Plane Waves

We want to find the expression for a plane that is perpendicular to k ⃗ k ⃗ , where k ⃗ k ⃗ is a vector in the direction of propagation of the wave.The plane is the set of points that has the same projection onto the vector k ⃗ k ⃗ That is any point r ⃗ r ⃗ that satisfies k ⃗ ⋅ r ⃗ = c o n s t a n t k ⃗ ⋅ r ⃗ = c o n s t a n t is a point on the planeNow consider the function ψ ( r ⃗ ) = A e i k ⃗ ⋅ r ⃗ ψ ( r ⃗ ) = A e i k ⃗ ⋅ r ⃗ we see that the magnitude of ψ ( r ⃗ ) ψ ( r ⃗ ) is the same over every plane that is defined by k ⃗ ⋅ r ⃗ = c o n s t a n t k ⃗ ⋅ r ⃗ = c o n s t a n t we want to construct harmonic waves, ie. they should repeat every wavelength along the direction of propagation so they should satisfy ψ ( r ⃗ ) = ψ ( r ⃗ + λ k ⃗ k ) ψ ( r ⃗ ) = ψ ( r ⃗ + λ k ⃗ k ) where λ λ is the wavelengththen we must have A e i k ⃗ ⋅ r ⃗ = A e i k ⃗ ⋅ ( r ⃗ + λ k ⃗ k ) = A e i k ⃗ ⋅ r ⃗ e i k ⃗ ⋅ k ⃗ λ / k = A e i k ⃗ ⋅ r ⃗ e i k λ A e i k ⃗ ⋅ r ⃗ = A e i k ⃗ ⋅ ( r ⃗ + λ k ⃗ k ) = A e i k ⃗ ⋅ r ⃗ e i k ⃗ ⋅ k ⃗ λ / k = A e i k ⃗ ⋅ r ⃗ e i k λ This is true if e i λ k = 1 = e i 2 π e i λ k = 1 = e i 2 π or λ k = 2 π λ k = 2 π k = 2 π λ k = 2 π λ This should have a familiar look to it! Finally we want these waves to propagate in time so you should be able to guess the answer from our work on mechanical waves ψ ( r ) = A e i ( k ⃗ ⋅ r ⃗ ∓ ω t ) ψ ( r ) = A e i ( k ⃗ ⋅ r ⃗ ∓ ω t )