JEE Advanced 2025 — Paper 1 — Question 5

$\mathrm{y}_{1}=\mathrm{y}_{0} \cos (\omega \mathrm{t}-\mathrm{kx})$

when wave going from Rarer to Denser,

$\mathrm{y}_{\mathrm{r}}=\mathrm{A}_{\mathrm{r}} \cos (\omega \mathrm{t}+\mathrm{kx}+\pi)$

$\mathrm{y}_{\mathrm{r}}=\mathrm{a}_{1} \mathrm{y}_{0} \cos (\omega \mathrm{t}+\mathrm{kx}+\pi)$

option (A) correct (B) For transmitted from point P

$\mathrm{y}_{\mathrm{t}}=\mathrm{A}_{\mathrm{t}} \cos \left[\omega \mathrm{t}-\mathrm{k}_{1} \mathrm{x}\right]$

$\frac{\mathrm{k}_{1}}{\mathrm{k}}=\sqrt{\frac{\mu_{1}}{\mu}}=\frac{\mathrm{k}_{1}}{\mathrm{k}}=\sqrt{\frac{4 \mu}{\mu}}$

$\mathrm{k}_{1}=2 \mathrm{k}$\ $\mathrm{y}_{\mathrm{t}}=\mathrm{a}_{2} \mathrm{y}_{0} \cos [\omega \mathrm{t}-2 \mathrm{kx}]$

option (B) incorrect

(C) when reflected from Q

$\mathrm{y}_{\mathrm{i}}=\mathrm{a}_{2} \mathrm{y}_{0} \cos [\omega \mathrm{t}-2 \mathrm{kx}]$

$\mathrm{y}_{\mathrm{r}}=\mathrm{a}_{3} \mathrm{y}_{0} \cos [\omega \mathrm{t}+2 \mathrm{kx}+\pi]$

option (C) incorrect

(D) when transmitted from Q

$\mathrm{y}_{\mathrm{t}}=\mathrm{a}_{4} \mathrm{y}_{0} \cos \left[\omega \mathrm{t}=\mathrm{k}_{2} \mathrm{x}\right]$

$\frac{\mathrm{k}_{2}}{2 \mathrm{k}}=\sqrt{\frac{16 \mu}{4 \mu}} \Rightarrow \mathrm{k}_{2}=4 \mathrm{k}$

$\mathrm{y}_{\mathrm{t}}=\mathrm{a}_{4} \mathrm{y}_{0} \cos [\omega \mathrm{t}-4 \mathrm{kx}]$

option (D) correct