Subjective Type

Energy is transmitted at rate $$ P_{1} $$ by a wave of frequency $$ f_{1} $$ on a string under tension $$ \tau_{1} . $$ What is the new energy transmission rate $$ P_{2} $$ in terms of $$ P_{1}$$, if , instead, the frequency is decreased to $$ f_{2}=f_{1} / 2 ? $$

Solution

We use $$ P=\dfrac{1}{2} \mu v \omega^{2} y_{m}^{2} \propto v f^{2} \propto \sqrt{\tau} f^{2} $$
If the frequency is halved, then $$ P_{2}=P_{1}\left(\dfrac{f_{2}}{f_{1}}\right)^{2}=P_{1}\left(\dfrac{f_{1} / 2}{f_{1}}\right)^{2}=\dfrac{1}{4} P_{1} $$

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