Question

anyone familiar with the "phenomenon of beats" in the context of differential equations?

Answer 1

anybody know how to break down the general solution to illustrate beats?

\[x= \frac{ \cos \omega_1 t - Acos(\omega_0t-\phi ) }{\omega^2_0 - \omega_1^2 }\]

\[\omega_1= driving frequence\]
\[\omega_0= naturaal frequency\]

Answer 2

'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: $$\sin(\omega_1 t)\sin(\omega_2 t)=\frac12[\cos((\omega_1-\omega_2)t)-\cos((\omega_1+\omega_2)t)]$$

Answer 3

this is the principle of heterodyning in radio receivers

Answer 4

http://www.phy.mtu.edu/~suits/beatgraph.gif
http://hyperphysics.phy-astr.gsu.edu/hbase/sound/imgsou/beat4.gif

Answer 5

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.

Answer 6

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

Answer 7

no idea

Answer 8

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

Answer 9

thanks...i'll try

Answer 10

This is so easy.

Answer 11

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'

Answer 12

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( ).