Waves and EM Waves

Frequency:
f = v/[lambda] (1/s)

Angular Frequency:
[omega] = 2[pi]f (rad/s)

Wavenumber:
k = 2[pi]/[lambda] (rad/m)

Speed of light:
c = 1/sqrt(e0u0) (m/s)


One dimensional wave equation of a function f with speed v:
d2F/dt2 = v^2 * d2F/dx2

Hence, the wave equation for an electromagnetic plane wave F is:
d2F/dt2 = c^2 * d2F/dx2
or
d2F/dt2 = 1/u0e0 * d2F/dx2

A general solution to the wave equation is:
F(x,t) = A*cos(kx - wt)
or
F(x,t) = A*cos(2[pi]*(x/[lambda] - ft)


Poynting vector:
S = 1/u0 * E x B (W/m^2)

In an EM wave, the E field is much larger than the B field because:
E = c*B

The average of the Poynting vector (average power/m^2) over a long period is:
ave(S) = 1/2u0c * E^2 = 1/2 * e0cE^2

Because:
S = 1/u0 * E x B = 1/u0 * EB * cos^2(kx - wt)
ave(S) = ave(1/u0 * EB) * ave(cos^2(kx - wt)) = 1/u0 * EB * 1/2
ave(S) = 1/2u0 * EB = 1/2u0 * E * E/c = 1/2u0c * E^2

The energy in an EM wave is carried equally by the E and B fields:
u = u(of E) + u(of B) = 1/u0 * B^2 = e0E^2

Thus, the average of the Poynting vector is numerically equal to c times the energy in either the E or B field:
ave(S) = c * u/2