\(Off[General::\(\*""spell1"\)]\)

Clear[Σk, Δk, Σω, 
      Δω]f = 
      Sin[(((Σk + Δk))/2) x - 
              (((Σω + Δω))/2) t]
             + Sin[(((Σk - Δk))/2) x - 
              (((Σω - Δω))/2) t]
             // TrigFactor

\(On[General::\(\*""spell1"\)]\)

f /. {
        (1/4) x (-Δk + Σk) - 
            (1/4) x (Δk + Σk) - 
            (1/4) t (
              -Δω + Σω) + 
            (1/4) t (Δω + Σω)
                 -> -(1/2) x  Δk + 
            (1/2) t  Δω, 
        (1/4) x (-Δk + Σk) + 
            (1/4) x (Δk + Σk) - 
            (1/4) t (
              -Δω + Σω) - 
            (1/4) t (Δω + Σω)
                 -> (1/2) x  Σk - 
            (1/2) t  Σω}

\(Δk = k1 - k2; 
    Δω = ω1 - ω2; 

    Σk = k1 + k2; 
    Σω = ω1 + ω2; 

    ttot = \(2  π\)\/Δω; \)

k1 = 10; k2 = 12; c = 1; ω1 = (c/k1); ω2 = (c/k2);

Do[Plot[{Sin[k1 x - ω1 t] + Sin[k2 x - ω2 t], 
          2  Cos[Δ(k/2) x - Δ(ω/2) t], 
          -2 Cos[
              Δ(k/2) x - Δ(ω/2) t]}, {x, 
          0, 10}], {t, 0, ttot, 5}]