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The following four waves are described by these equations (x, y, z form a rectangular coordinate system): (i) y1 = a sin(omega*(t - x/v)) (ii) y2 = a cos(omega*(t + x/v)) (iii) z1 = a sin(omega*(t - x/v)) (iv) z2 = a cos(omega*(t + x/v)) Which of the following statements are correct?

1. Superposing waves (i) and (iii) produces a travelling wave with amplitude a*sqrt(2).
2. Superposition of waves (ii) and (iii) is not possible.
3. Superposing waves (i) and (ii) produces a transverse stationary wave with maximum amplitude a*sqrt(2).
4. Superposing waves (iii) and (iv) produces a transverse stationary wave.

Correct answer: Superposing waves (i) and (iii) produces a travelling wave with amplitude a*sqrt(2).

Solution

Wave (i) is in y-plane, wave (iii) is in z-plane; both travel in +x direction with same frequency and amplitude. Their vector superposition at any point: resultant amplitude = sqrt(a² + a²) = a*sqrt(2), and the wave travels in +x direction. Statement A is correct. Superposition of (ii) and (iii): (ii) is in y-plane travelling in -x, (iii) is in z-plane travelling in +x. Since they are in perpendicular planes, superposition is perfectly possible (they don't cancel; they form an elliptically polarised wave). Statement B is incorrect. Superposing (i) and (ii): both are in y-plane, travelling in opposite directions (i in +x, ii in -x). This forms a stationary wave: y = a*sin(omega*t - kx) + a*cos(omega*t + kx). Max amplitude = a*sqrt(2) (since the two amplitudes are equal and the stationary wave has amplitude = sqrt(A²+B²) when combining sin and cos stationary patterns). Statement C is correct. Superposing (iii) and (iv): both in z-plane, opposite directions — forms a stationary wave in z. Statement D is correct.

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