# Standing Wave: How to show that $\frac{\mathrm df}{f}=0.5\frac{\mathrm dF}{F}$ where $f$ is frequency and $F$ is tension of the string?

How to do this?

Show that if the tension $F$ in a string is changed by a small amount $\mathrm dF$, the fractional change in frequency of a standing wave, $\frac{\mathrm df}{f}$ is given by:

$$\frac{\mathrm df}{f}~=~0.5\frac{\mathrm dF}{F}.$$

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If you pull the string from equilibrium then to the first order there will be a force $F$ pulling it back that is proportional to tension $T$ and the distance $u$ that the string is moved from the equilibrium $$F = -A T u$$ where $A$ is some constant not interesting to the further discussion.
For a piece of string with mass $m$ we get to the first order the equation for harmonic oscillator
$$0 = m \ddot{u} - F = m \ddot{u} + A T u$$ which can be solved by $u \sim \exp(i\omega t)$ giving us the relation $$\omega \sim \sqrt{T}$$ and differentiating both sides gives you the relation you wanted.