Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    Science Advisor
    Homework Helper

    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


    User Avatar
    Science Advisor

    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 :


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


    User Avatar
    Science Advisor
    Homework Helper

    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


    User Avatar
    Science Advisor
    Homework Helper

    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:

    sin wave moving to the right at twice the speed of the last pic:

    sin wave moving to the left at the same speed as the wave in the first pic:

    I hope that helps to visualise it.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook