thesis :

    hxxp://img294.imageshack.us/img294/5014/sineah8.png

    I understood the right thing?
     
  6. Apr 28, 2008 #5

    tiny-tim

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    Hi daemonakadevil! :smile:

    Sorry … no … your diagram is missing the point completely.

    You need a three-dimensional diagram … f(x,t) has a different value at every pair (x,t).

    Think of it as y = f(x,t).

    Then you need a vertical y-axis, and two horizontal axes for x and t.

    It should look like the sea! :smile:

    Then you compare moving x-wards with moving t-wards, and find the combination that keeps you "riding along on the crest of a wave …" :smile:
     
  7. Apr 29, 2008 #6
    Hello sir tiny-tim, can you please take little time for me and teach me all this visually means by mean of diagram? , thanks in advance :)
     
  8. Apr 29, 2008 #7

    tiny-tim

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    hmm … don't know how to do wavy diagrams on the computer …

    Draw x y and z axes, but label the y-axis t instead of y.

    Draw a wave along the x-axis (left-right), with a crest at the origin.

    Draw a diagonal line through the origin with slope 1/v … that's the line x = vt (and z = 0), so it represents speed v.

    Now draw more waves in the x-y-plane, parallel to the x-axis, all with crests where they intersect that line.

    If you're artistic, do a little shading in between to get rolling downlands! :smile:

    Now, from any point (t,x), draw another line parallel to the first line until it meets the t-axis. It does so at (t - x/v,0).

    The height at (t,x) is the same as at (t - x/v,0), isn't it?

    So if you know the height along the t-axis, you know it everywhere.

    In other words: the height is a function of t - x/v. :smile:

    So you can see that the height of the wave depends only on t - x/v. :smile:
     
  9. May 2, 2008 #8
    Hello !

    I don't know if it will help but I understand it that way :

    A wave function (what a sine function of space and time is) moves in space during time. If you take as a reference the time t where a certain point of this sine function (say, for a non-perfect sine function, a maximum, just for you to visualize better) can be found on the point x in space, which means you consider the value the function has on point x at the time t, so f(x,t), then you know that the value the function has at this moment at this point could be found at some time at the point 0 (origin). The wave function moves from the origin to the point x, at the speed v (celerity of the wave), so in a time x/v. It means that a time x/v before, every point of the wave you consider was a distance x "before" (it means a distance x in the opposite direction of the movement), including f(x,t). So f(x,t) = f(0,t-x/v), the function has the same value at any point and time x and t than it had a time x/v and a distance x "before".
    The point is to consider the function as a fixed form of something moving along the x axis during time.

    Tell me if I have not been clear enough !
     
  10. May 4, 2008 #9
    now i got the concept , thanks

    here is now what i visualise in my mind hxxp://img291.imageshack.us/img291/9711/sinesm3.png
     
  11. May 4, 2008 #10
    I plotted some graphs of sin waves moving through time (away from you on the green y axis). Just add in the h before.

    sin wave moving to the right:
    ttp://i30.photobucket.com/albums/c339/marmoset_rock/sinwavemovingright.jpg



    sin wave moving to the right at twice the speed of the last pic:
    ttp://i30.photobucket.com/albums/c339/marmoset_rock/sinwavemovingrightattwicethespeed.jpg


    sin wave moving to the left at the same speed as the wave in the first pic:
    ttp://i30.photobucket.com/albums/c339/marmoset_rock/sinwavemovingleft.jpg


    I hope that helps to visualise it.
     
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