Reflection of Waves on a String: Understanding Mass and Phase Differences"

[Aequiveri]

If I send a pulse down a string with mass (m1), consider what happens at a point where the mass of the string becomes (m2):

i) If m1 > m2, the wave is reflected with the same phase as the incident wave.
ii) If m1 < m2, the wave is reflected 180 degrees out of phase with the incident wave.

(Let's ignore any transmitted wave)

Why is this so? Is it an experimental fact, or is there some kind of theoretical/mathematical explanation for this behavior.

And I can see mathematically why it is the case that if m2 is infinitely larger than m1 (i.e. no transmitted wave - the end of the string is nailed down) the amplitude of the reflected wave is equal to the incident wave, but this seems physically inconsistent to me; if (ii) is true, shouldn't the reflected wave be equal to the "negative" amplitude of the incident wave?

I appreciate any insight you can give me. Thanks!

 

[Aequiveri]

Perhaps the Fresnel equations provide the necessary theoretical explanation?

 

[GRDixon]

the amplitude of the reflected wave is equal to the incident wave, but this seems physically inconsistent to me; if (ii) is true, shouldn't the reflected wave be equal to the "negative" amplitude of the incident wave?

I think the incident and reflected wave amplitudes ARE opposite; isn't the result a standing wave on the incident wave side of the barrier?