Two plane monochromatic waves propagating in the same direction with amplitudes 3 A and 5 A and differing in phase by \( \displaystyle\frac{2\pi}{3} \), superimpose. The amplitude of the resulting wave will be:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \sqrt{19}\,A \)
By the principle of superposition, two waves of the same frequency with amplitudes \( a_{1} \), \( a_{2} \) and phase difference \( \phi \) add like two vectors at angle \( \phi \).
\( \displaystyle a^{2} = a_{1}^{2} + a_{2}^{2} + 2a_{1}a_{2}\cos\phi \)
\( \displaystyle a_{1} = 3A, \quad a_{2} = 5A, \quad \cos\frac{2\pi}{3} = -\frac{1}{2} \)
\( \displaystyle a^{2} = 9A^{2} + 25A^{2} + 2(3A)(5A)\left(-\frac{1}{2}\right) = 34A^{2} - 15A^{2} = 19A^{2} \)
\( a = \sqrt{19}\,A \)