Electromagnetic Waves

Concept:Constructive interference occurs when the path difference between two split sound waves equals an integer multiple of the wavelength $\lambda$.Explanation:Given wavelength $\lambda = 0.29\ \mathrm{m}$ for all cases.For configuration (P):Path 1 (straight) $x_1 = l$, Path 2 (semicircle) $x_2 = \frac{\pi l}{2}$.Path difference $\Delta x = \frac{\pi l}{2} - l = l\left(\frac{\pi}{2}-1\right)$.Set $\Delta x = \lambda$ → $l = \frac{\lambda}{\pi/2 -1} \approx 0.51\ \mathrm{m}$. So (P) → (3).For configuration (Q):Path 1 length $x_1 = 2l$, Path 2 length $x_2 = l$.$\Delta x = l$ → set equal to $\lambda$ → $l = 0.29\ \mathrm{m}$. So (Q) → (4).For configuration (R):Path 1: $x_1 = l + \frac{\pi l \sqrt{2}}{2} = l\left(1+\frac{\pi}{\sqrt{2}}\right)$, Path 2: $x_2 = l$.$\Delta x = \frac{\pi l}{\sqrt{2}}$ → set equal to $\lambda$ → $l = \frac{\sqrt{2}\lambda}{\pi} \approx 0.13\ \mathrm{m}$. So (R) → (5).For configuration (S):Triangle angles: $45^\circ$, $105^\circ$, $30^\circ$. Using law of sines with side $l$,$a = \frac{l \sin 30^\circ}{\sin 105^\circ} = \frac{0.5l}{0.97} \approx 0.5155l$,$b = \frac{l \sin 45^\circ}{\sin 105^\circ} = \frac{0.7071l}{0.97} \approx 0.729l$.Path 1 length $x_1 = a+b \approx 1.2445l$, Path 2 length $x_2 = l$.$\Delta x = 0.2445l$ → set $0.2445l = 0.29$ → $l \approx 1.19\ \mathrm{m}$. So (S) → (2).Thus the matches are: P→3, Q→4, R→5, S→2.Answer:Option D: P→3, Q→4, R→5, S→2.