Please, answer all the following questions:

1. Using the definition of E, show that E in equation (2) is equivalent to E2 in equation (3). 2. Show that the time-average of ett , as defined below, tends to zero as T goes to infinity, = lim T -+ 0o eiwtdt = 0. 3. Plot the wave beats with the following parameters and estimate the period for the carrier wave and its envelope (corresponding to frequencies w and dw, respectively). E0 = 2; w = 15; dw = 1; f[t_] = 2E0 Cos[wt] Cos[dwt]; Plot[ f [t], {t, -5,10}] 4. Consider 2-wave interference equation (6) of the class notes. a) Determine the phase differences AQ = Q1- 02, for which the maximum (called constructive interference) and the minimum (called destructive interference) of the intensities occur, and b) derive the following expressions: Constructive interference: I = Imax = [VI, + VI2] Destructive interference: I = Imin = [VII -VI2] c) Show that for the waves of equal amplitudes I = I1 (1+cos(AQ) ) = 411 cos2(AQ/2), where AQ is the phase difference between the two waves