Find the relation between the group velocity `u` and phase velocity `upsilon` for the following disperison laws:
(a) `upsilon prop 1//sqrt(lambda_(1))`,
(b) `upsilon prop k`,
(c ) `upsilon prop 1// omega^(2)`,
Here `lambda, k`, and `omega` are the wavelength, wave number, and angular frequancy.

Explanation

(a) `v = a//sqrt(lambda), a =` constant
Then `u = v- lambda (dv)/(d lambda)`
`= (a)/(sqrt(lambda)) - lambda (-(1)/(2)a lambda^(-3//2)) = (3)/(2).(a)/(sqrt(lambda)) = (3)/(2)v`.
(b) `v = bk = omegak, b =` constant
so `omega = bk^(2)` and `u = (d omega)/(dk) = 2bk = 2v`.
(c) `v = (c)/(omega^(2)), c =` constant `= (omega)/(k)`.
so `omega^(3) = ck` or `omega = c^(1//3) k^(1//3)`
Thus `u = (d omega)/(dk) = c^(1//3) (1)/(3)k^(-2//3) = (1)/(3)(omega)/(k) = (1)/(2)v`