# Physics Cup – TalTech 2021 – Pr. 5, Hint 1.

According to the problem statement, the waves are coherent, and hence, you need to add the amplitudes (and not intensities). However, there is a problem: what happens with the phase of the wave when it is jumping to the other fibre? The answer given below is the consequence of the energy conservation law. Suppose a wave $A$ propagating along the first fibre is split at the junction into a straight-going wave $A'$ and a junction-crossing wave $A''$; denote the phase shift between $A''$ and $A'$ as $\Delta \phi_A$. Similarly, we introduce a wave $B$ propagating along the second fibre, and its straight-going and junction-crossing components $B'$ and $B''$, together with the phase shift $\Delta \phi_B$. Energy conservation law dictates that $\Delta \phi_A+\Delta \phi_B=\pi$ (showing this is a good exercise!). As an example, consider normal incidence of plane waves at the interface of two transparent media: if the wave $A$ comes from optically denser medium, $\Delta \phi_A=0$ (there are no phase shifts involved); then the wave $B$ must be coming from optically less dense medium in which case the reflected wave obtains a phase shift $\pi$ so that $\Delta \phi_B=\pi$, nicely yielding $\Delta \phi_A+\Delta \phi_B=\pi$.

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