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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:The reflected fraction is $1/9$, checking energy conservation. Squaring the displacement amplitude alone would give the wrong transmitted power.