Equation for the propagation of the crest of a wave

[songoku]

Homework Statement

The displacement y of a wave traveling in the positive x-direction is described by time t and position x of a medium as y = 3.0 sin {2π (0.50 t – 0.25 x)}. Find the equation of the propagation of crest

Relevant Equations

v = ω / k

x = x(0) + v.t

I am not sure what is meant by "equation of propagation of crest" but this is my attempt:

First, I find the velocity of wave:
v = ω / k = 0.5 / 0.25 = 2 m/s

Then I calculate wavelength:
k = 2π / λ
λ = 4 m

I imagine the crests will move to the right along with the wave so I try to use equation of motion: x = x₀ + v.t

The first crest will be located at the distance of λ/4 (1 m) from starting point (t = 0 and x = 0) so the equation becomes: x = 1 + 2t

But the answer is x = -1 + 2t

Where is my mistake?

Thanks

 

[PeroK]

The trick is to see what's happening at $t = 0$. You have a crest at $x = 1m$ at $t=0$. What is the y-value for $t = 0, x = 1$?

What about the book answer: $t = 0, x = -1$?

 

[songoku]

The trick is to see what's happening at $t = 0$. You have a crest at $x = 1m$ at $t=0$. What is the y-value for $t = 0, x = 1$?

What about the book answer: $t = 0, x = -1$?

When t = 0 and x = 1, the value of y is - 3

When t= 0 and x = -1, the value of y is 3

Based on the graph, the crest should be located at y = 3

But I don't understand the logic behind the working. At first I thought I should find equation of motion of crest where the position of crest would be measured with respect to origin. That's why I take x₀ = 1 instead of -1 because I thought if the equation is x = 1 + 2t then when t = 0, x = 1 means that the first crest is initially at distance of 1 m from origin.

If the equation is x = -1 + 2t then when t = 0, x = -1 m and I interpret this as the location of the crest is at 1 m to the left of origin which does not make sense to me

Thanks

 

[PeroK]

If the equation is x = -1 + 2t then when t = 0, x = -1 m and I interpret this as the location of the crest is at 1 m to the left of origin which does not make sense to me

Why does that not make sense? The origin is an arbitrary point on the path of the wave. The nearest crest to the origin at time $t = 0$ could be to the left or the right.

 

[songoku]

Why does that not make sense? The origin is an arbitrary point on the path of the wave. The nearest crest to the origin at time $t = 0$ could be to the left or the right.

Because if I draw a graph of sine function, the nearest crest to origin at t = 0 is either at (1, 3) or (-1, -3). I am not getting why it is located at (-1, 3)

 

[PeroK]

Because if I draw a graph of sine function, the nearest crest to origin at t = 0 is either at (1, 3) or (-1, -3). I am not getting why it is located at (-1, 3)

I you plug $t=0, x = 1$ into your equation you get: $y = 3\sin(-\frac \pi 2) = -3$. That's a trough.

Likewise at $t=0, x = -1$ you get: $y = 3\sin(\frac \pi 2) = 3$. That's a crest.

 

[songoku]

I think I understand. I had a wrong picture in my mind.

Thank you very much perok