\text{A transverse sinusoidal wave is being generated on a string under constant tension.}

\text{By what factor is the required power increased if the string is changed to one }\\ \text{ with half }\mu \text{ and an amplitude doubled? }

\displaystyle P_{\text{avg}}=\frac{1}{2}\mu v \omega^2 y_m^2

\displaystyle P_{\text{avg}}=\frac{1}{2}\mu \sqrt{\frac{\tau}{\mu}} \omega^2 y_m^2=\frac{1}{2}\sqrt{\mu} \sqrt{\tau} \omega^2 y_m^2

\displaystyle \text{If we use }\frac{\mu}{2} \text{and amplitude }2y_m

\displaystyle P_\text{new}=\frac{4}{\sqrt{2}}P_\text{avg}

\text{Abel's theorem}\\ \displaystyle W(x)=W(a) e^{-\displaystyle\int_a^x P(x) dx}