**Category:** Mechanics  
**Subtype:** Harmonic Motion  

**Problem Description:** This problem involves two points (1 and 2) that are oscillating harmonically along the x-axis. At the initial time \( t = 0 \), both points pass through the equilibrium position in the positive direction with the same speed. When the speed of point 1 becomes zero, the speed of point 2 has decreased by a factor of \(\sqrt{2}\). The task is to find the ratio of their displacements when they next have the same speed after the initial time.  

**Key Concepts:**  
- Harmonic motion and its properties  
- Relationship between speed and displacement in oscillatory motion  
- Understanding of phase differences in oscillating systems  

**Goal:** Determine the ratio of the magnitudes of their displacements at the next instance when their speeds are equal.