= Reflection at a string junction carrying a point mass
{title2=$r=[T(k_1-k_2)-iM\omega^2]/[T(k_1+k_2)+iM\omega^2]$}
Two semi-infinite strings with common <tension> $T$ and a point mass $M$ at their join have continuous transverse <displacement>, $A+B=C$. With time factor $e^{i\omega t}$ and a right-moving incident factor $e^{-ik_1x}$, <Newton's second law> at the join gives $k_1T(A-B)=(k_2T+iM\omega^2)C$. Solving gives the displayed <reflection coefficient> and $t=C/A=2k_1T/[T(k_1+k_2)+iM\omega^2]$. Their power fractions obey $|r|^2+(k_2/k_1)|t|^2=1$. Reversing the time convention complex-conjugates the coefficients. The limits $M=0$ and $M\to\infty$ recover a massless join and a fixed boundary, respectively.
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