Periodic wave direction question.

[NBAJam100]

I had actually posted this question earlier below one of my other questions but I am pretty sure it was hidden and no one saw it because it was nestled under a few responses... so here:

A periodic wave has the equation .15m*sin(10t+(pi)x)

Now i am getting a little bit mixed up about how to determine the direction of the wave... I am going to say this wave is moving in a positive direction because (pi)x is positive... meaning it is going forward... is that correct? So when i rewrite the equation it would be .15m*sin(10t+(pi)(x-vt)) to account for the positive change? Thanks for all of the help guys.

 

[James R]

The standard form of a harmonic wave is

$$y(x,t) = A \sin(kx \pm \omega t)$$

The speed of the wave is $v = \omega / k$.

The direction of motion depends on whether the sign in the sine function is positive or negative. Negative sign means the wave moves in the positive x direction; negative sign means it moves in the positive x direction.

To see this, consider the following function:

$$y(x) = A \sin(x)$$

Now, compare that to this:

$$y(x) = A \sin(x - 3)$$

What does the -3 do to the graph?

Now consider:

$$y(x) = A \sin(x - t)$$

As t increases, what happens to the graph?

Does that help?

 

[NBAJam100]

Ok, so i think i get it now... In your last example, as t increases, we would be subtracting more from the sine equation, so that means the wave is moving to the right (positive direction). We are accounting for that shift by subtracting the value, which is why (-) in the sine means a positive shift. So in my question, since we are adding "t", that means the wave is traveling in the negative direction?

I just wasnt sure if it mattered if the sign was in from of the t or the x value...

 

[James R]

That's right. In your example:

$$.15m*sin(10t+(pi)x) = .15m \sin(\pi x + 10t)$$

This is a wave that moves in the negative x direction, because of the plus sign.