= Solution
The reflected wave remains on the incident string, so its <wavenumber> magnitude is still $k_1$. Its outward <power> magnitude is
$$
\boxed{\frac{|P_r|}{P_i}=\frac{|A_r|^2}{|A_i|^2}=\frac19}.
$$
Thus $|P_r|+P_t=(1/9+8/9)P_i=P_i$, verifying <energy conservation>. With signed positive-$x$ fluxes, $P_r=-P_i/9$ and the same statement is $P_i+P_r=P_t$. The minus sign of the reflected <displacement> is a <phase> reversal, whereas the minus sign of its flux records travel in the opposite direction; these are different uses of a sign.
\Image[/past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2015-string-junction-amplitude-and-power.png]
{title=Signed displacement coefficients and conserved power fractions at a string junction}
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