Question

Assuming no energy loss in the wire, find the fraction of the incident power that is reflected at the junction and fraction of the same that is transmitted.

Answer 1

Since $v=\sqrt{\frac{T}{\mu }}$

${\mu }_1=\frac{T}{v_1^2}$

${\mu }_2=\frac{T}{v_2^2}$

$⟹\frac{{\mu }_1}{{\mu }_2}=\frac{v_2^2}{v_1^2}=4$

Thus $\frac{P_r}{P_i}=\frac{\frac{1}{2}{\mu }_1{\omega }^2{A}_r^2{v}_1}{\frac{1}{2}{\mu }_2{\omega }^2{A}_i^2{v}_2}=\frac{1}{9}$

Also $\frac{P_t}{P_i}=\frac{\frac{1}{2}{\mu }_1{\omega }^2{A}_t^2{v}_1}{\frac{1}{2}{\mu }_2{\omega }^2{A}_i^2{v}_2}=\frac{8}{9}$