A vacuum wave E₀(r, t) = x̂E₀ exp[i(k_yy + k_zz − ωt)] strikes a perfectly conducting surface.

(a) Write down the total electric field E which exists above the surface when the latter is defined by z = 0.
(b) To the field E found in part (a), add a linear combination of plane waves E'such that E + E' satisfies
the boundary conditions at a corrugated surface z = a sin(2πx/d) for the case when ωa/c << 1.
(c) Show that, if the incident angle is sufficiently close to grazing, E' decreases exponentially from the surface into the vacuum.