Phase change from mass in middle of a string

[Koranzite]

Homework Statement

There is a mass M in the middle of an infinite string, of the same linear density on both sides. The reflection coefficient is $r=-ip/(1+ip)$ where $p=ω^2 M/2Tk$
How does the phase change on reflection vary with M, for fixed ω and T?

Homework Equations

Phase change = Im(r)/Re(r)

The Attempt at a Solution

I got the expression $\varphi =arctan(1/p)$, which I can't see how it could be right, as it implies that a phase change of $\pi/2$ occurs when the mass is 0. I know that when the mass is 0, there should be no phase change, and it should tend to $\pi$ as the mass tends to infinity (but the arctan function has a maximum value of $\pi/2$). How can I resolve this to get an equation for the phase change that works?

 

[mark726]

Hello,

Your attempt at a solution is on the right track, but you are correct in noting that it does not account for the expected behavior when the mass is 0 or infinity. This is because the arctan function has a limited range of values, as you mentioned.

To resolve this, we can use the fact that the reflection coefficient r can also be written as r = e^(iφ), where φ is the phase change. This means that we can express the phase change as:

φ = Im(ln(r)) = Im(ln(-ip/(1+ip))) = arctan(p)

This expression gives us the correct behavior for both small and large values of p. When p is small (corresponding to a small mass M), the phase change is close to 0, which is what we expect since there should be little to no phase change when there is little mass present. When p is large (corresponding to a large mass M), the phase change tends to π, which is what we expect since a large mass should result in a significant phase change.

I hope this helps clarify the issue. Good luck with your research!