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interference

The expression u(x,t) = A1sin(ω1t-k1x+φ1) + A2sin(ω2t-k2x+φ2) can be simplified with standard techniques from goniometry. Here, we will treat the case A1=A2=A. For that case we only need to remember sin(a+b) = sin(a)cos(b) + cos(a)sin(b)      and      sin(-b) = -sin(b) We obtain then sin(a+b) + sin(a-b) = 2sin(a)cos(b) So, if we choose a+b = ω1t-k1x+φ1      and      a-b = ω2t-k2x+φ2 leading to 2a = (ω1+ω2)t - (k1+k2)x + φ1 + φ2      and      2b = (ω1-ω2)t - (k1-k2)x + φ1 - φ2 then we reach at the expression u(x,t) = 2Acos(b)sin(a)

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