## baru one year ago anyone familiar with the "phenomenon of beats" in the context of differential equations?

1. baru

anybody know how to break down the general solution to illustrate beats? $x= \frac{ \cos \omega_1 t - Acos(\omega_0t-\phi ) }{\omega^2_0 - \omega_1^2 }$ $\omega_1= driving frequence$ $\omega_0= naturaal frequency$

2. anonymous

'beats' occur when we have two sinusoids of similar frequency interfering with one another, seemingly moving in and out of phase with time to constructively/destructively interfere periodically, resulting in a oscillating wave whose amplitude *envelope* is also described by a sinusoid. the basis of 'beats' has to do with the fact we can express the product of $$\sin(\omega_1t),\sin(\omega_2t)$$ as: $$\sin(\omega_1 t)\sin(\omega_2 t)=\frac12[\cos((\omega_1-\omega_2)t)-\cos((\omega_1+\omega_2)t)]$$

3. anonymous

4. anonymous
5. baru

but how can I break down the equations i've typed out into products of 'sin'... the co-efficients of both the 'cos' terms are different.

6. Astrophysics

Trig identities maybe? Hmm $\cos(a-b)=\cos(a)\cos(b)+\sin(a)\sin(b)$

7. baru

no idea

8. Astrophysics

Maybe try messing around http://www.purplemath.com/modules/idents.htm here are some identities

9. baru

thanks...i'll try

10. anonymous

This is so easy.

11. anonymous

indeed: $$\cos(\omega_1t)-A\cos(\omega_t-\phi)$$ the beats occur when the forcing frequency does not match the natural frequency, and instead of consistent constructive interference we move in and out of constructive and destructive interference, resulting in an oscillatory motion with time-varying amplitude which is 'jerky'

12. baru

i get the idea of mismatched frequency...thats not what i'm talking about. in-order to confirm that the resulting plot is one with the amplitude varying sinusoidally, it has to expressed as a products of "sin" : Asin( )sin( ).