Returning to Section 9.1, letAnd where the wavefront shapes are not explicitly specified,...

Returning to Section 9.1, let

And

where the wavefront shapes are not explicitly specified, and  and  are complex vectors depending on space and initial phase angle. Show that the interference term is then given by

You will have to evaluate terms of the form

for T>> T (take another look at Problem 1). Show that Eq. (9.109) leads to Eq. (9.11) for plane waves.

Problem 1

The time average of some function f(t) taken over an interval T is given by

where t' is just a dummy variable. If T = 2π/ω is the period of a harmonic function, show that

when T = T and when T >> T.