Question

Question #6fa5d

Answer 1

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`