# Interference of Waves question

Given :

$$\phi_1(x,t)=A\cdot \cos(k_1x-\omega_1t)\hat{x},\phi_2(x,t)=A\cdot \cos(k_2x-\omega_2t)\hat{x}$$ with no difference phase,$$k_1-k_2 \ll k_1,k_2,\omega_1-\omega_2 \ll \omega_1,\omega_2 .$$

I have to find group velocity of $$\phi_{total}.$$

$$\phi_{total}=\phi_1(x,t)+\phi_2(x,t)=A\cdot \cos(k_1x-\omega_1t)+A\cdot \cos(k_2x-\omega_2t)=2A\Big(\cos(\frac{x(k_1+k_2)-t(\omega_1+\omega_2)}{2})\cos(\frac{x(k_1-k_2)-t(\omega_1-\omega_2)}{2})\Big)$$

I know that while the wave equation is $$\phi(x,t)=A\cos(kx-\omega t)$$ the group velocity is $$v_g=\frac{d\omega}{dk}$$.

$$\phi_{total}$$ is a multiplication of 2 "cos" function , so I get a little bit confused how to find the group velocity.

Any help is welcome, thanks !

You are being asked for the velocity of the slowest'' of the two cosines --- as that is factor that determines how fast the amplitude modulation is moving. Look at the factors. Does not one of them look like an approxiamtion to $$d\omega/dk$$?