Is the speed of a wave determined by the medium in which it travels, the frequency of the source, or both? I know that for a string of linear density $\mu$ and tension $T$, the wave speed is given by $v=\sqrt{\frac{T}{\mu }}$.
Additionally, the speed of any sinusoidal wave is given by $v=\lambda f$.
My question is: do both the frequency and the physical properties of the string determine the wave speed or just one of these?
 A: Those two equations tell you
1) The speed of a wave on a string depends on only tension and density of the medium, not the frequency of the source.
2) IF the frequency of the source if $f$, you can find the wavelength by $\lambda = v/f$. High frequency sources produce shorter wavelengths, and vice versa. You're NOT free to choose both the wavelength and the freqency; if the frequency of your source is $f$, it will necessarily have wavelength $\lambda =v/f$.
All the waves have the same speed, given by $\sqrt{\frac{T}{\mu}}$. The second equation is basically telling you that on this string, you can't produce a wave with ANY frequency and wavelength. You can only produce waves satisfying $\lambda f = v$.
A: Trust the math. The first equation gives a specific speed of the wave. Second equation gives a relation between speed, frequency and wavelength. You don't get much information out of 2nd equation however the 1st equation explicitly states that the velocity depends upon the medium. Just take sound for example, if the speed were dependent on frequency the time interval between hearing of two notes would be heavily dependent upon their respective frequencies. This would make music very messy but in real life it is not.
A: Most simple waves that you've studied probably obey the wave equation: https://en.wikipedia.org/wiki/Wave_equation#Scalar_wave_equation_in_one_space_dimension.  This equation describes many kinds of commonly-encountered waves quite well: e.g. vibrating guitar strings, water ripples, sound waves through air, light traveling though empty space.  These kinds of waves are determined entirely by the medium through which they travel (e.g. the density of the air or the thickness of the guitar string).
But waves traveling through more complicated materials can experience a phenomenon called dispersion and are no longer described by the simple wave equation.  Roughly speaking, a dispersive wave's speed depends on its frequency, and therefore on the frequency of its source.  More technically, it's actually not obvious how to even define the "speed" of a dispersive wave.  One can talk about the "phase velocity," which (roughly) means the speed at which tiny ripples move, or the "group velocity," which (roughly) means the speed at which a large envelope containing many smaller ripples moves.  The easiest way to get an intuition on the difference is probably just to search "group and phase velocity" on YouTube and look at some video clips.
