# Deciding amplitude for Beats

I have two harmonic sound waves of nearly equal angular frequencies $$\omega_1$$ and $$\omega_2$$, and whose equations(which I have particularly modified for convenience), are $$s_1=a.\cos\omega_1 t$$ $$s_2=a.\cos\omega_2t$$

Also, $$s_1,s_2$$ denote the transverse displacements.

Assuming that $$\omega_1>\omega_2$$, and using Principle of Superposition of Waves, we get that the resultant wave is $$S=2a\cos\frac{(\omega_1-\omega_2)t}{2}.\cos\frac{(\omega_1+\omega_2)t}{2}.$$

Now how am I to decide the amplitude of this resultant wave? In all books, the (variable )amplitude is taken to be $$\displaystyle A=2a\cos\frac{(\omega_1-\omega_2)t}{2}$$. How do I assert this?

I mean, the amplitude may as well be $$\displaystyle A=2a\cos\frac{(\omega_1+\omega_2)t}{2}$$. What decides the above given amplitude?

## 1 Answer

At the beginning of your post you said the frequencies $$\omega_1$$ and $$\omega_2$$ are nearly equal.

Hence, $$\omega_1 - \omega_2$$ is much smaller than $$\omega_1 + \omega_2$$.

From that you can conclude that $$\cos\frac{(\omega_1-\omega_2)t}{2}$$ oscillates much slower than $$\cos\frac{(\omega_1+\omega_2)t}{2}$$.

Therefore it makes sense to call this slowly varying part the amplitude, like the red envelope line in this diagram from Wikipedia:Beating Frequency. • Thank you so much, this makes sense for me! Also, it would be nice if you(or anyone else) could show some solid proof for this( I mean, something more rigorous than just using sense)! – fruitsauce Mar 2 '19 at 20:13