(a) Superpose two scalar waves: a plane wave u 1 exp(ikx) and a spherical wave (u 2 /r) exp(ik r + ). Show that

(a) Superpose two scalar waves: a plane wave u₁ exp(ikx) and a spherical wave (u₂/r) exp(ik · r + δ). Show that the locus of points where constructive interference occurs defines a family of parabolas.
(b) Superpose two equal-amplitude, time-harmonic, y-polarized plane waves. Their wave vectors satisfy |k₁| = k₂| and lie in the x-z plane as shown below. Is the sum of these two plane waves itself a plane wave? Is it transverse?
(c) Plot the time-averaged Poynting vector for the wave in part (b) as a function of position for θ = 0, θ = π/4, and θ = π/2.

X k 10 0 K Z