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belleamie

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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?