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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.

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  • $\begingroup$ Do you have the PDE? The $k$ and $\omega$ probably comes from the fourier transform of the PDE. $\endgroup$
    – Emil
    Oct 1, 2017 at 9:18
  • $\begingroup$ What's a PDE? I'm still an undergraduate so I'm still studying the basics. $\endgroup$ Oct 1, 2017 at 9:20
  • $\begingroup$ It is an equation that connects small difference quotiences in different variables to eachother (the time derivative and spatial derivatives mostly). You'll get there. $\endgroup$
    – Emil
    Oct 1, 2017 at 16:30

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The definition of something is what it means. The definition of the speed of a (sinusoidal) wave is the distance a wavefront moves per unit time.

Your first formula, $v=\frac{\omega}{k}$, is closely related to the definition of $v$ (though not strictly the definition). The presence of $\omega$ doesn't necessarily indicate a dependence on frequency, because $k$ may well depend on frequency (even though the definition of $k$ doesn't involve frequency!). In fact, for many types of wave the speed is independent of the frequency, so $k$ is proportional to $\omega$.

Your second formula, $v=\sqrt{\frac{T}{\mu}}$, relates the speed of the wave on a stretched string to the tension in the string and the string's mass per unit length. It is certainly not a definition of the speed of the wave. As you say, this formula implies that the speed of a transverse wave on a stretched string doesn't depend on the frequency, because we know that neither $T$ nor $\mu$ has any dependence on frequency. [The formula only holds as long as certain conditions apply, for example the wave amplitude is much less than the wavelength.]

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  • $\begingroup$ So w/k is a general case while other one is special case of stretched string? $\endgroup$ Oct 1, 2017 at 9:09
  • $\begingroup$ And for some analysis can these two be equated? $\endgroup$ Oct 1, 2017 at 9:12
  • $\begingroup$ That's right. Do make sure you understand the distinction between the $definition$ of something and $a\ fact\ about$ something! $\endgroup$ Oct 1, 2017 at 9:34

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