When you were in elementary school, you might have jumped rope with a friend where there were only two of you, so the rope had to be tied to a post while your friend rotated it and you jumped. I hope you took turns as that’s the nice thing to do.

Anyway, I bet you remember sending a pulse down the rope and it coming back at you upside down. This is a classic wave reflection. Consider tying to different ropes together and sending a sinusoidal wave down the end of one of them. What happens as the wave meets to junction of the two ropes? Does it depend on which rope is heavier or lighter? Why?

Consider now explicitly the general 1D solutions to the wave equations where the left rope is the one which has the initial wiggles,

$\tilde{f}_{left}(z,t) = \tilde{A}_Ie^{i(k_lz-\omega t)} + \tilde{A}_Re^{i(-k_lz-\omega t)}$

$\tilde{f}_{right}(z,t) = \tilde{A}_Te^{i(k_rz-\omega t)}$

Find the relationship between the amplitudes of the incident ($I$), reflected ($R$), and transmitted waves ($T$) as well as the phase relationships.

When does the reflected wave come back upside down? When is there no transmitted wave?

Discuss any assumptions you have to make regarding boundary conditions at the point where the strings are joined.