= Reflection of a string wave from a freely moving endpoint mass
{title2=$R=(Z+i m\omega)/(Z-i m\omega),\quad |R|^2=1$}
For a semi-infinite ideal string on $x\le0$ with <tension> $T$ and <linear mass density> $\mu$, let a free transverse endpoint <mass> be $m$. Use a <time-harmonic wave> $y=\operatorname{Re}\{A(e^{ikx}+Re^{-ikx})e^{-i\omega t}\}$, where $k=\omega\sqrt{\mu/T}>0$. The endpoint <boundary condition> from <Newton's second law> is $my_{tt}=-Ty_x$. It gives the title displacement <reflection coefficient>, with <characteristic impedance of a stretched string> $Z=\sqrt{T\mu}$. Since numerator and denominator have equal moduli, all mean incident <power> is reflected. The endpoint <kinetic energy> varies within a cycle, but its mean rate of change is zero. A free <mass> changes the reflected <phase>, and does not by itself provide <dissipation>. The limits $m\to0$ and $m\to\infty$ give $R=+1$ and $R=-1$ respectively at fixed positive <angular frequency>.
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