Wave speed of a stretched string

Main Question or Discussion Point

A bit confused here as to what wave speed is dependent on. At first I learned that v = λƒ, and a couple of pages later in my textbook I find that v = √(τ/μ). Also, I found that speed is only dependent on the properties of the medium, specifically its elasticity and mass. Where does wavelength and frequency come in to play if the medium is the only dependent?

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Nathanael
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The frequency of the wave depends on the source of the wave (how it is being generated). If you consider a wave on a rope, how frequently you wave your arm up and down will determine the frequency of the wave.
The speed of the wave depends on the tension in and density of the rope. So the medium determines the speed of the wave, but there are infinite combinations of frequency and wavelength that will give the correct speed.
For a given rope under a given tension, the wave will travel at a certain speed and the wavelength will depend on the frequency according to $\lambda=\frac{v}{ƒ}$.

The medium plays a very big role in wave speed. If the waves travel along a string, the lower the mass and greater the density of the string, the faster waves travel across the string. Waves also travel much more slowly in the air than they do underwater or along a dense string. I'm sure the source of the wave plays a role in its speed as well, but I'm not sure how. This source has some information on it: http://dev.physicslab.org/Document.aspx?doctype=3&filename=WavesSound_WavesAlongStrings.xml

Orodruin
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The medium plays a very big role in wave speed. If the waves travel along a string, the lower the mass and greater the density of the string, the faster waves travel across the string. Waves also travel much more slowly in the air than they do underwater or along a dense string. I'm sure the source of the wave plays a role in its speed as well, but I'm not sure how. This source has some information on it: http://dev.physicslab.org/Document.aspx?doctype=3&filename=WavesSound_WavesAlongStrings.xml
The source does not play any role in wave speed other than through the fact that some media have dispersion relations such that the wave velocity is frequency dependent. The relation v = λf can rather be thought of as relating the frequency and wave length for a given wave velocity, i.e., for a wave with wave velocity v, a wave of frequency f will have wave length λ = v/f. Of course, if you measure the wave length and frequency, you can infer what the velocity is (and thus obtain information on the internal properties of the medium).