Question

For a monochromatic plane wave given in the equation below, what does the fact that the amplitudes can be complex numbers represent physically?

Answer 1

\[E(z,t)=E_oe^{i(kz−wt)}, B(z,t)=B_oe^{i(kz−wt)}\]

Answer 2

This is just a compact way of expressing the wave. The only part that is in the 'real world' is the real part of these functions, the cos part.

Answer 3

so the complex part of the coefficients mean nothing? or does it just add in more solutions?
Say E_o=R+iJ; then you're saying you would do this?
\[E(z,t)=(R+iJ)(\cos(kz-wt)+i\sin(kz-wt) \text{ but we keep only the real part}\]\[E(z,t) = Re(R\cos(kz-wt)+ iR\sin(kz-wt) +iJ\cos(kz-wt)-J\sin(kz-wt)\]\[E(z,t)=Rcos(kz-wt)-Jsin(kz-wt)\]which of course allows for multiple solutions to the wave equation and the superposition principle to apply. So for physical meaning, do the complex coefficients just allow for wave superposition?

Answer 4

Right.

Note that if the E_0 or B_0 obviously give amplitude of the waves. But more than that, if have imaginary parts, this causes a phase shift. That is, write them in polar form and you can see how that works its way through.

Answer 5

For example, if \[ E_0 = Ae^{i\phi} \] where A is the amplitude of E_0, then
\[ E(z,t) = A e^{i(kz-wt + \phi)} \]
The \( \phi \) gives a phase shift to the wave.

Answer 6

Make sense?

Answer 7

yeah, Thanks alot man

Answer 8

complex numbers because the amplitudes are fliping very quickly with time