What Is the Correct Progressive Wave Equation for a Particle at Origin?

[hhm28]

Homework Statement

A wave moving in the positive Ox-direction has displacement of particle of 0 at the origin, O at time = 0.The displacement-distance graph showed a positive sine graph.
Write an expression for the variation of the displacement y with time t for the particle at O.

Homework Equations

The Attempt at a Solution

My book said that for the particle at O, the equation of motion of particle is y= A sin (ωt).
But shouldn't it be y= -A sin (ωt -kx) and when x=0, the equation became y=-A sin(ωt) since the particle at O must move down for the wave to propagate to the right hand side.

And if its y= A sin (ωt - kx), at t=0 and the x=λ/4, then displacement is negative. This is inconsistent with the positive sine graph of displacement-distance graph.

Someone please lend me a hand on this, Thanks a lot.

 

[HallsofIvy]

Do you not understand that "A" is an arbitrary constant which may be, itself, either poisiitive or negative>?

I can make no sense at all out of "if it's y= A sin (ωt - kx), at t=0 and the x=λ/4" There was no mention of λ before this. Where did that come from?

 

[hhm28]

Okay. I will provide more details. The positive sine graph of displacement-distance graph has amplitude of 3 and wavelength of 4 meter while frequency is 2.5 Hz.
Therefore, the displacement against t for particle at O given by book is y=3 sin (5ωt).

For your doubt, I assume that when t=0 and the x=λ/4, y= 3 sin (-pi/2) which is negative value. But the positive sine graph of displacement-distance graph at t=0, showed a positive displacement of 3 when its is λ/4 away from O.

Shouldn't it be y= -3sin wt for particle at O?

 

[haruspex]

If 'positive sine graph' means that for small x > 0, y > 0, then I agree: if the sine wave moves towards increasing x then at 0 it must be descending, so y = -|A|sin wt.