Solv'g Displacement Diff for 2 Points w/ 30 Deg. Phase Diff Given f, v

[tigger88]

I've got a review question on classical waves (SHM) where I'm given the frequency as f = 20 (sec^-1) and velocity v = 80m/s.
The question is: How far apart are two points whose displacements are 30 degrees apart in phase?

My own reasoning doesn't seem to agree with the solution I was provided.

Here is my thinking:
Phase = wt +/- kx
For the first point, let x = x1 and t = t.
For the second point, let x = x2 (and t = t)
So then the phase difference would be (wt-kx1) - (wt-kx2) = k(x2-x1).
Then, given v and f, lambda = v/f = 4 m, so k = 2pi/lambda = pi/2.
So phase difference, which is given to be 30 degrees, or pi/6 rad, becomes:
pi/6 = (pi/2)(x2-x1) -where solving for x2-x1 gives an answer of 1/3.

The solution I was given says that the phase difference = 2pi(x2-x1) = pi/6, which gives x2-x1 = 1/12.

I don't understand how the 2pi comes into it!
Could someone tell me where my thinking went wrong?

 

[alphysicist]

Hi tigger88,

Your answer looks right to me. Another way of thinking about it would be to say that two points with displacement 360 degrees apart in phase would be one wavelength apart in distance, and since 30 degrees is 1/12 of 360 degrees, the distance you're looking for is 1/12 of a wavelength.

 

[baba89]

I would like to clarify that the phase difference is not equal to the difference in displacement. The phase difference refers to the difference in the phase angle between two points on a wave, while the difference in displacement refers to the difference in their physical positions.

To solve this question, we can use the formula for phase difference, which is given by:
Δφ = 2πΔx/λ
where Δφ is the phase difference, Δx is the difference in displacement, and λ is the wavelength.

Substituting the given values, we have:
Δφ = 30 degrees = π/6 radians (since 1 degree = π/180 radians)
λ = v/f = 80/20 = 4m
Δx = (λ/2π)Δφ = (4/2π)(π/6) = 1/3m

Therefore, the two points are 1/3m apart in terms of their displacement. This means that if one point is at x1 = 0, the other point would be at x2 = 1/3m.

I hope this clarifies your confusion and helps you understand the concept of phase difference better. Remember, always double check your units and make sure you are using the correct formulas for the given quantities.