Two sinusoidal waves of the same frequency are to be sent in the same direction along a taut string. One wave has an amplitude of 6.0 m, the other 10.0 m. A) What phase difference φ1 between the two waves results in the smallest amplitude of the resultant wave? B) What is that smallest amplitude? C) What phase difference φ2 results in the largest amplitude of the resultant wave?D) What is that largest amplitude? E) What is the resultant amplitude if the phase angle is (φ1 - φ2)/2?

A. 180° B. 4 m C. 0° D. 16.0 m E. 11.7 m

Explanation:

Let y₁ = 6.0 m represent the amplitude of the first wave and y₂ = 10.0 m represent the amplitude of the second wave

A. The phase difference for the smallest amplitude is when the waves move in opposite directions. That is, θ₁ = 180° = π.

B. The smallest amplitude is thus y₂ - y₁ = 10.0 m - 6.0 m = 4.0 m

C. The phase difference for the largest amplitude is when the waves move in same directions. That is, θ₂ = 0° .

D. The largest amplitude is thus y₂ + y₁ = 10.0 m + 6.0 m = 16.0 m

E. When the phase difference is (θ₁ - θ₂)/2 = (180° - 0°)/2 = 90°. At this point, the waves are perpendicular to each other. So the resultant amplitude is thus y = √(y₁² + y₂²) = √(6² + 10²) = √(36 + 100) = √136 = 11.7 m