# Are these definitions for transverse wave velocity on a string consistent?

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.

• Do you have the PDE? The $k$ and $\omega$ probably comes from the fourier transform of the PDE.
– Emil
Oct 1, 2017 at 9:18
• What's a PDE? I'm still an undergraduate so I'm still studying the basics. Oct 1, 2017 at 9:20
• 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.
– Emil
Oct 1, 2017 at 16:30

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.]
• That's right. Do make sure you understand the distinction between the $definition$ of something and $a\ fact\ about$ something! Oct 1, 2017 at 9:34