Why Are Sine Waves a Function of (t-(x/v))?

[daemonakadevil]

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!

 

[tiny-tim]

Hi daemonakadevil!

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.

 

[HallsofIvy]

Suppose you are "surfing" on top of the wave A sin(\theta(x,t)). First, in order that that "look" like a regular sine wave, \theta(x,t) 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, \theta(x,t)= Bx+ Ct. 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+ Ct<sub>1[\sub]), we must have Bx0= Bx1[/sup]+ Ct1 (or add a multiple of 2\pi 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.</sub>

 

[daemonakadevil]

Here is what i understood from your thesis :

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

I understood the right thing?

 

[tiny-tim]

Hi daemonakadevil!

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!

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 …"

 

[daemonakadevil]

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]

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!

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.

So you can see that the height of the wave depends only on t - x/v.

 

[Celunas]

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 !

 

[daemonakadevil]

now i got the concept , thanks

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

 

[marmoset]

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.