Wave Problem (time for a point to move half a wavelength)

[terryds]

Homework Statement

If a wave y(x, t) = (6.0 mm) sin(kx + (600 rad/s)t + θ) travels along a string, how much time does any given point on the string take to move between displacements y=+2.0 mm and y=-2.0 mm?

Homework Equations

ω=2πf (but it's not necessary in this problem, this problem just requires algebra, I think)

The Attempt at a Solution

For a point to travel between y=+2.0mm to y=-2.0mm, the distance x is (y_2 - y_1)/(2A)* 0.5λ = 1/6 λ (This is what I think since if it's 6.00mm (amplitude) to -6.00mm (-amplitude) it'll be 0.5λ)

So, x_2 = x_1 + (1/6) λ

So, I write equation for (x_1,t_1) and (x_1 + (1/6) λ, t_2)

6 sin(kx_1 + 600t_1 + θ) = 2 => kx_1 + 600t_1 + θ = arc sin (2/6) ...... (1)
6 sin(k(x_1 + λ/6) + 600t_2 + θ) = -2 => kx_1 + π/3 + 600t_2 + θ = arc sin (-2/6) ...... (2)

Subtracting (2) and (1), we get

600 (t_2 - t_1) + π/3 = -0,6796

t_2 - t_1 = -2.8781 * 10^-3 sWhere did I get wrong? Why t_2 - t_1 is negative though I have relate x_2 to x_1?
I see the solution manual the answer is 0.011 s, but it assumes that x_1 = x_2, and it subtracts (1) and (2), NOT (2) and (1). I really don't get it. As long as the time ticks, the position of the point changes so the x changes, right ?

 

[PeroK]

I see the solution manual the answer is 0.011 s, but it assumes that x_1 = x_2, As long as the time ticks, the position of the point changes so the x changes, right ?

A point on the string is defined by a single x-coordinate. If you have two different x-coordinates then you have two different points on the string.

 

[terryds]

A point on the string is defined by a single x-coordinate. If you have two different x-coordinates then you have two different points on the string.

So, does it mean that x-coordinate of point traveling in a wave doesn't depend on the time? I'm confused.

Or, does the x in the wave formula means the initial x-coordinate (t=0) of a point, not x-coordinate as function of t?

 

[PeroK]

So, does it mean that x-coordinate of point traveling in a wave doesn't depend on the time? I'm confused.

Or, does the x in the wave formula means the initial x-coordinate (t=0) of a point, not x-coordinate as function of t?

If you have a wave in a string, each point in the string moves up and down. That is, each particle in the string is moving up and down. The string itself isn't moving in the direction of the wave.

PS
$y = f(x, t)$ tells you the "vertical" displacement, $y$, of each point in the string, $x$, at each time, $t$.

 

[DrClaude]

So, does it mean that x-coordinate of point traveling in a wave doesn't depend on the time? I'm confused.

Or, does the x in the wave formula means the initial x-coordinate (t=0) of a point, not x-coordinate as function of t?

The problem statement says

how much time does any given point on the string take to move between displacements y=+2.0 mm and y=-2.0 mm?