Wave dispersion: modulation envelope, dispersion relation, beaded string

[belleamie]

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?

 

[Pieter Kuiper]

These problems are about group velocity $$v_g = {\rm d}\omega/{\rm d}k$$. You need to write the superposition as a product of average and beat frequency, using those relations there are for sin a + sin b.

Here is some help, with a nice simulation:
http://webphysics.davidson.edu/faculty/dmb/bernstein/qmwave/section2b.html