= Energy per wavelength of a transverse string wave
{title2=$E_\lambda=\mu A^2\omega^2\lambda/2$}
For a small-slope <travelling wave> $y=A\cos(kx-\omega t+\phi)$ on a uniform string with <linear mass density> $\mu$ and <tension> $T$, the <wave equation on a string> requires $\omega^2=Tk^2/\mu$. Its <elastic energy of a small-slope string> and <kinetic energy density> are equal, giving total <energy density> $u=\mu A^2\omega^2\sin^2(kx-\omega t+\phi)$. Its spatial average is $\mu A^2\omega^2/2$, and integrating over a full <wavelength> $\lambda=2\pi/|k|$ gives the title formula. Omitting $\lambda$ produces an <energy density>, not an <energy>; units and a direct one-period <integral> detect the mistake independently.
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