I have a general query about velocity of waves. We have a formula velocity of waves: $$ V = f\lambda,\quad f= \mathrm{frequency}, \ λ= \mathrm{wavelength} $$ But in string waves it's $$ V = \sqrt{\frac{F}{\mu}}, \quad F =\text{tension force}, \ μ=\text{linear mass density} $$ I am confused with other formulas also (like sound wave velocity). Is the $V = f\lambda$ the general one, i.e can we use it in all cases, including sound waves and string vibrations?. If so, what is the need of the second formula above?
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2 Answers
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$v = f \lambda$ is indeed the general case and can be used for water waves, waves on a string, electromagnetic waves or pretty much anything you like. In contrast, $v = \sqrt{F/\mu}$ is particular to waves on a string.
Note that there are different ways of defining the velocity of a wave depending on what property of the wave you are interested in. The velocity in the expressions above is the phase velocity of the wave.
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The general expression of the velocity of waves is : $$v=\frac{\lambda}{T}=\lambda.\nu$$ Where $T$ is the period of the wave and $\nu$ its frequency.
But in some special cases, we can use other formulas.
$$v=\sqrt{\frac{F}{\mu}}$$ Gives the velocity of waves propagating in a string.
$$v=\sqrt{\frac{\gamma RT}{M}}$$ Gives the velocity of an acoustic wave propagating in a gas. $$v=\sqrt{gh}$$ Gives the velocity of a wave propagating in a fluid.
Hence you can notice that every expression has its own uses, and we can derive them all using the necessary laws. But after all the general expression is the first one.