= Standing-and-travelling decomposition of a reflected string wave
For real negative displacement reflection amplitude $r=-a$, with $0<a<1$, use
$$
A[\cos(kx-\omega t)-a\cos(kx+\omega t)]
=(1-a)A\cos(kx-\omega t)+2aA\sin kx\sin\omega t.
$$
This separates a travelling field and a standing field algebraically. Their individual <powers> cannot be treated as independent incident-power fractions because the two fields interfere. Reflected and transmitted <power> should instead be calculated from travelling-wave amplitudes with the appropriate <characteristic impedance> factors.
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