In the figure shown, $S_{1} a n d S_{2}$ are two identical point sources of sound which are coherent and out of phase by $180 °$ . Taking $S_{1}$ as centre, two circular arcs $l$ and m of radii 1 $m$ and 2 $m$ are drawn. Taking $S_{2}$ as centres, two circular arcs $p$ and $q$ are drawn having radii $2 m$ and 4 $m$ respectively. It is given that wavelength of wave produced by each source is $4.0 m .$

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Maximum amplitude is observed at A and C.

Maximum amplitude is observed at B and D.

Maximum amplitude is observed at A only.

Minimum amplitude is observed at B only.

answer is D, C.

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Detailed Solution

Amplitude on are $l$ due to $S_{1} = 2 a$
Amplitude on are $q$ due to $S_{2} = \frac{a}{2}$
At point $A : Δ ϕ = \frac{2 π}{λ} Δ x + π = \frac{2 π}{4} (2) + π = 2 π$
$∴ a_{A} = \sqrt{a^{2} + {(\frac{a}{2})}^{2} + 2. a . \frac{a}{2}} . \cos 2 π = \frac{\sqrt{5}}{2} a$
At $B; Δ ϕ = π [∵ n o p a t h d i f f e r e n c e]$
$∴ a_{B} = 0$
$A t C; Δ ϕ = \frac{2 π}{4} (1) + π = \frac{3 π}{2}$
$∴ a_{c} = \sqrt{a^{2} + {(2 a)}^{2} + 2. a .2 a \cos (\frac{3 π}{2}) =} \sqrt{5 a}$
$A t D : Δ ϕ = \frac{2 π}{4} (3) + π = \frac{3 π}{2} + π = \frac{5 π}{2}$
$a_{D} = \sqrt{{(\frac{a}{2})}^{2} + {(2 a)}^{2} + 2 (\frac{a}{a})} (2 a) \cos \frac{5 π}{2} = \frac{\sqrt{17}}{2} a$