While studying velocity of a Transverse wave I came across 2 definition

1. v=w/k

2. v=(T/u) ^1/2

Are these two definitions pointing towards the same velocity or is there a fundamental difference as 1) suggests dependence on frequency while 2) doesn't.

I've found two definitions for the velocity of a transverse wave on a stretched string: $$ \begin{align} v & = \frac{\omega}{k} \tag{1} \\[10px] v & = \sqrt{\frac{T}{u}} \tag{2} \end{align} $$

Question: Are these two definitions mutually consistent, or is there a fundamental difference?

I'm asking because $\operatorname{Eq.}{\left(1\right)}$ suggests dependence on frequency while $\operatorname{Eq.}{\left(2\right)}$ doesn't.

While studying velocity of a Transverse wave I came across 2 definition

1. v=w/k 2)v=(T/u) ^1/2 Are these two definitions pointing towards the same velocity or is there a fundamental difference as 1) suggests dependence on frequency while 2) doesn't.

While studying velocity of a Transverse wave I came across 2 definition

1. v=w/k

2. v=(T/u) ^1/2

Are these two definitions pointing towards the same velocity or is there a fundamental difference as 1) suggests dependence on frequency while 2) doesn't.