# Wave reflection, boundary conditions?

When a wave-pulse is reflected such that initially it was moving in the denser medium, and towards the other side of the interface lay the rarer medium, will there have been a phase change post reflection in the reflected wave travelling in the opposite direction in the same medium?

By denser, I assume you mean with a larger refractive index.

The answer is established from the Fresnel equations giving the ratio of the electric field amplitudes. The reflection amplitudes for (for example) light travelling from glass to air can be either positive, negative or complex depending on the angle of incidence and the polarisation state of the light compared with the angle of incidence.

Up to the critical angle of incidence, the reflection amplitude coefficient is a real number which gives the ratio of the reflected to incident E-field amplitudes.

For s-polarised light (E-field polarised perpendicular to the plane of incidence and incident, reflected and transmitted waves initially assumed to be in the same direction) the reflection coefficient is always a positive number and the initial sign convention adopted means that this represents no phase change on reflection.

For p-polarised light (H-field polarised perpendicular to the plane of incidence and incident, reflected and transmitted H-fields initially assumed to be in the same direction) the reflection coefficient is negative and then changes to positive above the Brewster angle. But the convention adopted for the H-fields above means that a negative coefficient for p-polarised light means that there is not a phase change (in the sense that the components of the E-field parallel to the interface are in the same direction), whilst a positive number means that they are reversed.

A plot of the coefficients for these cases (and adopted sign conventions) is shown below (from hyperphysics). You can also play with different scenarios using this applet I designed with Geogebra. Fresnel equations on Geogebratube

When you go beyond the critical angle and get total internal reflection, then although it has a modulus of 1, the amplitude reflection coefficient becomes complex and there is an angle-dependent phase shift between the incident and reflected E-fields. The phase shift is different for s-polarised and p-polarised light at the same incidence angle. You can find some analytical calculations and expressions for these phase shifts here. This phase shift difference means that unpolarised light becomes elliptically polarised upon total internal reflection.