Question

Two strings of linear mass densities u and 9u are stretched under same tension. A wave travelling on the lighter string towards the heavier string gets partially reflected and transmitted at the junction. Then fraction of incident wave energy getting transmitted to the heavier string is

Answer 1

For your convenience, please refer to the following diagram first, and then continue with the theoretical part from below.

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[contd. from diagram]

As per the given diagrams,

$E_{t} = x . E$                                           $v_{1} =$ √(T/μ)    ,   $v_{2} =$ √(T/9μ)

∴ $A_{t} = | \frac{2v_{2} }{v_{1} +v_{2} }| =$ [√(T/9μ)] ÷ [√(T/9μ)+√(T/μ)] = $\frac{2}{3}$ ÷ $\frac{1}{3} + 1$ = $\frac{1}{2}$

and, Energy ∝ $A^{2}$ μ $v$

∴ [E ∝ $A^{2}$ × μ × √(T/μ)] ÷ [$E_{t}$ ∝ $A^{2}t$ × 9μ × √(T/9μ)] ⇒ $\frac{E}{E_{t} } =$ [$A^{2}$ × μ ×× √(T/μ)] ÷ [$(\frac{1}{2} A^{2})$ × 9μ × √(T/9μ)] = $\frac{1}{2}$ ∝ A = $\frac{4 *3}{9}$

Thus, $E_{t} = \frac{3}{4} E$