m1=3; m2=1; c=1; a=1;
w1[k_] = Sqrt[2c(m1+m2) + Sqrt[4 c^2(m1+m2)^2-4(m1 m2) (2 c^2 (1-Cos[k a]))]]
w2[k_] = Sqrt[2c(m1+m2) - Sqrt[4 c^2(m1+m2)^2-4(m1 m2) (2 c^2 (1-Cos[k a]))]]
Plot[{w1[k],w2[k]},{k,-Pi, Pi}]

m=1; c=1; a=.5;
w[k_] = Sqrt[2c (1-Cos[k a])]
Plot[w[k],{k,-2 Pi, 2 Pi}]