Verifying A Cosines Addition Equation with Beats

[don_anon25]

The problem asks me to show that the addition of two cosines with different wavelength and frequencies gives a solution with beats.

Mathematically, I need to verify that A cos (k1x-w1t)+A cos (k2x-w2t) is equivalent to A cos (.5(k1+k2)x-.5(w1+w2)t) cos (.5(k1-k2)x-.5(w1-w2)t)

I converted the second equation into exponential form (cos (kx-wt)=1/2(e^i(kx-wt)+e^-i(kx-wt)), multiplied the resulting binomials together, and simplified to get the first equation. My problem is with the constant, A. How do I deal with it? I need A/2 for the first equation, but simplication of the second yields A/4. How to resolve this?

Any help greatly appreciated!

 

[Fermat]

use the trig formula,

$$\cos A + \cos B = 2\cos(\frac{A+B}{2})\cdot\cos(\frac{A-B}{2})$$

 

[George Jones]

You may wonder from where Fermat's indentity comes.

Sum the 2 standard identities

cos(a + b) = cosa cosb - sina sinb

cos(a - b) = cosa cosb + sina sinb

and make the substitutions A = a + b and B = a - b.

Regards,
George