# Why is the reflected wave going from a less dense material to a denser material 180 deg out of phase with the incident wave, but not the other way?

For example, say a light ray going from air into glass. Air has a refractive index of $$n_{air}=1$$ and a characteristic impedance,$$z_1=377\Omega /n_{air}=377\Omega$$. But glass has a characteristic impedance of $$z_2=377\Omega/3=125.67\Omega$$.

The reflection coefficient, $$\Gamma$$, which is the ratio of the reflected wave's voltage to the incident ray's voltage, has the formula $$\frac{z_2-z_1}{z_2+z_1}$$. In the case of a ray going from air to glass, it will produce a negative result, indicating that the reflected wave is the negative of the incident wave, or in other words, 180 deg out of phase with the incident wave. But if the ray goes from glass to water, it will produce a positive result, indicating no phase different. But why? (Could you explain it in a more intuitve way. I am not that big a math person)

It should be clear that the formula $$(z_2-z_1)/(z_1+z_2)$$ changes sign when $$z_1$$ and $$z_2$$ are swapped. That's the mathematical reason for the sign change. For a physical reason, it's a general property of waves. If you subtracted the reflected wave out of the picture, the two sides would have waves of different amplitudes. That doesn't work physically because the waves have to match at the boundary. So a reflected wave is created. There are two cases, if the incoming wave is too high, the reflected wave will have to be negative. And if the incoming wave is too low, the reflected wave will need to be positive.