Consider the component of the incident wave with an angular frequency of ω, which has the waveform u inc ( x , t ) = A e i ( k 1 x − − ω ω t ) , A ∈ ∈ C . At t = 0, the incident reaches the boundary between the two media at x = 0. Therefore, the corresponding reflected wave and the transmitted wave will have the waveforms u refl ( x , t ) = B e i ( − − k 1 x − − ω ω t ) , u trans ( x , t ) = C e i ( k 2 x − − ω ω t ) , B , C ∈ ∈ C . The continuity condition at the boundary is u inc ( 0 , t ) + u refl ( 0 , t ) = u trans ( 0 , t ) , u x inc ( 0 , t ) + u x ref ( 0 , t ) = u x trans ( 0 , t ) . This gives the equations A + B = C , A − − B = k 2 k 1 C = c 1 c 2 C , and we have the reflectivity and transmissivity B A = c 2 − − c 1 c 2 + c 1 , C A = 2 c 2 c 2 + c 1 . When c 2 < c 1, the reflected wave has a reflection phase change of 180°, since B / A < 0. The energy conservation can be verified by B 2 c 1 + C 2 c 2 = A 2 c 1 . The above discussion holds true for any component, regardless of its angular frequency of ω.