Surface Plasmon’s consider a semi-infinite plasma on the positive side of the plane z = 0. A solution of Laplace's equation Δ²φ = 0 in the plasma is φ_i (x, z) = A cos kx e ^{– kz} whence E_si = kA cos kx e ^–kz E_si = kA sin kx e ^–kz. 

(a) Show that in the vacuum φ₀ (x, z) = A cos kx e^kx for z < 0 satisfies the boundary condition that the tangential component of E be continuous at the boundary; that is, find E_x0.

(b) Note that D, = e(w)Ei; Do = E,. Show that the boundary condition that the normal component of D be continuous at the boundary requires that ε(w) = – 1, whence from (10) we have the Stern-Ferrell result: w²_s = ½ w²_p for the frequency w , of a surface plasma oscillation