= Solution
The transmitted and incident waves have the same <angular frequency> and <tension>, but the transmitted <wavenumber> is twice as large. The time-average string-power formula therefore gives
$$
\boxed{\frac{P_t}{P_i}=\frac{k_2|A_t|^2}{k_1|A_i|^2}=2\left(\frac23\right)^2=\frac89}.
$$
The extra factor $k_2/k_1$ is essential. Squaring the <displacement> transmission coefficient alone would give $4/9$, which is not the transmitted <power> fraction in strings with different <linear mass densities>.
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