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How to prove the expression for speed of transverse wave in a string that is :- $$ v= \sqrt{\frac Tμ }$$ Without loss of mathematics rigor, that is without using infinitesimals or differentials in the realm of standard analysis?

Here,

T is tension on string,

$\mu$ is mass per unit length of a uniform density string.

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    $\begingroup$ Can you explain what you don't like about the standard derivation? $\endgroup$
    – knzhou
    Commented Feb 6, 2019 at 16:03
  • $\begingroup$ FWIW non-standard analysis, which is based on the explicit notion that infinitesimals really are numbers, is a much simpler model for translating physics into math than the limit-based definitions of "standard analysis." $\endgroup$
    – alephzero
    Commented Feb 6, 2019 at 17:20
  • $\begingroup$ @Knzhou The use of differential , that is to take a small element, what is small formally, if you can make that proof more rigorous maybe through epsilon delta or something else I would be grateful. $\endgroup$ Commented Feb 6, 2019 at 17:21
  • $\begingroup$ @alephzero I am not acquainted with non standard analysis till now, but I think the formalization in classical mechanics took place way before the birth of non standard analysis that is before 1975 , so I hope there should be some formal solution of my referred problem, if you can write an answer giving a formal solution I would be grateful. $\endgroup$ Commented Feb 6, 2019 at 17:24

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