= Solution
The transverse-wave impedance of a string is $Z=\sqrt{T\mu}$ for common tension $T$. The density ratio four gives $Z_2/Z_1=2$. Continuity of displacement and transverse <force> gives transmitted amplitude ratio $t=2Z_1/(Z_1+Z_2)=2/3$. The <power transmission coefficient> includes the impedance ratio:
$$
\boxed{\mathcal T=\frac{Z_2}{Z_1}|t|^2=\frac89.}
$$
The reflected fraction is $1/9$, checking <energy> conservation. Squaring the displacement amplitude alone would give the wrong transmitted power.
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