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(C) The resultant amplitude $A_R$ of the superposition of waves with phase angles $\phi_1, \phi_2, \phi_3, \phi_4$ is given by the vector sum of indiv

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$3$

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(C) The resultant amplitude $A_R$ of the superposition of waves with phase angles $\phi_1, \phi_2, \phi_3, \phi_4$ is given by the vector sum of individual amplitudes $A_0$ (where $I_0 = k A_0^2$).Let the waves be represented as complex numbers: $z_1 = A_0 e^{i0} = A_0$,$z_2 = A_0 e^{i\pi/3}$,$z_3 = A_0 e^{i2\pi/3}$,and $z_4 = A_0 e^{i\pi} = -A_0$.The sum is $S = A_0(1 + e^{i\pi/3} + e^{i2\pi/3} - 1) = A_0(e^{i\pi/3} + e^{i2\pi/3})$.Using the identity $e^{i	heta} = \cos 	heta + i \sin 	heta$,we have $S = A_0 [(\cos \pi/3 + i \sin \pi/3) + (\cos 2\pi/3 + i \sin 2\pi/3)]$.$S = A_0 [(1/2 + i\sqrt{3}/2) + (-1/2 + i\sqrt{3}/2)] = A_0 (i\sqrt{3})$.The resultant intensity $I = |S|^2 = A_0^2 |i\sqrt{3}|^2 = I_0 (3) = 3I_0$.Thus,$n = 3$.

Four harmonic waves of equal frequencies and equal intensities $I_0$ have phase angles $0, \pi / 3, 2 \pi / 3$ and $\pi$. When they are superposed,the intensity of the resulting wave is $nI_0$. The value of $n$ is

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