Three light waves combine at a point find resultant amplitude and phase angle

CianM

1. The problem statement, all variables and given/known data
Three light waves combine at a point where their electric field components are

E₁ = E_o[tex] sin \omega[/tex]t

E₂ = E_o[tex] sin (\omega[/tex]t - 2[tex]\pi[/tex]/3)

E₃ = E_o[tex] sin (\omega[/tex]t + [tex]\pi[/tex]/3)

Find the resultant amplitude of the electric field E_R at that point and it's phase angle[tex]\beta[/tex]
Write the resultant wav int the form E = E_R[tex] sin(\omega[/tex]t + [tex]\beta[/tex])

2. Relevant equations



3. The attempt at a solution

Am I right in assuming that first you add E₁+E₂ then add E₁₂ + E₃ using double angle formulas? Or am I going about this completely the wrong way?

denverdoc

What about adding all three at once, using dbl angle formula to break out the phase shifts and collecting like terms. The results should be the same.

CianM

Well I already did the way I suggested and the answer I got was :
E_R = 2E_o sin ([tex]\omega[/tex]t)cos([tex]\pi[/tex]/3)
taking cos([tex]\pi[/tex]/3) = 1/2
then equals E_R = E_osin( [tex]\omega [/tex]t) .
Is this right?

denverdoc

yep, some careful examination of the problem shows that the -phase shift term is equal and opposite to the positive shift term, cancelling out, leaving your result. (in other words the two phase angles sum to pi)

CianM

Ok. Thanks for your help!