= Characteristic impedance of a stretched string
{title2=$Z=\sqrt{T\mu}$}
For the lossless small-slope <wave equation on a string>, <tension> $T$ and <linear mass density> $\mu$ give <wave speed> $c=\sqrt{T/\mu}$. With transverse <force> $F_y=-Ty_x$ exerted across a cut toward increasing $x$, a rightgoing displacement $y=f(x-ct)$ has <velocity> $v_y=-cy_x$. Its positive <mechanical impedance> is therefore $Z=F_y/v_y=T/c=\sqrt{T\mu}$. A leftgoing component has $F_y/v_y=-Z$ under the same fixed orientation. A rightgoing <time-harmonic wave> with peak displacement <wave amplitude> $|A|$ carries mean <power> $Z\omega^2|A|^2/2$.
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