Sum of 2 EM Waves w/ Same Phase & Amp but Diff Freq

[fluidistic]

Homework Statement

Describe the sum of two EM waves that have the same initial phase and same amplitude but different frequencies such that $$\omega _1 >> \omega _2$$.

Homework Equations

$$E=E_0 \cos (kx -\omega t + \alpha)$$.

The Attempt at a Solution

I summed them up and reached, after an approximation, that $$E_1+E_2 \approx 2 E_0 \cos \left (kx -\frac{\omega _1t}{2} + \alpha \right ) \cos \left ( \frac{\omega _ 1 t}{2} \right )$$. I don't know how to simplify further. It seems that the amplitude is the sum of both amplitudes and I'm not sure yet what is the frequency. It should be almost $$\omega _1$$, intuitively. I just don't know how to show it.
Any help is appreciated.

 

[physicsworks]

Did you consider the fact that
$$k=\frac{\omega}{c}$$
and that's why k is different for the two plane waves with two different frequencies?

 

[fluidistic]

Did you consider the fact that
$$k=\frac{\omega}{c}$$
and that's why k is different for the two plane waves with two different frequencies?

Thanks, actually I didn't consider this. I will redo the exercise.