Question

Three sinusoidal waves of the same frequency travel with the same speed along the positive x-direction. The amplitudes of the waves are \(a, a/2, a/3\), and the phase constants of the waves are \(\pi/2, \pi, 3\pi/2\), respectively. If \(A_m\) and \(\phi_m\) are the amplitude and phase constant of the wave resulting from the superposition of the three waves, which of the following is/are correct?

• A. \(A_m = \dfrac{5a}{6}\)
• B. \(\phi_m = \dfrac{\pi}{2} + \tan^{-1}\left(\dfrac{3}{4}\right)\)
• C. \(A_m = \dfrac{7a}{6}\)
• D. \(\phi_m = \tan^{-1}\left(\dfrac{3}{4}\right)\)

Hint:

When adding waves of the same frequency, use phasor addition to determine resultant amplitude and phase.

Answer 1

The Correct Option is A, B

Step 1: Represent the waves as phasors. 
The resultant wave amplitude is obtained from vector addition of phasors. Let \(A_1 = a e^{i\pi/2}\), \(A_2 = (a/2)e^{i\pi}\), \(A_3 = (a/3)e^{i3\pi/2}\). 
 

Step 2: Calculate net amplitude. 
\[ A_m = \left|A_1 + A_2 + A_3\right| = \left| ai - \frac{a}{2} - \frac{ai}{3} \right| = \frac{a}{6}\sqrt{3^2 + 4^2} = \frac{5a}{6} \]

Step 3: Determine phase. 
\[ \tan(\phi_m - \pi/2) = \frac{3}{4} \Rightarrow \phi_m = \frac{\pi}{2} + \tan^{-1}\left(\frac{3}{4}\right) \]

Step 4: Conclusion. 
Thus, \(A_m = \dfrac{5a}{6}\) and \(\phi_m = \dfrac{\pi}{2} + \tan^{-1}\left(\dfrac{3}{4}\right)\).