Wave dispersion relation and group vs phase velocity

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belleamie
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Hi I'm studying for a test, and in the suggested reading book review has a few equations that they talk about but I'm not don't really understand how it jumps from one thing to another? the book is very vauge... I've broken the parts i don't understand into A,B,C (I used w = omega)

A)IT shows a graph, explain that the end of a string is given a transverse displacement phi=cosw1t+cosw2t where the two frequencies are almost equal and w1>w2 the resultant motion is a traveling wave of angular frequency (w1+w2)/2, modulated by n envelope which is a traveling wave of (w1-w2)/2 There the speed of this envelop is (w1-w2)/(k1-k2) ...? I don't understand how they got that?

B) A system with dipersion relation w=ak^r...a and r are constants because v(sub g)=xv(sub phi) at all wave frequencies. i duno where then got the other variables v(sub g)? i know that v(sub phi) =c(1+ak^2)^1/2 but i don't understand how they relate?

C) a beaded string above cut off, the dependence of k on frequency is given by w=w(sub c) cosh1/2ka showing a graph, How does k depend on the frequency? i know a beaded string can exhibit high freq cut off and that the part od the system vibrates in anti phase with each other...and k=(pi/a)-ik where k can be found as a function by replacing k=pi/a in w/w(sub c)= sin (1/2 Ka-i1/2ka) but I'm not sure how?
 

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A) comes from the trig for cosw1t+cosw2t = (cos at)(cos bt) with
a=(w1+w2)/2 and b=(w1-w2)/2.
 
B) v_{phi}=w/k=ak^r/r=ak^{r-1}.
v_g=dw/dk=rak^{r-1}=rv_{phi}.
The x must be a misprint.
 
I'm not sure what you're asking. If you have w as a function of k in your first eq.,
can't you just solve that for k? In this and in (B), you may be confusing two different situations.