1. The problem statement, all variables and given/known data
Consider the superposition of two waves;
[itex]\zeta_1 + \zeta_2 = \zeta_{01} e^{i(kr_1  wt)} + \zeta_{02} e^{i(kr_2  wt + ∅)} [/itex]
where [itex] ∅ [/itex] is a phase difference that varies randomly with time. Show that the timeaverages satisfy;
[itex]<\zeta_1 + \zeta_2^2> = <\zeta_1^2> + <\zeta_2^2> [/itex]
2. Relevant equations
(1) If it wasn't clear, The two waves are;
[itex] \zeta_1 = \zeta_{01} e^{i(kr_1  wt)} [/itex] and
[itex]\zeta_2 = \zeta_{02} e^{i(kr_2  wt + ∅)} [/itex]
3. The attempt at a solution
Unless I have my definition of timeaverage wrong. I can't seem to get this to work.
[itex]\zeta_1 + \zeta_2^2 = (\zeta_1 + \zeta_2)(\zeta_1^* + \zeta_2^*) = \zeta_1^2 + \zeta_2^2 + \zeta_1\zeta_2^* + \zeta_2\zeta_1^* = \zeta_1^2 + \zeta_2^2 + 2\zeta_{01}\zeta_{02}cos(k(r_1  r_2)  ∅)[/itex]
Then, I believe, the time average is given by;
[itex]\frac{1}{T}\int^T_0 {\zeta_1^2 + \zeta_2^2 + 2\zeta_{01}\zeta_{02}cos(k(r_1  r_2)  ∅)} dt[/itex]
However, I don't see how this turns in to the form I desire. It would require that the last term (containing the cosine) is timeaveraged to zero. Can this be the case? Also, can [itex] ∅ [/itex] still even be considered a function of time when it varies RANDOMLY?
