The Geometry of the “Head Start”: Visualizing Temporal Delay
What happens when two waves move at the same speed but start at different times? In this experiment, we explore how “Phase” creates the invisible gap between signals.
In our previous experiments, every wave began its journey at exactly zero. But in the real world—from stereo audio to alternating current (AC)—waves often start “late” or “early.” This shift in the starting line is known as the Phase Shift.
By adding a constant to our input angle, we aren’t changing the circle’s size or its speed; we are simply rotating the starting point of our vector. It is the geometry of a head start.
Lab Results: The Temporal Phase Displacement
If you look at the Phase Shift chart, you’ll notice that while the two waves are identical in shape and speed, they no longer line up vertically. The yellow wave appears to have “jumped ahead” of the white base wave. This horizontal separation represents a difference in timing.
In this experiment, we introduced a Phase constant of \(C = \pi/2\). While the Amplitude (height) and Frequency (peaks per second) remain unchanged, the starting position of the rotation has been offset.
The Key Takeaway:
- The Input: The Initial Angle \(C\) of the rotating vector.
- The Output: A Horizontal Shift along the time axis.
By adjusting the phase, we move from observing a single pulse to understanding Synchronization. This is the geometry behind how we perceive “stereo” sound—our ears use this tiny phase displacement to calculate exactly where a sound is coming from in space.
Note: This relationship is a fundamental concept in Trigonometric Projections and serves as a classic example of Simple Harmonic Motion. By understanding how a rotating vector corresponds to a horizontal shift in time, we can bridge the gap between static geometry and the dynamic oscillations found in physics and engineering.
The Mathematical Model: The Offset Pulse
The starting position and timing of the wave are now controlled by the phase constant C.
By adding \( C = \pi/2 \), we shift the entire horizontal path of the wave. This adjustment doesn’t change the wave’s shape or speed, but rather its timing relative to the origin.
Name: Source Code: Manim Implementation *