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Answer 1 · 2018-02-13T12:19:58

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Phase velocity#,v_p = omega/k#

#therefore omega = v_pk#

Group velocity #,v_g = (domega)/(dk#

#v_g = (d(v_pk))/(dk)#

#= v_p (dk)/(dk) + k(dv_p)/(dk)#

#v_p + (2pi)/lambda (dv_p)/(d((2pi)/lambda))#

#v_p + 1/lambda(dv_p)/(d((1)/lambda))#

#v_p + 1/lambda(dv_p)/(d((1)/lambda))(dlambda)/(dlambda)#

#"But " d/(dlambda^-1) = -lambda^2#

#therefore v_g = v_p - lambda (dv_p)/(dlambda)#

if #v_p = c_1 + c_2lambda#

then #v_g = c_1 + c_2lambda - lambda (d(c_1 + c_2lambda))/(dlambda)#

#= c_1 + c_2lambda - lambda(0+c_2)#

#= c_1 + c_2lambda - c_2lambda#

#v_g = c_1#