1. **Problem statement:** Two sources X and Y emit waves of wavelength $0.40$ m with a constant phase difference of $\pi$ radians. Point P is $5.2$ m from X and $4.6$ m from Y. Each wave has amplitude $A$ at P. Find the amplitude of the resultant wave at P. 2. **Relevant formula:** The resultant amplitude $A_r$ when two waves interfere with a phase difference $\Delta \phi$ and equal amplitudes $A$ is given by: $$A_r = 2A \left| \cos \left( \frac{\Delta \phi + \Delta k \Delta r}{2} \right) \right|$$ where $\Delta r = r_X - r_Y$ is the path difference and $\Delta k = \frac{2\pi}{\lambda}$ is the wave number. 3. **Calculate path difference:** $$\Delta r = 5.2 - 4.6 = 0.6 \text{ m}$$ 4. **Calculate wave number:** $$\Delta k = \frac{2\pi}{0.40} = 5\pi$$ 5. **Calculate total phase difference at P:** $$\Delta \phi_{total} = \pi + \Delta k \Delta r = \pi + 5\pi \times 0.6 = \pi + 3\pi = 4\pi$$ 6. **Calculate resultant amplitude:** $$A_r = 2A \left| \cos \left( \frac{4\pi}{2} \right) \right| = 2A |\cos(2\pi)| = 2A \times 1 = 2A$$ 7. **Interpretation:** The resultant amplitude at P is $2A$. **Final answer:** D. $2A$