Definition:
The phase velocities of the incident, reflected, and transmitted fields must be equal on the boundary. Another way to represent this relationship for the incident and transmitted fields is:
[tex]\beta_{1}sin(\theta_{i}) = \beta_{2}sin(\theta_{t} \Rightarrow sin(\theta_{t}) = \frac{\beta_{1}}{\beta_{2}}sin(\theta_{i}[/tex]

Question:
Could someone elaborate on the definition above-

The boundary refers to the boundary between the two media between which the light is propagating. Equal "on the boundary" means that right at that interface between the two media, the velocity of the wave is continuous. There is no abrupt discontinuity. The waves that are incident upon, reflected from, and transmitted through the boundary all have the same speed.