Beat frequency when there are phase constants

[nietzsche]

Homework Statement

If we have two vibrations with angular frequencies ω₁ and ω₂ with ω₁≈ω₂. Then we will have beats with beat frequency ω₁-ω₂.

But suppose we have two different phase constants, for example

x₁ = Acos(ω₁t + ϕ₁) and
x₂ = Acos(ω₂t + ϕ₂).

What happens to the beat frequency?

My intuition tells me that the beat frequency will remain the same, but is there a way of showing this?

 

[Redbelly98]

Yes, the beat frequency is independent of the phase difference. To show that, you can use the following rule:

If we have two vibrations with angular frequencies ω₁ and ω₂ with ω₁≈ω₂. Then we will have beats with beat frequency ω₁-ω₂.

Note, there should be an absolute-value-sign, so it's |ω₁-ω₂|