Here we consider the interference between two harmonic waves at t0 (a- 1pt) using matlab produce a plot of (x,に0) according to eq. (1) on the interval a-p.30] for kı-2, k2-1.8 b - 2pts) by using the rules for the sum of trigonometric functions show that (x, t-0) 2 cos k+ with k± = (kit k2)/2 > 0. the interference of the two waves can therefore be expressed as a product of cosine functions with different propagation numbers il. hint: to apply one of the commonly listed sum-to-product trig identities, you may want to express sin(k2r) in term of cos function with a shifted argument c- 2pts) for ki k2 the average propagation number k+(k+ k2)/2 ki controls the fast carrier oscilations that arise in your plot from part (a), whereas the "spread" k- -(ki - k2)/2 < < k+ describes the slower spatial envelope that modulates the carrier oscillation. the spatial periods l+ of the carrier and envelope fringes follow from the relations k± 2π, which requires the harmonic waves to repeat in phase after a period. demonstrate that the spatial periods l+ ~ 2π/h and l-~ 4n/(ki-k2) indeed appear in your numerical results from part (a)