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 alot.

 

[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.

 

[Nickie]

Thank you for your question. The progressive wave equation describes the motion of a wave as it travels through space and time. In this case, we are looking at a wave moving in the positive Ox-direction with a displacement of 0 at the origin (x=0) and a positive sine graph for displacement-distance.

The equation for the displacement y at any point x and time t can be written as y(x,t) = A sin (ωt - kx), where A is the amplitude, ω is the angular frequency, and k is the wave number. In this case, since the wave is traveling in the positive Ox-direction, we can set k=0 and the equation becomes y(x,t) = A sin (ωt).

At time t=0, we can see that the equation reduces to y(x,0) = A sin (0) = 0, which is consistent with the displacement of 0 at the origin. As the wave propagates, the displacement at any point x and time t can be determined by substituting the values of ω and k into the equation.

It is important to note that the displacement-distance graph shows the displacement of the wave at a fixed point in space over time. This means that at x=0, the displacement-distance graph will always show a positive sine graph, regardless of the value of t. The equation y(x,t) = A sin (ωt) represents the displacement of the particle at the origin (x=0) over time, as you mentioned.

I hope this helps clarify the progressive wave equation and its application in this scenario. Keep up the good work in your studies!