Need help with sine waves Equation!

Main Question or Discussion Point

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!
 

Answers and Replies

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:
 
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.
 
Here is what i understood from your thesis :

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

I understood the right thing?
 
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:
 
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 :)
 
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:
 
14
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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 !
 
now i got the concept , thanks

here is now what i visualise in my mind hxxp://img291.imageshack.us/img291/9711/sinesm3.png
 
42
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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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